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2005-01-0398
SAE TECHNICAL
PAPER SERIES
Influence of Chassis Characteristics on
Sustained Roll, Heave and Yaw Oscillations
in Dynamic Rollover Testing
Aleksander Hac
Delphi Corporation
Reprinted From: Vehicle Dynamics and Simulation 2005
(SP-1916)
2005 SAE World Congress
Detroit, Michigan
April 11-14, 2005
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Printed in USA
2005-01-0398
Influence of Chassis Characteristics on Sustained Roll, Heave
and Yaw Oscillations in Dynamic Rollover Testing
Aleksander Hac
Delphi Corporation
Copyright © 2005 SAE International
ABSTRACT
In dynamic rollover tests many vehicles experience
sustained body roll oscillations during a portion of road
edge recovery maneuver, in which constant steering
angle is maintained. In this paper, qualitative explanation
of this phenomenon is given and it is analyzed using
simplified models. It is found that the primary root cause
of these oscillations is coupling occurring between the
vehicle roll, heave and subsequently yaw modes
resulting from suspension jacking forces. These forces
cause vertical (heave) motions of vehicle body, which in
turn affect tire normal and subsequently lateral forces,
influencing yaw response of vehicle. As a result,
sustained roll, heave and yaw oscillations occur during
essentially a steady-state portion of maneuver. Analysis
and simulations are used to assess the influence of
several chassis characteristics on the self-excited
oscillations. The results provide important insights, which
may influence suspension design.
INTRODUCTION
Dynamic rollover testing has received a lot of attention in
the past several years. In an effort to build safer vehicles
and to reduce the number of fatalities resulting from
rollover accidents, vehicle manufacturers and suppliers
pursue design changes, which could lead to improved
rollover resistance. These goals cannot be achieved
without adequate testing procedures. Concurrently, the
need to evaluate and rank rollover resistance of vehicles
and pass this information to consumers led to recent
introduction by National Highway Traffic Safety
Administration (NHTSA) of a dynamic rollover test as
part of a New Car Assessment Program (NCAP). In this
test, vehicle driven on a level, dry surface is steered by a
robot at high rates and large amplitudes of steering angle
first in one direction, then the other in order to emulate
driver steering input in an emergency road edge recovery
maneuver. The vehicle fails the test when it experiences
a Two Wheel Lift Off (TWLO) of 2 inches (about 5 cm) or
more. After the second turn, the steering angle is
maintained for about four seconds. During this portion of
maneuver, many vehicles experience sustained, and
sometimes even increasing, body roll oscillations. In
some instances, these oscillations are so severe that
TWLO occurs not in transient, but during steady state
part of the maneuver. Concurrently, generally weaker
oscillations occur in lateral acceleration and yaw rate
responses of vehicle. Similar behavior is observed in
severe J-turn maneuvers. To the author’s best
knowledge no satisfactory explanation of this
phenomenon has been offered.
In this paper an explanation of these oscillations is given
and a quantitative description is presented using
simplified models of vehicle dynamics. It is found that the
primary root cause of these sustained vibrations is
coupling occurring between the vehicle roll, heave and
subsequently yaw modes resulting from suspension
jacking forces. These forces cause significant vertical
(heave) motions of vehicle body during heavy cornering.
These vertical motions result in fluctuations of tire normal
forces. Since in the steady-state portion of the dynamic
rollover test the vehicle is at the limit of adhesion, where
lateral tire forces are approximately proportional to
normal forces, changes in normal forces exert direct
influence on the lateral forces, and subsequently yaw
response of vehicle. As a result, sustained roll, heave
and yaw oscillations occur during essentially a steadystate portion of maneuver. These vibrations are analyzed
in this paper in more detail.
The paper is organized as follows. First, the studied
phenomenon is described quantitatively and illustrated
using vehicle test data and then reproduced in
simulations using a validated vehicle model. It is shown
that in addition to roll oscillations, significant body heave
oscillations occur. The effect of several parameters on
amplitude of oscillations is then examined. It is found that
reducing the jacking effects in suspension can nearly
eliminate persistent body roll and heave oscillations.
Using simplified vehicle models, local stability of the
vehicle system in the neighborhood of the operating point
is analyzed based on system equations linearized about
that point. Influence of several chassis characteristics on
the self-excited oscillations is examined. The results
provide important insights, which may influence
suspension design.
2
Meas. Lat. Accel. [m/s ]
400
200
0
-200
-400
0
2
4
6
8
40
20
0
-20
-40
0
10
measured
simulated
5
0
-5
-10
0
2
4
6
8
2
4
Time [s]
6
8
10
Roll Angle [deg]
In order to study this phenomenon, vehicle tests were
performed using a large pick up truck. The test vehicle is
shown in Figure 1.
Hand Wheel Angle [deg]
NHTSA dynamic rollover test, known as a “road edge
recovery maneuver” or “fishhook” test is designed to
measure vehicle propensity to tip up or rollover during an
emergency maneuver. The result of the test, along with
the Static Stability Factor (SSF), are combined to yield
the rollover stability star ranking of the vehicle, although
the contribution of the dynamic test to the overall ranking
is relatively small. In the test, vehicles steered by a robot
are swerved in rapid succession in two directions on
smooth, level and dry pavement. Following the second
turn, the steering angle is kept constant for about four
seconds. Details of this test, as well as results obtained
for several vehicles have been published by NHTSA
(Forkenbrock et al., 2002). One unexpected result of
these tests was that, rather than approaching a steadystate condition during the portion of the test with constant
steering angle, most vehicles experienced quite severe,
sustained body roll oscillations. These were observed
regardless of whether the ESC systems were available
or enabled on tested vehicles or not. In some cases, the
roll oscillations were so severe that the two-wheel lift off
(TWLO), as defined by NHTSA, was achieved during the
steady-state portion of the maneuver. In rare instances
the roll oscillations were increasing.
maneuver obtained using a validated vehicle model are
also shown. The results of simulation match the test data
quite well; more importantly, the model accurately
predicts the roll angle oscillations. It should be
emphasized that in the simulations the road was
assumed to be perfectly smooth and the surface
coefficient of adhesion was uniform. Thus the resulting
oscillations cannot be explained by variations in road
surface, or by road roughness. It is also noted that many
vehicles tested by NHTSA (Forkenbrock et al., 2002)
experienced much more severe body roll oscillations
than those observed in Figure 2.
Yaw Rate [deg/s]
QUALITATIVE DESCRIPTION
2
4
Time [s]
6
5
0
-5
-10
0
8
Figure 2. Measured and Simulated responses of Vehicle
in a Fishhook Test Performed at 40 mph
In order to obtain further insight, vehicle roll angle and
vertical position of the body in the same maneuver are
shown in Figure 3.
The load rack above the truck bed was used to vary the
height of vehicle center of mass. The weight of the
vehicle equipped with all safety and test equipment
approached the gross weight. A representative test result
in a fishhook maneuver performed at an entry speed of
40 mph (about 18 m/s) is shown in Figure 2. The same
entry speed is used throughout this section.
The measured steering angle, lateral acceleration at
vehicle center of gravity, yaw rate and roll angle are
shown. It is seen that vehicle experiences significant
body roll oscillations and some oscillations in lateral
acceleration and yaw rate responses during the portion
of the maneuver when the steering angle is held
constant. The results of simulation for the same
5
0
measured
simul ated
-5
-10
0
1
2
3
1
2
3
4
5
6
7
8
4
5
6
7
8
2
Body Heave [cm]
Figure 1. Test Vehicle
Roll Angl e [deg]
10
0
-2
-4
-6
-8
0
Time [ s]
Figure 3. Roll and Heave Motions Of Vehicle Body
During Fishhook Maneuver
40
2/3 nominal
nominal
1.5 nominal
5
0
-5
-10
0
2
4
6
5
0
-5
-10
0
Yaw Rate [deg/s]
2
Meas. Lat. Accel. [m/s ]
0
-5
-10
0
2
4
6
20
0
-20
-40
0
8
20
2
4
6
8
2
4
Time [s]
6
8
5
10
0
-10
2
4
Time [s]
6
8
0
-5
-10
0
0
-20
2
4
6
8
Figure 5. Effects of Roll Centers Heights and Nonlinearity
in Suspension Stiffness Characteristics on Vehicle
Behavior in Fishhook
5
Body Heave [cm]
Roll Angle [deg]
10
40
nominal
linear
zero rollc
5
-20
0
20
-40
0
8
10
Body Heave [cm]
10
Yaw Rate [deg/s]
Meas. Lat. Accel. [m/s 2]
In order to investigate the effects of vehicle suspension
parameters on the studied phenomenon, several of them
were varied in simulations. To illustrate, the results for
vehicle with nominal damping, two thirds of nominal
damping and one and a half of the nominal damping at
all four corners are shown in Figure 4.
performance. Among other parameters considered, the
height of roll centers and, to a lesser extent, nonlinearity
in suspension stiffness characteristics proved to have the
largest influence on vehicle behavior. Within the typical
range of values, reducing the height of roll centers and
making the stiffness characteristic more linear both
reduced the amplitude of body oscillations, although they
increased the steady-state roll angle, about which the
oscillations occur. An illustrative example is shown in
Figure 5. Here the results for nominal vehicle, vehicle
with linear suspension stiffness characteristic (versus
progressive characteristic for nominal suspension), and
vehicle with nominal stiffness characteristic and roll
centers at the road level are shown.
Roll Angle [deg]
According to SAE sign convention, positive vertical
position corresponds to the body being down relative to
the static equilibrium. In the test data, the body roll angle
and the vertical position were determined from three
height sensors placed in three corners of the body. Thus,
the “measured” vertical position at the body center of
gravity is a derived value and may be inaccurate; hence
the discrepancy between the traces of body heave
obtained from test data and simulation. During a portion
of the maneuver performed at a constant steering angle,
the vehicle body is subjected to approximately periodic
oscillations in both roll and heave of approximately the
same frequency of about 1.6 Hz. Both types of
oscillations appear to be coupled; furthermore, vertical
oscillations appear to be coupled with the lateral
acceleration (not shown in Figure 3), as expected. A
possible explanation is that the vertical body motions
cause changes in vertical tire forces, which induce
fluctuations in lateral forces that in turn affect lateral
acceleration and subsequently sustain body roll
oscillations.
2
4
Time [s]
6
8
0
-5
-10
-15
0
2
4
Time [s]
6
8
Figure 4. Effect of Suspension Damping on Vehicle
Response in Fishhook Maneuver
As expected, reducing the suspension damping
increases the amplitude and time duration of oscillations,
while increasing it has an opposite effect. Simulations
performed for wider range of damping variations showed
similar trend, with only minimal and quickly damped
oscillations when damping is increased to double the
nominal value. However, for the vehicle considered here,
oscillations cannot be entirely eliminated by variation of
suspension damping within the range, which is normally
judged acceptable for good body isolation (ride)
Reducing the heights of roll centers increases the peak
and steady-state values of roll angle, but it also
significantly reduces amplitude of body heave vibrations
and roll oscillations at steady-state. The effect of
suspension nonlinearity is subtler, but making the
suspension stiffness characteristic linear also reduces
the roll and heave oscillations and makes them appear
better damped. This is not surprising, since reducing
suspension stiffness at the operating point, while keeping
damping unchanged, increases the damping ratio.
Reducing suspension nonlinearity and the heights of roll
centers are both factors contributing to reduction in the
suspension jacking forces. These are the unbalanced
vertical components of the suspension forces, which
occur during cornering on a smooth road surface and
tend to lift the vehicle body above the static equilibrium.
Reducing the height of the roll centers reduces the
vertical components of the resultant forces in the rigid
suspension arms during cornering. Reducing nonlinearity
in the suspension stiffness characteristic makes the
compression of the outside suspension closer to the
extension of the inside suspension, thus reducing the
lifting of body center. This is explained in more detail in
the next section.
In maneuvers performed on smooth roads, jacking
forces in the suspension constitute a primary coupling
mechanism between the body roll and heave modes.
Nonlinearities in suspension damping can also contribute
an unbalanced vertical force during transient maneuvers,
but this effect is generally smaller and occurs only during
transients.
10
5
0
-5
-10
0
2
4
6
8
C
0
-20
-40
0
Body Heave [cm]
Roll Angle [deg]
5
0
-5
4
Time [s]
6
As described earlier, during steady-state cornering
jacking forces arise primarily due to two sources: vertical
components of forces transmitted by suspension rigid
links and nonlinearities in suspension stiffness
characteristics. The first effect is illustrated in Figure 7.
hroll
8
R
Fz
2
4
6
F
8
0
hrollc
Fy
γ
2
2
In this section a simplified analytical model is developed,
which describes the coupling between the roll and heave
body modes and subsequently the yaw plane motion.
The model needs to be simple enough to facilitate
studies of the system stability and to describe explicitly
the effects of important design parameters on vehicle
body roll oscillations in fishhook tests. At the same time,
the model should capture the important aspects of
vehicle behavior in a steady-state turn at the limit lateral
acceleration, as the vehicle experiences in the second
phase of fishhook maneuver. In particular, the effects of
suspension jacking forces, which couple the roll and
heave modes must be included in the model.
20
10
-10
0
ANALYSIS
40
1/2 st.com.
nominal
2 st.com.
Yaw Rate [deg/s]
Meas. Lat. Accel. [m/s 2]
The effects of other parameters of the vehicle were
considered in simulations. For example, the influence of
front steer compliance is illustrated in Figure 6.
Performance of a vehicle with nominal steer compliance
is compared to one with one half and double the nominal
compliance. Within a reasonable range of values the
effect of steer compliance is small. However, even when
the front steer compliance is twice the nominal value, the
body roll and heave oscillation do not decrease, even
though the steady-state values of lateral acceleration and
roll angle are reduced. This is somewhat surprising,
since the oscillations generally decrease as severity of
maneuver
is
reduced.
Additional
simulations
demonstrated that in maneuvers performed at higher
speeds, increasing front steer compliance may even lead
to increased body roll oscillations in the steady-state
portion of maneuver. This effect is further analyzed in the
next section.
state portion of maneuver. In order to verify this
conjecture, provide further insights, and establish
quantitative relationships between vehicle parameters
and the investigated phenomenon, stability analysis of
vehicle during steady-state limit cornering was
conducted. It is described in the next section.
tw/2
-2
-4
-6
-8
Figure 7. Jacking Force Exerted by Suspension Links
0
2
4
Time [s]
6
8
Figure 6. The Effect of Front Steer Compliance on
Vehicle Response in Fishhook Maneuver
It is concluded that the most likely primary cause of
sustained body roll oscillations in steady-state portion of
the maneuver is coupling between the vehicle roll, heave
and subsequently yaw modes resulting from suspension
jacking forces. These forces cause vertical (heave)
motions of vehicle body, which in turn affect tire normal
and subsequently lateral forces, influencing yaw
response of the vehicle. As a result, sustained roll, heave
and yaw oscillations occur during essentially a steady-
Lateral forces generated during cornering maneuvers are
transmitted between the body and the wheels through
relatively rigid suspension links. In general, these
members are not parallel to the ground; therefore the
reaction forces in these elements have vertical
components, which usually do not cancel out during
cornering, resulting in a vertical net force, which pushes
the body up. It is known (Gillespie, 1993; Reimpell and
Stoll, 1996) that forces transmitted between the vehicle
body and a wheel through lateral arms are dynamically
equivalent to a single force, which reacts along the line
from the tire contact patch to the roll center of
suspension. The roll center is by definition the point in
the transverse vertical plane at which lateral forces
applied to the sprung mass do not produce suspension
roll. This is illustrated in Figure 7 for a double A arm
suspension. If the tire lateral force is Fy, then the jacking
force, Fz, is
Fz = F y tan γ
(1)
where γ is the inclination angle of the line connecting tire
contact patch with the roll center to the horizontal. Note
that in SAE sign convention the force lifting the body up
is negative.
The second jacking effect is due to non-linear spring
characteristics. Suspension stiffness characteristics are
usually progressive; that is, stiffness increases with
suspension deflection in order to maintain good ride
properties with a full load and to minimize bottoming of
suspension. During cornering maneuvers, a progressive
characteristic of suspension permits smaller deflection in
compression of the outside suspension than deflection in
extension of the inside suspension. As a result, height of
vehicle center of gravity increases. This effect can be
particularly significant for fully loaded vehicle, for which
the suspension can become fully compressed during
heavy cornering, thus entering the region of high
nonlinearity. This effect is illustrated in Figure 8.
Fz
∆zcomp
∆zext
∆Fz
O
∆z
A
∆Fz
Figure 8. Jacking Effect Resulting from Non-linearity in
Suspension Stiffness Characteristic
The nominal operating point, corresponding to static
equilibrium of lightly loaded vehicle is the point O, and
the operating point with full load is A. In Figure 8
suspension extension is positive and compression
negative. During cornering, part of the roll moment,
which is not balanced by a roll bar, Mrolls, is balanced by
the suspension springs, resulting in a vertical load
transfer of ∆Fz = Mrolls/ts, where ts is either the lateral
distance between the suspension springs (for a rigid
axle) or a track width multiplied by suspension linkage
ratio. Due to nonlinear spring characteristics, this change
in spring load brings about extension of the inside
suspension by ∆zext, which is larger in magnitude than
compression of the inside suspension, ∆zcomp. As a
result, in addition to body roll, the center of the body will
experience a vertical displacement of
∆z =
∆z ext − ∆z comp
(2)
2
Thus for any value of roll moment (in static conditions)
the corresponding values of roll angle and vertical rise of
body center can be determined. This vertical
displacement can be expressed in terms of suspension
force at the operating point. Carrying out these
calculations for front and rear suspensions, the jacking
effect at vehicle center of gravity can be determined and
expressed in terms of vertical force in response to body
roll angle:
Fzs = k zφ (φ )φ
(3)
Here Fzs is the jacking force due to the non-linearity in
suspension stiffness and kzφ denotes the coefficient
relating the jacking force to roll angle. Note that since the
jacking force is always negative in SAE sign convention,
kzφ is negative for positive roll angles.
STABILTY ANALYSIS OF ROLL AND HEAVE MOTION
As stated earlier, our objective here is to develop a
simplified model, which would permit one to study the
effects of design parameters on stability of roll and heave
modes during limit cornering. According to Liapunov’s
indirect method, local stability of a nonlinear system can
be determined, under quite general conditions, by
studying the stability of linearized model, as long as the
linearized system is not marginally stable (Vidyasagar,
1978, Section 5.4). In this section we first introduce a
simple non-linear model describing the lateral motion of
vehicle along with the body heave and roll, which takes
into account fundamental couplings between these
modes. Subsequently, we linearize equations for this
model around the operating point corresponding to
steady-state limit cornering; using this linearized
equations, we formulate necessary and sufficient
conditions for the asymptotic stability (or instability) of the
system. Since the parameters in the equations of motion
depend on the characteristic parameters of the vehicle
and the suspension system, the effects of these design
parameters on stability in steady-state limit cornering can
be examined.
The lateral, roll and heave modes can be approximately
described by the following equations:
ma y + m s (hroll − z )φ = F yf cos δ + F yr
I xx φ + cφ φ + k φ φ = −m s (hroll − z )(a y − gφ )
(
)
m s z + c z z + k z z = − F yLF − F yRF cos δ tan γ
(
)
(4)
f
− F yLR − F yRR tan γ r + k zφ (φ )φ − m s g (1 − cos φ )
In the above equations m denotes vehicle mass, ms the
body (sprung) mass, hroll is the height of body center of
mass above the roll axis at equilibrium, Ixx the body roll
moment of inertia about the roll axis, Fyf and Fyr are the
lateral forces of front and rear axle, δ is the front wheel
steering angle, cφ and kφ are the roll damping and roll
stiffness of suspension, cz and kz are the suspension
damping and stiffness in vertical direction (in heave),
FyLF, FyRF, FyLR, and FyRR are the lateral tire forces, γf and
γr are the inclination angles of lines connecting the tire
contact patches to the roll centers for front and rear
suspensions, respectively, kzφ is the term describing the
jacking force due to suspension stiffness nonlinearity,
and g is gravity acceleration. Variable ay denotes lateral
acceleration, φ - the body roll angle, and z vertical
position of the body with respect to static equilibrium.
SAE sign convention is used in equations (4). The
stiffness parameter values kφ and kz include the effects of
suspension and tire compliance and they correspond to
linearized values about the operating point.
In order to simplify the terms involving the differences in
lateral tire forces appearing in the last of equations (4), it
is assumed that the tire lateral forces in limit cornering
are proportional to the normal forces. That is, for the
front axle
F yLF − F yRF
F yf
=
FzLF − FzRF
Fzf
(5)
where subscript z refers to vertical forces. Similar
equation holds for the rear axle. In addition, the normal
force for the left front tire can be approximated by the
following equation
FzLF =
m 
mb 
 g − s z + κ
2L 
m 
(
mh
a y − gφ
tw
f
)
(6)
(7)
2mκ f h
tw g
tan γ f , Ar =
(
)
2m 1 − κ f h
tw g
tan γ r
where µ y = a y max / g and aymax is the maximum steadystate acceleration, which vehicle can sustain on dry road.
The sign of expression on the right hand side is the
same as that of ay. Substituting equations (7) and (9) into
equations (4) yields
ma y + m s (hroll − z )φ = µ y (mg − m s z)
I xx φ + cφ φ + k φ φ = − m s (hroll − z )(a y − gφ )
(
) (
m s z + c z z + k z z = − A f + Ar a y a y − gφ
+ k zφ (φ )φ − m s g (1 − cos φ )
)
(8)
It can further be assumed that in the limit cornering, the
sum of all lateral tire forces is proportional to the total
normal load. That is
(10)
The system of equations (10) can now be linearized
about the operating point (ay0, φ0, z0), which corresponds
to steady-state limit cornering. Let us denote
a y = a y 0 + ∆a y , φ = φ 0 + ∆φ ,
z = z o + ∆z
(11)
where the symbol ∆ signifies presumably small
incremental variables. Carrying out the linearization
procedure (that is, substituting (11) into (10), canceling
equal terms on both sides of equations, and neglecting
higher order terms in incremental variables), the
following linear equations are obtained:
m∆a y + m s hroll1 ∆φ = − µ y m s ∆z
I xx ∆φ + cφ ∆φ + kφ1 ∆φ − m s a y 2 ∆z = − m s hroll1 ∆a y
(
)
[
(
)
(12)
]
+ k zφ + A f + Ar a y 0 g − m s g sin φ 0 ∆φ
The following notation is used in the above equations:
hroll1 = hroll − z 0 , k φ1 = k φ − m s g (hroll − z 0 )
a y1 = a y 0 −
1
gφ 0 , a y 2 = a y 0 − gφ 0
2
(13)
Note that since z0 < 0, hroll1 > hroll. Eliminating the
incremental lateral acceleration, ∆ay, from equations (12)
yields a system of two second-order linear differential
equations with two unknowns. After taking the Laplace
transform on both sides of these equations, the following
fourth-order characteristic equation is obtained:
b4 s 4 + b3 s 3 + b2 s 2 + b1 s + b0 = 0
Here Af and Ar are the following coefficients:
Af =
(9)
m s ∆z + c z ∆z + k z ∆z = −2 A f + Ar a y1 ∆a y
Here L denotes vehicle wheelbase, b is the distance of
vehicle center of mass to the rear axle, h the height of
vehicle center of mass above the ground, and κf is the
fraction of the total suspension roll stiffness contributed
by the front suspension. In addition to the static
component, this equation reflects the influence of the
quasi-static load transfer due to cornering and the effect
of body heave. Analogous equations can be written for
the remaining tires. Using equations (5) and (6) and their
analogues for remaining tires, as well as the stated
assumption that the lateral forces are proportional to the
normal forces, the following equations are obtained:
(FyLF − FyRF )cos δ tan γ f = A f a y (a y − gφ )
(FyLR − FyRR )tan γ r = Ar a y (a y − gφ )
F yf cos δ + F yr = µ y (mg − m s z)
(14)
In order to abridge the subsequent equations, the
following notation is introduced:
2
m s2 hroll
m
1
, b22 = 2 A f + Ar a y1 s hroll1
m
m
m
m s1 = m s − 2 A f + Ar a y1 µ y s
(15)
m
k zφ 1 = k zφ + A f + Ar a y 0 g − m s g sin φ 0 ,
(
I xx1 = I xx −
(
(
)
)
)
Now the coefficients in equation (14) are given by:
b0 = k z k φ 1 − m s a y 2 k zφ1 , b1 = cφ k z + k φ 1c z ,
b2 = I xx1 k z + k φ 1 m s1 + cφ c z − µ y k zφ 1
m s2 hroll1
m
− m s a y 2 b22 ,
b3 = I xx1 c z + cφ m s1 , b4 = I xx1 m s1 − µ y b22
(16)
m s2 hroll1
m
Stability of the linear system of equations (12) is
determined by the roots of the characteristic equation
(14). All of them must have negative real parts for the
system to be asymptotically stable. The presence of
exponentially stable, but weakly damped modes can also
be detected. Since the coefficients of equations (12) and
(14) depend on the parameters of the vehicle and on
suspension design parameters, their effects on system
stability can be investigated. To illustrate, the effect of
location of roll centers on vehicle stability is shown in
Figure 9. These results were obtained for nominal
vehicle parameters and lateral acceleration at the
2
operating point ay0 = 8.5 m/s .
Rear Roll Center Height [m]
UNSTABLE
0.3
0.2
STABLE
0.1
0
0
0.1
0.2
0.3
0.4
Front Roll Center Height [m]
It has been known based on quasi-static analysis (Hac,
2002) that in order to maximize rollover resistance, roll
centers cannot be too high, since then the positive effect
of reducing steady-state roll angle may be offset or even
dominated by the negative effect of increase in height of
the body center of mass during cornering due to jacking
forces. The analysis in this paper demonstrates that the
heights of roll centers are also limited by the requirement
of stable roll response in the steady-state portion of the
fishhook test. Although for the vehicle considered here,
the heights of roll centers that can induce unstable
behavior are larger than typically occurring in light
vehicles, even lower values can contribute to stable, but
poorly damped response.
It should be noted that when both roll centers are at the
ground level and the suspension characteristic is linear,
then the coefficients Af = Ar = 0 (equation 8) and kzφ= 0.
Consequently, there is no coupling between the roll and
heave modes according to equation (12) and heave
mode is not excited at all; as a result, the system is
asymptotically stable.
Similarly the effects of other vehicle design parameters,
which affect parameters in equation (12), on stability of
vehicle in steady state limit cornering can be
investigated. These parameters include suspension
stiffness and damping, roll stiffness distribution between
front and rear axles, vehicle mass (payload), roll moment
of inertia, etc. Although the effect of tires is not directly
included, it is reflected to some extend in the steady
state lateral acceleration at the limit, ay0. For example, it
can be shown that increasing this value by use of more
aggressive tires makes the linearized system less stable.
0.5
0.4
linearized equations (12). Instability of the linearized
model implies only local instability of the nonlinear
vehicle model, since as the amplitude of oscillations
increases, the assumption that the incremental variables
are small is violated. Thus, the nonlinear system may not
become unstable, but develop a limit cycle. On the other
hand, significant oscillation may occur when the
linearized system is stable, but with poorly damped
mode(s). In this case vibrations induced during the
transient phase of fishhook maneuver will not increase,
but may decay very slowly.
0.5
Figure 9. Influence of Roll Center Locations on Stability
of Linearized System
Increasing the height of roll centers (with other
parameters unchanged) can make the vehicle unstable.
The size of the stability region, however, is sensitive to
the operating point, in particular to the lateral
acceleration, ay0. The line separating the stable and
unstable regions is a straight line since the effects of roll
center heights are contained in parameters hroll, Af and
Ar, with the first being a linear function of roll center
heights and the latter appearing only as a sum in
It is noted that during Fishhook maneuvers vehicle body
experiences not only roll and heave, but also pitch
motion. This has primarily two causes: longitudinal
deceleration of vehicle during hard cornering and
differences in proportion between front and rear jacking
forces relative to weigh distribution. The effect of body
pitch was neglected in the interest of simplicity. Taking it
into account would not provide significant new insights,
but would make close form solution too complex.
ANALYSIS OF EFFECT OF STEER COMPLIANCE
Vehicle response in the steady-state cornering at the
limit can be affected by steering compliance, which
introduces coupling between the lateral acceleration and
front steering angle. Considering a vehicle model that
describes yaw plane motion along with the roll and heave
body motions yields equations that are too complex for
explicit analysis. Fortunately, significant insights
regarding this coupling mechanism can be gained by
considering vehicle motion in the yaw plane only. The
equations of motion are:
(
)
m v y + v x Ω = F yf cos δ + F yr
= F a cos δ − F b
I zz Ω
yf
yr
(18)
where δ0 is the front steer angle without compliance and
KF is the lateral force steer compliance coefficient, which
is assumed constant. Assuming that the incremental
steer angle due to compliance, K F F yf , is much smaller
than δ0, the function cos δ can be approximated by a
Taylor expansion limited to linear terms. This yields
cos δ = cos δ 0 + K1 F yf ,
K1 = K F sin δ 0
(19)
Note that the presence of compliance reduces the
steering angle, δ, compared to δ0, but increases cosδ,
thus increasing the forcing term in equation (17).
Substituting equation (19) into (17) and subsequently
linearizing these equations about the operating point (vy0,
Ω0), the following system of equations for incremental
variables is obtained:
(
)
(
)
= ∆F a (cos δ + 2 K F )− ∆F b
I zz ∆Ω
yf
0
1 yf 0
yr
m ∆v y + v x ∆Ω = ∆F yf cos δ 0 + 2 K 1 F yf 0 + ∆F yr
β0
Cf0 = tanβ0
(17)
Here Ω denotes vehicle yaw rate, vx and vy are the
longitudinal and lateral velocities of vehicle center of
mass, a and b are the distances of vehicle center of
mass to the front and rear axles, respectively, Izz denotes
vehicle yaw moment of inertia and the meaning of other
symbols remains the same as before. Due to front steer
lateral force compliance, the front steer angle is
δ = δ 0 − K F F yf
Fyf
(20)
In the above Fyf0 is the front axle lateral force at the
operating point. Let us assume further that the cornering
stiffness of both tires of front and rear axles at the
operating point are Cf0 and Cr0, respectively. The
cornering stiffness at the operating point can be
interpreted as a tangent of the inclination angle of the
lateral force versus slip angle characteristic at the tire
slip angle corresponding to the operating point. This is
conceptually illustrated in Figure 10 for the front axle.
The cornering stiffness in the linear range of the tire
characteristic is Cfl and at the operating point
corresponding to severe cornering it is Cf0. The latter is
much smaller than the former.
Cfl = tanβl
βl
αf
Figure 10. Front Axle Lateral Force Characteristic
The incremental front axle lateral force, ∆Fyf, is related to
the front axle incremental slip angle, ∆αf, as follows:
 ∆v y + a∆Ω
∆F yf = −C f 0 ∆α f = −C f 0 
− ∆δ f
vx





(21)
Since, however, due to equation (18)
∆δ f = − K F ∆F yf
(22)
equations (21) and (22) yield
∆F yf = −
Cf0
∆v y + a∆Ω
1+ C f 0KF
vx
(23)
Similarly for the rear axle,
 ∆v y − b∆Ω 

∆F yr = −C r 0 ∆α r = −C r 0 

vx


(24)
Substituting equations (23) and (24) into the state
equations (20) and performing some manipulations
yields
∆v y = −
=
∆Ω
C f 1 + Cr0
mv x
C r 0 b − C f 1a
I zz v x
C r 0 b − C f 1a 

∆v y + − v x +
 ∆Ω
mv x


∆v y −
C f 1a 2 + C r 0 b 2
I zz v x
(25)
∆Ω
In the above equations
C f 1 = C f 0 cos δ 0
1 + 2 K F tan δ 0 Fyf 0
1+ C f 0KF
(26)
Equations (25) lead to a second order characteristic
equation, from which stability conditions can be derived.
However, these equations have the same form as those
describing a linear bicycle model (Wong, 1993).
Therefore, when Cf1 and Cr0 are both positive, the
sufficient condition for asymptotic stability at all speeds is
that the value of Cr0b-Cf1a, which is directly related to the
undesteer gradient, be positive. Using equation (26) it is
straightforward to show that the following two conditions
must be satisfied in order to make the understeer
gradient of the linearized system negative:
C r 0 b − 2aF yf sin δ 0 < 0
KF >
C r 0 b − C f 0 a cos δ 0
2aF y 0 sin δ o − C r 0 b
(27a)
(27b)
For realistic values of parameters these conditions
cannot be satisfied. However, the handling model
becomes less stable as a result of steer force
compliance if the presence of compliance increases the
value of coefficient Cf1, since in this case the value of
Cr0b-Cf1a, and consequently the understeer gradient, are
reduced. According to equation (26), Cf1 = Cf0cosδ0 when
the steer force compliance, KF = 0. Thus the presence of
compliance increases the apparent front cornering
stiffness value, Cf1, if
C f 0 < 2 F y 0 tan δ 0
(28)
In the linear range of tire operation this condition is not
satisfied since Cf is much larger than Fy (for example, for
the vehicle considered here, Cf ≈ 110,000 N/rad and Fy is
always less than 20,000 N). Thus in the linear range, the
steer force compliance reduces the front cornering
stiffness coefficient, Cf1, improving stability. When the
vehicle is at the limit, however, as is the case in the
fishhook maneuver, the front cornering stiffness can be
lower than Cf by an order of magnitude and may even be
close to zero, as illustrated in Figure 10. At the same
time, the value of 2Fy0tanδ0 can be substantial when the
front steering angle is very large. For example, for the
vehicle considered here in the fishhook maneuver
o
2Fy0tanδ0 ≈ 2*15,000N*tan20 ≈ 11,000 N, which can be
larger than Cf0 at the limit. Thus, the presence of front
steer compliance can significantly reduce the stability
margin of vehicle in the yaw plane for large steering
angles experienced in fishhook maneuver. This is
particularly true for maneuvers performed at higher
speeds, when front tire slip angle becomes large,
resulting in small values of front axle cornering stiffness
at the operating point. Therefore, yaw plane motion of
the vehicle becomes more sensitive to changes in
normal loads, thus reinforcing the coupling between the
body roll and heave and vehicle yaw modes.
The observation that the presence of steer force
compliance may reduce vehicle stability margin by
increasing the apparent front axle cornering stiffness of
the linearized model at large steering angles may seem
surprising. This can be heuristically explained as follows.
The impact of the steer force compliance on the
apparent front cornering stiffness of the linearized model
is the result of two influences, one of which acts to
increase, the other to reduce, the cornering stiffness
(and consequently the lateral force). The net result
depends on which effect dominates. In fact, the steer
compliance reduces the cornering stiffness of the
linearized system, as seen in equation (23). However,
the steer compliance also reduces the steering angle, δ,
and thus increases cosδ, as expressed in equation (19).
The latter effect is significant only for large steering
angles. Since the front axle lateral force in equation (17)
is a product of Fyf, which is reduced, and cosδ, which
increases, the net result depends on which effect
dominates. For very large steering and slip angles of
front axle, the steering compliance may provide a
positive feedback of lateral force, thus increasing the
coupling between vertical and yaw modes of vehicle and
contributing to instability.
CONCLUSION
In this paper sustained body roll oscillations experienced
by vehicles during a steady-state portion of road edge
recovery maneuver were explained and analyzed. It was
found that these oscillations occurring during hard
cornering arise primarily because of coupling between
the body roll and heave, and subsequently vehicle yaw
modes resulting from suspension jacking forces. These
forces result in vertical motions of vehicle body, which
cause fluctuations in tire normal forces and consequently
tire lateral forces. The lateral forces directly influence
lateral acceleration of vehicle, which in turn affects body
roll. The presence of front steer compliance may
reinforce the coupling between the vertical and yaw
modes and make the vehicle yaw motion less stable
under the conditions analyzed here, but for realistic
parameter values this is a higher order effect.
Stability analysis of coupled roll, heave and yaw motions
demonstrated that increasing roll center heights, which
increases jacking forces and can be used as means of
reducing steady-state roll angle, generally reduces
dynamic stability of vehicle roll and heave motion in limit
cornering by reinforcing the coupling between the roll
and heave modes, which is the primary mechanism
responsible for sustained oscillations. Thus, in addition to
other considerations, the heights of roll centers are
limited by the requirement of stable roll response in the
steady-state portion of the fishhook test. Other
suspension design characteristics, which lead to
increase of jacking forces, such as extremely
progressive suspension stiffness characteristic at the
operating point (which may be the case when
suspension bottoms) also contribute to oscillatory
response. As expected, increasing suspension damping
makes vehicle response more stable, but the levels of
damping necessary to effectively suppress the
oscillations may be, for some vehicles with high center of
gravity, difficult to reconcile with requirements of ride
comfort in normal conditions.
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