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Vehicle System Dynamics: InternationalJournal of Vehicle
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A combined use of phase plane andhandling diagram method to
studythe influence of tyre and vehiclecharacteristics on
stabilityFlavio Farroni a , Michele Russo a , Riccardo Russo a ,
Mario Terzo a
& Francesco Timpone aa DII Department of Industrial
Engineering , University of Naples ,Federico II Via Claudio 21,
80125 , Napoli , ItalyPublished online: 24 May 2013.
To cite this article: Flavio Farroni , Michele Russo , Riccardo
Russo , Mario Terzo & FrancescoTimpone (2013) A combined use of
phase plane and handling diagram method to study the influenceof
tyre and vehicle characteristics on stability, Vehicle System
Dynamics: International Journal ofVehicle Mechanics and Mobility,
51:8, 1265-1285, DOI: 10.1080/00423114.2013.797590
To link to this article:
http://dx.doi.org/10.1080/00423114.2013.797590
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Vehicle System Dynamics, 2013Vol. 51, No. 8, 12651285,
http://dx.doi.org/10.1080/00423114.2013.797590
A combined use of phase plane and handling diagram methodto
study the influence of tyre and vehicle characteristics on
stabilityFlavio Farroni, Michele Russo, Riccardo Russo*, Mario
Terzo and Francesco Timpone
DII Department of Industrial Engineering, University of Naples
Federico II Via Claudio 21,80125 Napoli, Italy
(Received 10 September 2012; final version received 15 April
2013 )
This paper deals with in-curve vehicle lateral behaviour and is
aimed to find out which vehicle physicalcharacteristics affect
significantly its stability. Two different analytical methods, one
numerical (phaseplane) and the other graphical (handling diagram)
are discussed. The numerical model refers to thecomplete
quadricycle, while the graphical one refers to a bicycle model.
Both models take into accountlateral load transfers and nonlinear
Pacejka tyreroad interactions. The influence of centre of
masslongitudinal position, tyre cornering stiffness and front/rear
roll stiffness ratio on vehicle stability areanalysed. The
presented results are in good agreement with theoretical
expectations about the aboveparameters influence, and show how some
physical characteristics behave as saddle-node
bifurcationparameters.
Keywords: vehicle stability; phase plane; handling diagram;
cornering stiffness; roll stiffness
1. Introduction
The study of vehicle handling behaviour is of fundamental
importance in order to improvevehicle safety, especially as it
concerns the loss of stability in the lateral direction,
resultingfrom unexpected lateral disturbances like side wind force,
tyre pressure loss, -split brakingdue to different road pavements
such as icy, wet, and dry surfaces, etc.
In recent years, the interest for vehicle stability has been
increasing, and consequentlythe study of the local stability has
become a fundamental discipline in the field of vehicledynamics,
being a vehicle a strongly nonlinear system mainly because of tyres
behaviour.
During short-term emergency situations, the average driver may
exhibit panic reactionand control authority failure, and he may not
generate adequate steering, braking/throttlecommands in very short
time periods. In order to make vehicles more safe under this point
ofview, two different approaches can be followed, consisting in
improving active and passivesafety. Improving active safety means
to equip the vehicle with proper and effective lateralstability
control systems, which may compensate the driver during the panic
reaction time bygenerating the necessary corrective yaw
moments.[14] Improving passive safety means toinvestigate vehicle
constructive and geometric parameters which mainly influence its
stability
*Corresponding author. Email: [email protected]
2013 Taylor & Francis
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1266 F. Farroni et al.
and to evaluate the effects of their variation with the ultimate
goal of maximising the intrinsicstability regardless of the control
systems.
However, in both cases, in order to improve vehicle stability,
it is necessary to obtain a deepknowledge of all the possible
stable and unstable behaviours arising as a consequence of
acorrective action.
The study of the driving stability of a vehicle travelling on a
curve in steady-state equilibriumconditions is the first and most
important step in order to understand the effects of
perturbationsunavoidably occurring during vehicle motion. Such a
study may have a dual application: firstof all, it can contribute
to develop and optimise control logic of the vehicle subsystems,
asfor example, the steering (active steering); secondly it may be
useful to identify the physicalquantities affecting stability and
to find out their optimal set under this point of view.
So it becomes fundamental to dispose of suitable stability maps,
which typically are consti-tuted by curves in a phase plane
delimiting zones in which the system of equations describingthe
vehicle dynamics admits stable and strongly attractive solutions
(stable region).
In the recent years, several ways to construct these maps have
been proposed.A great numberof studies in literature are mainly
oriented to the development of control systems strategiesaimed to
improve vehicle stability and so in such papers the authors
proposed methods basedon extremely simplified vehicle and tyre
models.
Typical simplifications consist in considering a bicycle and/or
in to linearise the tyreroadinteraction neglecting the saturation
effects in the tyre lateral forces.
Chung and Yi,[5] adopting a bicycle model and linearising
tyreroad interaction forces,proposed a method to define a stability
region in sideslip angle yaw rate plane aimed todevelop a Vehicle
Stability Control (VSC) program.
Koi and Song,[6] adopting a bicycle model, defined stability
regions and verified theirextension using field tests and results
provided by a detailed multibody model.
Ono et al.,[7] using a bicycle model and a simplified Pacejka
tyre model, showed that vehicleloss of stability is caused by a
saddle-node bifurcation which depends heavily on the rear tyreside
force saturation and proposed a control strategy for steering.
Also Catino et al. [8] focused on steering control strategies
and confirmed the above resultsadopting the same simplifications
and using the continuation method.
In Escalona and Chamorro,[9] a method is reported for the
stability analysis of the steadycurving of vehicles based on
equations of motion that are obtained using multibody dynamics.
A study about vehicle dynamics and its in-plane stability,
considering different friction onthe front and rear axle, has been
conducted by Shen et al. [10] which introduced the so-calledjoint
point locus approach to find system equilibrium points and their
associated stabilityproperties, finding that the difference between
the front and the rear steering angles plays akey role. In their
paper again aimed to active steering control, the authors adopted a
bicyclemodel together with a simplified Pacejka tyre model and
linearised the relations to calculatesideslip angles.
In his doctoral thesis, Nguyen [11] analysed the behaviour of a
complete four wheel, two-axle vehicle model, including roll
dynamics and lateral load transfer and adopting the Pacejkamagic
formula to model tyreroad interactions. A number of phase
trajectories are plotted inthe sideslip angle yaw rate plane
varying steering angle and velocity. In the paper, however,the
stability maps are not defined.
Dai and Han [12] investigated the motion and stability of a
four-wheel steering vehicle,schematised by means of a bicycle, with
nonlinear lateral tyre forces modelled using a cubicexpression,
taking into account the effects of the steering mechanism, with
particular referenceto the rear steering consequences.
No studies have been found aimed to find out which vehicle
physical characteristics affectsignificantly its stability and the
influence of their variation on the stable region extension.
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Vehicle System Dynamics 1267
The present paper pursues precisely this purpose and hence it is
necessary to use a sufficientlydetailed vehicle model, allowing to
take into account all the physical parameters on which
toinvestigate. The stability of a complete four wheel, two axle,
vehicle model will be discussed.The Pacejka magic formula will be
adopted to model the pure tyreroad interactions, lateralload
transfers will also be considered.
Two different methods, both based on nonlinear models, will be
discussed: a numericalmethod consisting in the evaluation of the
state matrix eigenvalues and a graphical methodinspired to the so
called handling diagram construction.[1316]
The proposed combined use of the two methods to the vehicle
stability analysis allowsto broaden the investigation field and to
get additional information about the nature of theequilibrium
points; so, the physically expected influence of some vehicle
parameters (suchas front and rear roll stiffness, centre of mass
position and tyre cornering stiffness) on thestability region
extension can be justified also from a mathematical point of
view.
2. Vehicle model
In order to study vehicle steady-state equilibrium in lateral
dynamics and its stability condi-tions, a quadricycle planar model
has been adopted. The model is characterised by two statesreferred
to 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 referredto the earth-fixed axis system, and U
(longitudinal velocity) and V (lateral velocity) are itscomponents
in the vehicle axis system; r is the yaw rate evaluated in the
earth-fixed system,Fxij and Fyij are, respectively, longitudinal
and lateral components of the tyreroad interactionforces with
reference to the tyre middle plane.
The front and the rear wheel track are indicated with tf and tr
; the distances from the frontand rear axle to the centre of
gravity are represented by a and b, respectively. The steer angleof
the front tyres is denoted by , while the rear tyres are assumed to
be non-steering.
Steering angle and longitudinal velocity represent manoeuvre
parameters.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
Figure 1. Coordinate systems.
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1268 F. Farroni et al.
following:
m(V + Ur) = (Fy11 + Fy12) cos + Fy21 + Fy22Jzr = (Fy11 + Fy12)a
cos (Fy21 + Fy22)b + (Fy11 Fy12)
t
2sin ,
(1)
where m is the vehicle total mass, Jz is its moment of inertia
with respect to the z-axis.It is worth highlighting that Equations
(1) are also valid in the hypothesis of full negligi-
bility of both aerodynamic actions and tyre rolling resistance,
independently from the drivingwheels.
2.1. Tyre model
One of the most critical aspects in vehicle dynamics modelling
is the determination of thetyreroad interaction forces. The
underlying physical phenomena are rather complex and, fortheir
description, is often necessary referring to empirical models, the
most renowned of whichis undoubtedly the Pacejkas magic
formula.[16,17]
It expresses the interaction forces as a function of the
longitudinal slip ratio, of the slipangle and of the vertical load
by means of nonlinear functions. The formula contains a numberof
parameters {P}, which have no clear physical meaning, usually
identified starting fromexperimental data in order to reproduce
properly tyre physical behaviours. In the presentpaper, parameters
of common passenger tyres will be employed, analysing, in
particular, theonly lateral dynamics.
The functional expression of the Pacejka pure lateral
interaction is the following:
Fyij = fy(Fzij, ij; {P}), (2)
where is the slip angle, Fz is the vertical load, i is an index
defining the axle (1 for front, 2for rear), j is an index defining
the side (1 for left, 2 for right).
Pacejka coefficients implemented in the model provide the
lateral force characteristicsshown in Figure 2 for three different
load conditions.
Figure 2. Lateral force interaction curves for three different
vertical loads.
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Vehicle System Dynamics 1269
As concerns the previously described vehicle model, the slip
angle of each tyre is given by
11 = tan1(
V + raU r(t/2)
)
12 = tan1(
V + raU + r(t/2)
)
21 = tan1(
V rbU r(t/2)
)
22 = tan1(
V rbU + r(t/2)
).
(3)
Moreover, the vertical load on each wheel can be seen as the sum
of a static load contributionand a dynamic variable load due to
inertia forces on the vehicle body.
With reference to a static approach for the vehicle roll motion,
normal forces can be definedas [18]:
Fz11 =mgb
2(a + b) 1t
(d
2(a + b)Urb +k1k
(h d)(Ur))
Fz12 =mgb
2(a + b) +1t
(d
2(a + b)Urb +k1k
(h d)(Ur))
Fz21 =mga
2(a + b) 1t
(d
2(a + b)Ura +k2k
(h d)(Ur))
Fz22 =mga
2(a + b) +1t
(d
2(a + b)Ura +k2k
(h d)(Ur))
,
(4)
where g is the gravity, d is the height of the intersection
point between the roll-axis and thevertical plane passing through
the y-axis, h is the height of the centre of gravity, k1 and k2are,
respectively, the front and the rear axle roll stiffness (taking
into account also roll stiffnessdue to suspension elements) and
their sum is indicated by k = k1 + k2 .
3. Equilibrium points and local stability analysis
Probably the simplest way to investigate on the stability of
such a complex nonlinear system isthe step by step time integration
method, although it does not allow a complete understandingof the
influence exerted by the different system parameters.
An analytical approach, on the contrary, would allow one to
clearly describe each parametersinfluence on vehicle behaviour. In
the following, two different ways to solve the analyticalproblem
will be discussed.
3.1. Numerical method (phase plane)In order to analyse the
stability problem in the (, r) plane, the nonlinear model (1)
ispreliminarily posed in the form as follows:
=(r + 1
Um[(Fy11 + Fy12) cos() + Fy21 + Fy22]
)1
1 + tg2r = 1
Jz
[(Fy11 + Fy12)a cos() (Fy21 + Fy22)b + (Fy11 Fy12) t2 sin()
].
(5)
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Being
= tan1(
VU
). (6)
System (5), with the expressions (2)(4) is in the form:
x = f (x). (7)
In which f is a nonlinear function of the state x = { r }
obtained taking into account, differentlyfrom the widely used
approaches, nonlinearities due to tyreroad interactions (2) and to
slipangles (3), together with the load transfers (4).
The equilibrium points are such that:
f (x) = 0. (8)
The analysis of the eigenvalues of the linearised system allows
to evaluate the stabilityproperties of the equilibrium points.
In order to find the zeros of Equation (5), a numerical
procedure is recursively adoptedusing, as initial guesses, all the
possible pair of values , r with and r varying in the range(5,5) in
step of 0.5.
Solutions characterised by a positive value of the vertical load
on one or more tyres aredischarged, not being an acceptable
condition for the vehicle model (loss of contact).
Equation (5) is then numerically linearised by computing its
Jacobian matrix in the neigh-bourhood of each found solution and
the eigenvalues of the matrix are analysed so qualifyingthe 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
nonlinear Equation (5) is then stepby step time integrated, by the
RungeKutta method, starting from suitable initial
conditions;moreover step by step time integration allows to plot
phase trajectories that, even if slightly dif-ferent from the ones
obtainable integrating a complete system (accounting for roll
dynamics),are suitable for the purposes of the work and do not
affect its generality.
Considering the typical roll stiffness and roll moment of
inertia values, it can be easilyunderstood that the dynamics
connected with them result negligible if compared with lateraland
yaw dynamics. In a comparative analysis as the proposed one, the
negligibility of suchkind of transient dynamics does not invalidate
the results in terms of influence that the analysedparameters exert
on stability.
3.2. Graphical method (handling diagram)In normal working
conditions of a vehicle performing a steady-state lateral
manoeuvre, it ispossible to suppose little steering angles and
longitudinal velocity much greater than othervelocity components (U
V and U rt/2).
As a consequence, the slip angles become:
11 = 12 = 1 = V + arU21 = 22 = 2 = V brU
(9)
and so the slip angles of the two wheels of the same axle can be
assumed equal.
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Vehicle System Dynamics 1271
In the steady-state conditions, neglecting the term (Fy11
Fy12)(t/2) as commonly can befound in literature,[18] Equation (1)
becomes
mUr = Fy1 + Fy20 = Fy1 a Fy2 b,
(10)
where Fyi = Fyi1 + Fyi2 . Equation (10) describes the
steady-state motion conditions for anonlinear bicycle vehicle
model.
By means of proper equations which take into account suspension
parameters (e.g. rollstiffness, roll centre) and vehicle
geometry,[19,20] it is possible to obtain the lateral loadtransfers
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 correspondingvalue of the load transfer;
moreover, 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 generating twoforces, 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, characterisea point of the so called Effective
Axle Cornering Characteristic. Iterating this process fordifferent
values of the axle lateral force it is possible to build, point by
point, the wholeEffective Axle Cornering Characteristic (Figure
3).
The Effective Axle Cornering Characteristic curves are usually
represented in a non-dimensional form obtained by dividing them by
the vertical static load acting on each axle(Figure 4).
The solution of the motion equations system (10), representing
the steady-state equilibriumconditions, can be found by means of a
graphical method: the handling diagram [13] whichgives the
possibility to analyse the steady-state equilibrium of a nonlinear
bicycle model.
Figure 3. The thick curve (Front effective axle cornering
characteristic for K1 = 10, 000 Nm/rad) has been builtpoint by
point by means of the thin curves corresponding to the sum of the
interaction curves for different loadtransfers.
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1272 F. Farroni et al.
Figure 4. Dimensionless axle cornering characteristics for K1 =
K2 = 10, 000 Nm/rad.
To build the handling diagram, it is necessary to manipulate
properly Equation (10), thereforeobtaining the following:
Ur = 1m
[Fy1(1) + Fy2(2)] = AyFy1(1)a = Fy2(2)b
(11)
and, consequently:
Ayg
= U2
g(a + b) [ (1 2)]Ayg
= Fy1(1) (a + b)mgb
= Fy2(2) (a + b)mga
.
(12)
Figure 5. Handling diagram.
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Vehicle System Dynamics 1273
In the so-called handling diagram phase plane (Ay/g, (1 2)),
each vehicle equilibriumpoint corresponds (Figure 5) to the
intersection point between the curves defined in Equation(12). The
first equation of (12) is represented in that plane by a straight
line with parameters and U setting position and inclination of the
line and it is characteristic of the manoeuvre.The second curve
depends on the axle effective characteristics (i.e. lateral forces
Fy1(1) andFy2(2)) and on the vehicle body characteristics: m, a,
and b.
As an example, in Figure 5 two equilibrium points can be seen,
however nothing can besaid a priori on the nature of such
points.
4. Comparison between the two methods
As said above, given the vehicletyres pair, all the equilibrium
points must belong, in thehandling diagram plane, to a
characteristic curve described by the second equation of (12).So,
adopting the numerical method described in Section 3.1, for a great
number of differentpairs and U (i.e. different straight line in the
above plane) and saving, for all equilibriumpositions, the
quantities:
1 2 = 11 + 122 21 + 22
2Ayg
= Urg
(13)
the characteristic curve may be reconstructed point by point as
done in Figure 6, in which agood correspondence between numerical
and graphical solutions may be noted.
The little differences between the results provided by the two
methods occur only in corre-spondence of the greater (1 2) values
and may be attributed to the further simplificationsintroduced by
the graphical method in Equations (9) and (10).
Some considerations can be made: handling diagram offers the
great advantage to providequickly steady-state solutions of the
motion equations, but analysing its curves is not possibleto
discriminate univocally between the different equilibrium
natures.
Figure 6. Comparison between numerical and graphical method.
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1274 F. Farroni et al.
Moreover, the handling diagram allows to synthetically describe,
by means of a setof characteristic curves, the complete vehicle
behaviour independently from the particularmanoeuvre.
The combined use of the two methods allows on one side a
physical understanding of theresults obtained in the phase plane,
on the other the definition of handling diagram
characteristiccurves, which clearly indicate the link between the
vehicle parameters and the number/natureof the equilibrium points
in terms of stability. On the other side, such combined use gives
thepossibility to find out, thanks to the manoeuvre line, the
working conditions under which acertain equilibrium point has been
numerically obtained.
5. Stability analysis results
The influence of tyre and vehicle parameters will be analysed in
this section following thecombined approach. In particular, the
interest will be focused on parameters such as the centreof mass
position, cornering stiffness and roll stiffness.
In what follows, two different vehicle configurations are
considered (Table 1), vehicles Aand B share all the parameter
values except for the longitudinal position of the centre of
gravity(i.e. a and b parameter values are swapped).
The two vehicles can be equipped with three different tyres
(Figure 7).
Table 1. Vehicle parameters.
Parameter Vehicle A Vehicle B
a (m) 0.85 1.35b (m) 1.35 0.85tc (front/rear) (m) 1.6 1.6h (m)
0.55 0.55d (m) 0.4 0.4m (kg) 1500 1500Jz (kg m2) 1800 1800
Figure 7. Lateral force interaction curves for three different
tyres.
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Vehicle System Dynamics 1275
Curves in Figure 7 are relative to three different tyres in the
same conditions of vertical load(3000 N) and adherence. All the
tyres exhibit the same maximum value in terms of lateralforce (Fy)
while they show decreasing cornering stiffness passing from A to
C.
5.1. Influence of the centre of mass positionA first case study
deals with the influence of the centre of gravity position on
vehicle stability;two different manoeuvres have been considered for
the two vehicles A and B, both equippedwith tyres B and
characterised by the same value of the front and rear anti-roll
stiffness(10,000 Nm/rad). Two different manoeuvres have been
considered: a light manoeuvre (d =0.0134 rad, U = 8 m/s) and a
severe manoeuvre (d = 0.2215 rad, U = 60 m/s).
Adopting the numerical method, the results illustrated in Tables
2 and 3 have been obtainedfor vehicle A and B, respectively.
As concerns the light manoeuvre, a set of five different
solutions has been obtained forvehicle A, while vehicle B exhibits
only three solutions. The five solutions are a stable focus(i.e. a
pair of complex conjugate eigenvalues with negative real part), two
unstable foci (i.e. apair of complex conjugate eigenvalues with
positive real part) and two saddle points (i.e. a pairof real
eigenvalues of opposite sign). In Figures 8 and 9, some phase
trajectories are plottedin order to highlight the meaning of each
characteristic point.
Observing these phase portraits, it is possible to state that
for vehicle A (Figure 8) the stableregion extends to the whole
phase plane: the vehicle is able to reach the stable focus in
spiteof the presence of saddle points and unstable foci.
In the case of vehicle B, the three solutions are a stable node
(i.e. a pair of real negativeeigenvalues) and two saddle points. In
the phase portrait (Figure 9), it is possible to recognisethe
classical stable region in the neighbourhood of the node limited by
the phase trajectoriespassing through the saddle points.
For what concerns the severe manoeuvre, in both cases a set of
three different solutionshas been found; in each set of solutions
the stable solution are foci, while the unstable onesare saddle
points.
Table 2. Equilibrium points for vehicle A in two different
manoeuvres.
Manoeuvre Equilibrium nature b (rad) r (rad/s) Eigenvalues
Light Stable focus 0.00453 0.04756 12.71774 1.97914iSaddle
points 0.12788 0.68469 6.93441; 3.89110
0.16012 0.68905 6.16329; 3.55567Unstable focus 0.20871 0.62028
1.14940 2.08993i
0.25014 0.62781 0.94722 1.93767iSevere Stable focus 0.03451
0.06607 1.99386 7.22612i
0.42060 0.07793 0.63280 1.60226iSaddle points 0.29001 0.08719
4.19317; 4.92110
Table 3. Equilibrium points for vehicle B in two different
manoeuvres.
Manoeuvre Equilibrium nature (rad) r (rad/s) Eigenvalues
Light Stable node 0.001279 0.05015 14.86573; 10.56936Saddle
points 0.12665 0.57856 4.88539; 15.70561
0.15424 0.57683 4.54647; 14.79700Severe Stable focus 0.05402
0.08256 0.55651 5.78332i
Saddle points 0.23609 0.07043 1.24018; 1.982820.26275 0.07000
6.27382; 8.03310
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Figure 8. Phase trajectories for the vehicle A, light
manoeuvre.
Figure 9. Phase trajectories for the vehicle B, light
manoeuvre.
Observing the phase planes, in the case of vehicle A (Figure
10), the numerical methodshows two stable solutions, but from a
technical point of view only the first one (b = 0.03451;r =
0.06607) represents the realistic solution, being the yaw rate r
coherent with the steeringangle (i.e. both positive).
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Vehicle System Dynamics 1277
Figure 10. Phase trajectories for the vehicle A, severe
manoeuvre.
Similar considerations can be made for the pair of saddle points
as shown for vehicle B.Moreover, by the comparison of the stable
solution values for the vehicles A (realistic
solution) and B (Tables 2 and 3), it can be noted a more stable
behaviour of vehicle A denotedby a more negative real part of the
eigenvalues.
Looking at the phase trajectories in Figures 10 and 11, a
difference in the attractiveness ofthe stable foci for vehicle A
and B is also noted. In the case of vehicle B, in fact, the
phasetrajectory reaches its equilibrium point after a greater
number of orbits with respect to vehicleA. Finally, in both
figures, it can be noted the typical effect on the phase
trajectories exertedby saddle points: first the trajectories tend
to the saddle point and then, suddenly, they turnaway.
Figures 12 and 13 report the results provided by the graphical
method for the two differentmanoeuvres. In the figures, each
equilibrium point has been marked in accordance with theresults
provided by the numerical method for what concerns their nature
(Tables 2 and 3).
In the handling diagram plane, the two vehicles are
characterised by results in goodagreement with the ones obtained
using the numerical method. In fact, the same numberof equilibrium
points can be distinguished case by case for each vehicle.
The two handling curves (vehicle A and B) are symmetrical with
respect to the verticalaxis because the two axle cornering
characteristics shown in Figure 4 are swapped becauseof the
symmetry of the static loads in the two different configurations.
In this connection, it isinteresting to note that by changing the
vehicle centre position, the handling curve completelychanges,
allowing an immediate evaluation of the different vehicle
directional behaviour.Particularly in the linear zone of the
handling curve, the vehicle A takes an oversteeringbehaviour, while
vehicle B clearly exhibits an understeering tendency.
Moreover, the handling diagram confirms that intersections along
the linear part of thehandling curve are characterised by stable
conditions (stable focus in vehicle A and stablenode in vehicle
B).
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Figure 11. Phase trajectories for the vehicle B, severe
manoeuvre.
Figure 12. Handling diagram for vehicle A in Table 1.
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Vehicle System Dynamics 1279
Figure 13. Handling diagram for vehicle B in Table 1.
Table 4. Equilibrium points for vehicle B equipped with two
different tyres.
Tyres Equilibrium nature (rad) r (rad/s) Eigenvalues 1 2 (rad)
Ay/gA 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 0.4810Saddle
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 0.1000Saddle
point 1 0.1394 0.5552 4.6176; 8.5780 5.7324 0.5663Saddle point 2
0.1780 0.5414 4.4662; 8.2141 7.8055 0.5522
5.2. Influence of the tyre cornering stiffnessIn order to
investigate about the influence of tyres, a second case study
refers to a manoeuvrewith = 0.0201 rad and U = 10 m/s, performed by
vehicle B (Table 1) with the same frontand rear anti-roll stiffness
(10,000 Nm/rad) and equipped with two different tyres (tyres Aand C
in Figure 7).
Figures from 14 to 17 illustrate the results provided by the two
methods, while in Table 4the numerical results are summarised.
Figures 14 and 15 show the phase portraits (, r), respectively,
for A and C tyres equippedvehicles. Stable (node) and unstable
(saddle) equilibrium points can be noted together withsystem
trajectories delimiting the stable regions. The results provided by
the handling diagrammethod in the same conditions are shown in
Figure 16, in which it is possible to distinguishthe straight line
representing the manoeuvre and the two sets of curves for two
different tyres.
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Figure 14. Phase portrait of the vehicle B equipped with A
tyre.
Figure 15. Phase portrait of the vehicle B equipped with C
tyre.
Figure 17 represents a zoom of Figure 16 focused on the linear
region of the handling curves.Once again, in Figures 16 and 17, the
equilibrium points have been marked in accordance withthe results
provided by the numerical method for what concerns their
nature.
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Vehicle System Dynamics 1281
Figure 16. Handling diagram for the vehicle B equipped with A
and C tyre.
Figure 17. Handling diagram for the vehicle B equipped with A
and C tyre (zoom of Figure 16).
Both Figures 14 and 15 show three equilibrium points (i.e. two
saddle points and one stablenode). The same equilibrium points can
be seen in the handling diagram (Figures 16 and 17)as intersections
between the manoeuvre straight line and the handling curves. In
particular,saddle points are located in the nonlinear part of the
handling curves, while the nodes arelocated in its linear part and
are clearly visible in Figure 17.
Vehicles equipped with C tyre exhibits a phase portrait
characterised by a wider stable region(Figure 15) than the one
exhibited (Figure 14) by the vehicle equipped with A tyre. This
canalso be observed in the handling diagram (Figure 16) observing
the greater distance betweenthe two saddle points relative to the
same handling curve. At the same time, vehicles equippedwith A type
tyre is characterised by a more attractive stable node as can be
seen by comparingthe eigenvalues reported in Table 4 for both
cases. This attractive property is due to the greatercornering
stiffness that, in the presence of a perturbation of the system
state and consequently
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Figure 18. Manoeuvre: = 0.1 rad, U= 10 m/s. Saddle point
(square) and Stable node (circle) for threedifferent Roll Stiffness
set : (a) K1 = 5000 Nm/rad, K2 = 50, 000 Nm/rad; (b) K1 = K2 = 27,
500 Nm/rad;(c) K1 = 50, 000 Nm/rad, K2 = 5000 Nm/rad.
Figure 19. Handling diagram for three different values of the
ratio between front and rear axle roll stiffness.
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Vehicle System Dynamics 1283
of the side slip angles, makes the tyre A to generate a higher
reaction force variation, as alsonoticeable in the handling diagram
(Figure 17), and so a more effective return to the startingstable
condition.
The results of this second case study confirm that, also for
what concerns stability, the tyreshould be characterised by high
cornering stiffness and by an increasing trend of the lateralforce
up to high saturation values.
5.3. Influence of the roll stiffnessA third case study deals
with the investigation about the effect of the anti-roll stiffness
onvehicle stability; in particular, Figure 18 shows in the already
described (r) plane the effectof changes in the front /rear
stiffness distribution. With reference to a manoeuvre
charac-terised by U = 10 m/s and = 0.1 rad, three different
front/rear stiffness setup have beenconsidered; in particular,
Figure 18(a) refers to K1 = 5000 Nm/rad K2 = 50, 000 Nm/rad,Figure
18(b) refers to K1 = 27, 500 Nm/rad K2 = 27, 500 Nm/rad, Figure
18(c) refers toK1 = 50, 000 Nm/rad K2 = 5000 Nm/rad. Vehicle B,
equipped with B tyres, has beenconsidered for this case study.
Analysing Figure 18(a) or equivalently 19, only one equilibrium
point can be noticed,distinguishable by a negative value of r and
then of Ay. It represents an unstable analyticalsolution, very hard
to be physically reproduced by an actual vehicle; in fact the only
intersectionfound in the handling diagram plane belongs to a branch
of the vehicle curve not coherentwith the manoeuvre inputs.
Figure 20. Handling diagram for two different values of the
ratio between front and rear axle roll stiffness (zoomof Figure
19).
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Figure 21. Manoeuvre: = 0.1 rad, U = 10 m/s. Saddle () Node (O)
Bifurcation diagram. The Bifurcationparameter is the ratio:
K1/K2.
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 withmanoeuvre
inputs and moves towards the central zone of the stable region
(Figures 18(b)and 18(c)) confirming the positive effects and
increasing of the front axle stiffness.
In the handling diagram plane (Figure 19), this effect can be
highlighted considering theintersection between the manoeuvre line
and the linear part of the vehicle curve. Such inter-section occurs
for lower values of lateral acceleration with the increase of the
front/rear rollstiffness ratio (Figure 20). This kind of result
confirms that the vehicle loss of stability man-ifests itself with
the sliding of the rear axle and suggests that it is possible to
vary the wholevehicle tendency to instability acting properly on
the front and rear axle stiffness. For example,adopting active
control systems, such as active anti-roll bars and active
suspensions, it wouldbe possible to change properly the front/rear
roll stiffness ratio, so assuring a larger stabilitymargin.
For what is said above, the front/rear roll stiffness ratio can
be seen as a saddle nodebifurcation parameter. In Figure 21, the
bifurcation diagram has been reported with referenceto the two
states (, r). The front/rear roll stiffness ratio, corresponding to
the bifurcationpoint of Figure 21, would make, in the handling
diagram plane, the manoeuvre straight linetangent to the vehicle
curve.
6. Conclusions
Two different analytical methods, the phase plane and the
handling diagram, have beenemployed and their results have been
combined to better analyse the in-curve vehicle lateralstability.
The vehicle model is fully nonlinear, taking into account the
lateral load transfers andthe nonlinear Pacejka tyreroad
interactions. The combined use of the two methods allows aphysical
understanding of the results obtained in the phase plane,
highlighting the link betweenthe vehicle parameters and the
number/nature of the equilibrium points.
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Vehicle System Dynamics 1285
The study of the influence on vehicle stability of parameters,
such as the centre of themass longitudinal position, tyre cornering
stiffness and front/rear roll stiffness ratio, has beencarried out.
The obtained results confirm theoretical expectations about the
influence of theabove physical parameters, and show the possibility
for some of them to be saddle nodebifurcation parameters.
Moreover, the investigated influence of the front/rear roll
stiffness, not widely studied inliterature from an analytical point
of view, suggests the employment of active/semi-activeanti-roll bar
and suspensions not only for enhancing vehicle handling, but also
for stabilityimprovement.
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