On the Frequency Domain Analysis of Tire Relaxation Effects on

On the Frequency Domain Analysis of Tire Relaxation Effects on
Proceedings of the ASME 2009 International Mechanical Engineering Congress & Exposition
IMECE2009
November 13-19, Lake Buena Vista, Florida, USA
Proceedings of the ASME 2009 International Mechanical Engineering Congress & Exposition
IMECE2009-12643
November 13-19, Lake Buena Vista, Florida, USA
IMECE2009-12643
ON THE FREQUENCY DOMAIN ANALYSIS OF TIRE RELAXATION EFFECTS ON
TRANSIENT ON-CENTER VEHICLE HANDLING PERFORMANCE
Justin Sill
Clemson University- International Center for
Automotive Research
Greenville, SC, US
Beshah Ayalew
Clemson University-International Center for
Automotive Research
Greenville, SC, US
m
r
V
δ
UA
ABSTRACT
This paper presents an elegant frequency domain approach
that can be used to analyze lateral vehicle dynamics for
transient understeer and oversteer performance. Commonly
used steady-state understeer analysis techniques are not able to
expose some effects, such as tire relaxation, in on-center
transient maneuvers. The approach presented here addresses
such transient issues using a simple two degree of freedom
handling model coupled to a model for tire lateral dynamics. In
addition to the usual yaw rate and lateral acceleration transfer
functions, this paper proposes using an understeer angle transfer
function as an easy-to-interpret metric to evaluate transient oncenter handling. Using the approach, it is shown that at low
vehicle velocities, the inclusion of tire relaxation introduces
dramatically different system dynamics by introducing highly
undamped poles into the coupled system for both an
understeering and an oversteering vehicle.
INTRODUCTION
The lateral stability of vehicles has long been analyzed
through simple analytical methods. The dominating approach
has been an understeer analysis which can characterize the
vehicle’s steady-state tendency to understeer or oversteer within
the linear handling range [1-6]. The traditional steady-state
understeer analysis techniques can aide in the design of the
vehicle, the selection of tires, and design of suspension
geometry and components. However, these methods exclude
transient effects typical of on-center driving, i.e, those
characterized by small steering perturbations from straight
ahead driving. The realm of on-center handling has become
increasingly important because it typifies the most common
driving conditions that have low lateral acceleration levels
(lower than 0.2 Gs). In such on-center maneuvers, tire
relaxation effects play an important role [7], particularly for
lower vehicle velocities.
To characterize the on-center transient performance of a
vehicle there have been some experimental studies relating the
yaw rate and lateral acceleration responses to steering input in
order to determine relative understeer/ oversteer changes [8, 9].
Such experimental methods have been successfully used to
study the behavior of the entire system, but cannot be used to
isolate the effect of tire dynamics on the vehicle performance.
This paper analyzes the transient behavior of on-center
handling including tire dynamics. This is done using an elegant
frequency domain approach applied to a reduced linear single
track model of a two-axle vehicle. In particular, the paper
considers the effect of different prevailing lateral tire-relaxation
lengths on the front and rear tires of a vehicle on its transient
understeer performance.
Key Words: transient understeer, tire relaxation, on-center
handling, transient handling, vehicle transfer functions,
understeer angle
NOMENCLATURE
αf
Front Tire Slip Angle
αr
Rear Tire Slip Angle
a
Distance from CG to Front Axle
b
Distance from CG to Rear Axle
Ay
Lateral Acceleration
β
Side Slip
L
Vehicle Wheelbase
λf
Front Tire Relaxation Length
λr
Rear Tire Relaxation Length
Cα f
Front Axle Cornering Stiffness
Cα r
Rear Axle Cornering Stiffness
Jz
Vehicle Yaw Inertia
K yf
Front Tire Lateral Stiffness
K yr
Rear Tire Lateral Stiffness
Vehicle Mass
Yaw Rate
Vehicle Forward Velocity
Road Wheel Steer Angle
Understeer Angle
Copyright © 2009 by ASME
1
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The rest of the paper is organized as follows. The first
section gives a background on the simplified vehicle handling
model without tire-relaxation effects. This is then followed by
the derivation of the equations for the case with tire-relaxation
effects. The models are then simulated mainly in the frequency
domain and a discussion is given on two different cases of a
nominally understeering and oversteering vehicle. Finally, the
observations of the study are summarized in the concluding
session of the paper.
BACKGROUND
The Handling Model. The free-body diagram of the
widely used handling (bicycle) model is shown in Figure 1 [1,
2, 4-6, 10-13].
δ
Fyf
L
b
The tire forces are defined by the cornering stiffness and
the respective tire slip angles, αf and αr ,which in turn are
functions of vehicle side slip angle, β, yaw rate, r, and road
wheel steering angle, δ (see, for example, [4]).
Fyf = −Cα f α f
(1)
a
r −δ
V
b
αr = β − r
V
αf = β +
(6)
+ Cα f Cα r L2 + mV 2 ( Cα r b − Cα f a ) 
The lateral acceleration and yaw rate transfer functions are
determined to be:
Ay
[ J V ] s 2 + [Cα r bL ] s + [VCα r L]
(7)
= Cα f V z
δ
∆1 ( s )
r
δ
( s ) = Cα f V
[ amV ] s + [Cα r L]
∆1 ( s )
(8)
These transfer functions can be analyzed to determine a
vehicle’s handling stability and performance.
m b
a  α f − αr
−

=
L  Cα f Cα r 
Ay
gL
Vc =
K us
K us =
Figure 1. Model Free-Body Diagram
Fyr = −Cα rα r
+  mV ( Cα f a 2 + Cα r b 2 ) + VJ z ( Cα f + Cα r )  s
Understeer Gradient. In steady-state cornering
maneuvers, the above simple model can further be reduced to
the determination of the understeer coefficient or gradient, Kus.
This metric is widely used in determining the oversteering and
understeering tendency of vehicles as well as the calculation of
critical and characteristic velocities [1, 4-6, 12].
a
Fyr
∆1 ( s ) = V 2 mJ z  s 2
(2)
(3)
(4)
With the side slip angle and the yaw rate as the states of
vehicle and with the front road wheel steering angle as input,
the equations of motion reduce to:
− aCα f + bCα r 
 Cα f + Cα r
 Cα f 
−
− 1

2


β 
mV
mV
  β  +  mV  δ (5)
 =
2
2
 r   −aCα f + bCα r − a Cα f + b Cα r   r   aCα f 


 J 
Jz
J zV
 z 


The characteristic equation of this system is given by:
(9)
(10)
The definition of an understeering and oversteering vehicle
can be determined from Equation (9). An understeering vehicle
has a larger front tire slip angle than that of the rear leading to a
positive value for the understeer gradient. Conversely, an
oversteering vehicle has a larger rear tire slip angle than that of
the front leading to a negative value for understeer gradient.
Experimental Methods. There exist several test
methods to quantify transient handling performance of a
vehicle. These methods include maneuvers with steer inputs
characterized by a step, a sinusoid [14], as well as random or
sine sweeps where all frequency ranges are explored [8, 9].
The most used experimental method involves the analysis
of a sine sweep maneuver. This maneuver is completed at
constant speed and steer amplitude over a range of possible
driver input frequencies (approximately 0 to 4 Hz). The time
histories of steer angle, yaw rate, and lateral acceleration are
processed through FFTs (Fast Fourier Transforms) to yield the
vehicle’s frequency domain response. This processed data is
used to identify four metrics including the vehicle’s steady-state
yaw rate gain, yaw rate natural frequency, yaw rate damping,
and lateral acceleration phase delay at one hertz of steer
excitation. These metrics are plotted in a rhombus plot (also
known as spider chart) as shown in Figure 2. This is then used
to compare relative differences of vehicles or configurations.
The tendency of a vehicle, as compared to some baseline (Q), to
2
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Copyright © 2009 by ASME
be more responsive (P) or more stable (R) can easily be
visualized.
Yaw rate natural frequency
Yaw NF
(Heading
responsiveness)
(heading
responsiveness)
Understeer
(more stable)
understeer
(more stable)
P
Q
R
Yaw SS Gain
Yaw
rateeasiness)
SS gain
(heading
(heading easiness)
Acc Phase @ 1Hz
Lat. Lat
Acc.
Phase
@ 1 Hz
(following
controllability)
(following
controllability)
oversteer
(more responsive)
Oversteer
(more responsive)
Yaw Damping
(directional
Yaw Damping
damping)
(directional
damping)
Figure 2. Four Parameter Evaluation Method for Lateral Transient
Response [8]
The rhombus plot technique is a powerful tool for
analyzing objective test data to determine differences in
transient understeer performance for on-center handling and has
inspired the frequency domain analysis considered in this paper.
The method adopted here enhances the above approach by
demonstrating the use of a third transfer function, the understeer
angle transfer function. As will be shown below, trends in this
transfer function appear easier to interpret.
DERIVATION OF EQUATIONS
The above handling model (lateral & yaw vehicle motions)
can be used in conjunction with a first order transient tire
model. From Figure 1, the basic equations of motion can be
written in the form:
(
)
mV β + r = Fyf + Fyr
(11)
J z r = aFyf − bFyr
(12)
The tire transient behavior is modeled by a first order
dynamic system as [15]:
τ f Fyf + Fyf = −Cα f α f
(13)
τ r Fyr + Fyr = −Cα rα r
(14)
Where, the tire time constants τf and τr are defined by the
relaxation length and velocity as [16]:
τf =
τr =
λf
V
λr
V
(15)
(16)
This definition of tire transient behavior was used for
simplicity though it should be noted that there are many ways in
which to account for tire dynamics including higher order
models [4, 7, 17-19]. It should also be noted that when the tire
time constants are zero the tires are represented by only
cornering stiffnesses and slip angles thus reducing this vehicle
model to the traditional steady-state cornering handling model
described above.
After substitutions of the time constants and slip angles, the
tire force equations can be expressed as shown below:
Cα f V 
a
 Fyf V
(17)
Fyf = −
 β + r −δ  −
λf 
V
 λf
C V
b  FyrV
Fyr = − α r  β − r  −
λr 
V  λr
(18)
The complete model can then be represented in state space
form with the side slip angle, yaw rate, front axle lateral force,
and rear axle lateral force as state variables.
1
1 

−1
 0
mV mV 


 0 
a
b  β  
 β  

0
0
−
  
  r   0  (19)
J
J
z
z
r
  
   +  Cα f V  δ
 = C V
Cα f Va
V

 F
F
αf
yf
  −
0   yf   λ f 
−
−

λf
λf
 Fyr   λ f
  
  Fyr   0 
  


 Cα rV
Cα rVb
V 
0
− 
−
λr
λr 
 λr
The outputs of interest include the vehicle’s side slip, yaw
rate, lateral acceleration, and the understeer angle. The lateral
acceleration and understeer angle are defined by:
Ay =
Fyf + Fyr
m
(20)
a
b  L

 
(21)
UA = α f − α r =  β + r − δ  −  β − r  = r − δ
V
V  V

 
The understeer angle was derived from the subtraction of
the front and rear tire slips angles which in turn are functions of
the vehicle side slip angle and location of the vehicle center of
gravity as given by Equations (3) and (4). It is interesting to
note that the derived understeer angle is related to the yaw rate
by the constant ratio of wheelbase to velocity and a shift related
to the steer input.
The above output equations as well as the side slip and yaw
rate are expressed in terms of state-space output matrices as:
1 0
 β  0 1
 r  
  
  = 0 0
 Ay  
UA 
L
 0 V
0
0
1
m
0
0
0   β   0 
   
1   r  +  0 δ
 
m   Fyf   0 
 F   −1
0   yr   

3
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(22)
Copyright © 2009 by ASME
The transfer functions of the vehicle’s lateral acceleration,
yaw rate, and understeer angle with respect to steer angle input
can be found from this state-space representation of the model.
The characteristic equation of the system in Equation (19)
is:
∆ 2 ( s ) = λ f λr J z m  s 4 + VJ z m ( λ f + λr )  s 3
+  m ( Cα f a 2 λr + Cα r b2 λ f ) + J z ( Cα f λr + Cα r λ f ) + V 2 mJ z  s 2 (23)
+  mV Cα f a ( a − λr ) + Cα r b ( b + λ f ) + VJ z ( Cα f + Cα r )  s


2
2


+ Cα f Cα r L + mV ( Cα r b − Cα f a ) 
The yaw rate, lateral acceleration, and understeer angle
transfer functions are given, respectively, by:
[ amλr ] s 2 + [ amV ] s + [Cα r L]
r
(24)
( s ) = Cα f V
∆2 ( s )
δ
Ay
δ
)
( s ) = Cα f V
[ J z λr ] s3 + [ J zV ] s 2 + [Cα r bL] s + [VCα r L]
∆2 ( s)
λf =
+  mb ( Cα f aλr − Cα r bλ f ) − J z ( Cα f λr + Cα r λ f + mV 2 )  s 2
(
)
+  mV Cα f a ( λr + b ) − Cα r b ( λ f + b ) − J zV ( Cα f + Cα r )  s


UA
δ
(s) =
+ Cα f mV
2
( a − b )
∆2 ( s)
(26)
For comparison, the understeer angle with respect to steer
angle transfer function for the model without relaxation
(without tire dynamics) is defined here by:
 − mJ zV 2  s 2
+  mVb ( Cα f a − Cα r b ) − J zV ( Cα f + Cα r ) s
UA
δ
(s) =
+  Cα f mV 2 ( a − b ) 
∆1 ( s )
(27)
Cα f
(28)
K yf
C
λr = α r
K yr
(29)
Note that the tire stiffnesses, Kyf & Kyr, are the same values
for both front and rear axles. This assumption of the
insensitivity of lateral stiffness to load as compared to cornering
stiffness (due to the dominant effect of structural tire
components [16]), yields different relaxation lengths for the
front and rear axles for both vehicles, because of its dependency
on cornering stiffness in Equations 28 & 29.
Understeering Vehicle. The understeer vehicle, as
defined in Table 1, was analyzed using the models developed
above with and without tire relaxation. The yaw rate and lateral
acceleration transfer functions, defined in Equations (7, 8, 24,
& 25), for various vehicle speeds, including 30, 60, 90, 120, &
150 kph, are shown in Figures 3&4.
It should be noted that when the relaxation length of the
front and rear tires are set to zero the model with relaxation
lengths will reduce to the vehicle model without relaxation.
With Tire Relaxation
Without Tire Relaxation
10
5
0
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
0
Phase (deg)
RESULTS & DISCUSSION
In this section, the transfer functions derived above are
used to study the transient response of the vehicle. An
understeering as well as an oversteering vehicle with different
rear tire cornering stiffnesses are analyzed. The relevant values
are given in Table 1.
Units
Oversteer
Vehicle Mass
1581
kg
Yaw Inertia
2686
kg-m2
Wheelbase
2.7
m
Weight Distribution
63/37
%
Tire Lateral Stiffness
150
N/mm
Front Tire Cornering Stiffness
1504
N/deg
Front Relaxation Length
0.575
m
Rear Tire Cornering Stiffness
1043
687
N/deg
Rear Relaxation Length
0.398
0.262
m
Understeer Gradient
4.83
-9.17
deg/G
Characteristic/Critical Velocity
200
145
kph
The relaxation lengths for the front and rear tires were
calculated based on a tire lateral stiffness for the front and rear
tires as well as the respective cornering stiffnesses.
(25)
 − mJ z λ f λr  s 4 +  − mJ zV ( λ f + λr )  s 3
Value
Understeer
Yawrate/Steer (deg/s/deg)
(
Table 1. Vehicle Parameters
Parameter
-90
Figure 3. Yaw Rate Transfer Function of Understeering Vehicle with and
without Tire Relaxation for Increasing Velocity (Arrows)
4
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Phase (deg)
0
With Tire Relaxation
Without Tire Relaxation
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
-90
Figure 4. Lateral Acceleration Transfer Function of Understeering
Vehicle with and without Tire Relaxation for Increasing Velocity (Arrows)
Model without Tire Relaxation
2
With Tire Relaxation
Without Tire Relaxation
1.5
1
Imaginary
Understeer Angle/Steer (deg/deg)
As can be seen from the figures, the magnitude and phase
delay for the model with and without tire relaxation does show
some differences. For low frequencies, below 0.5 Hz, the
models show similar response, however for higher frequencies
the phasing is dramatically different. It is difficult to interpret
these differences and their significance from the perspective of
a driver. So it is proposed to use the unconventional understeer
angle transfer function, defined in Equations (26 & 27), to
investigate the model performance (See Figure 5).
vehicle side slip angle vanishes. The rear tire slip angle also
approaches zero. This is evident from Equations (3) and (4).
The effect of tire relaxation on the understeer angle of the
vehicle is more evident at low vehicle speeds, as can be seen by
the difference in the models at the lowest speed of 30 kph. As
the velocity is increased, the understeer response of the model
including tire relaxation approaches that of the model excluding
tire relaxation.
It can also be observed that the understeer transfer function
shows a peak frequency at low velocities that could be
significant. This peak is located at 3-3.5 Hz at 30 kph, which
lies in a frequency range in which a driver maybe sensitive. It
can also be observed that the model with relaxation length has
values of understeer angle greater than one for frequencies
above 2 Hz. This indicates an increased measure of understeer
when tire relaxation is included.
The effect of velocity on tire relaxation can be further
analyzed by computing the poles of the characteristic equations
(6) and (23) for increasing velocity. The loci of the poles of the
model with and without tire relaxation are shown in Figure 6.
As expected for an understeering vehicle, the model without tire
relaxation includes only two imaginary poles [4]. However, the
model with relaxation transitions from four imaginary poles to
two imaginary and two real ones as the vehicle velocity is
increased.
0.5
0
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
Phase (deg)
90
25
20
20
15
15
10
10
5
5
0
-5
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
Figure 5. Understeer Angle Transfer Function of Understeering Vehicle
with and without Tire Relaxation for Increasing Velocity (Arrows)
It can be observed that the understeer angle approaches a
magnitude of one degree per degree of steering for high
frequencies. The low or near zero values of yaw rate at such
frequencies causes the understeer angle to approach the steering
input. This is because, at high frequencies, the understeer angle
becomes a function of primarily the front tire slip angle which
in turn approaches the steer angle as both the yaw rate and the
0
-5
-10
-10
-15
-15
-20
-20
-25
-40
0
Model with Tire Relaxation
25
Imaginary
Lat. Accel/Steer (Gs/deg)
0.8
-30
-20 -10
Real
0
10
-25
-40
-30
-20 -10
Real
0
10
Figure 6. Loci of the Poles for Understeering Vehicle with and without
Tire Relaxation for Increasing Velocity (Arrows)
The effect of relaxation length is reduced as velocity
increases as shown by the convergence of the left and right plots
to similar imaginary poles. This is consistent with previous
studies showing that the effect of tire relaxation on vehicle
handling is most important at low velocities [7].
The initial imaginary poles (near zero velocity) that are on
the imaginary axis for low speeds are quite different from the
poles of the model without tire relaxation. The characteristic
5
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4
+  m ( Cα f a 2λr + Cα r b 2λ f ) + J z ( Cα f λr + Cα r λ f )  s 2
+ Cα f Cα r L2 
(30)
It can be shown that the roots of this equation are pure
imaginary for the data in Table 1 verifying the above
observation that at low speeds, tire-relaxation introduces
dominant oscillations. The impact of the system poles can
further be seen in the vehicle’s response to a step steer
maneuver of one degree road wheel angle at increasing
velocities of 30, 60, 90, & 120 kph (Figure 7).
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
-90
0.5
Time (sec)
0.6
0.4
0.2
0
0.5
Time (sec)
1
-1
-2
1
Figure 8. Yaw rate Transfer Function of Oversteering Vehicle with and
without Tire Relaxation for Increasing Velocity (Arrows)
0
0.5
Time (sec)
1
1
W/ Relaxation
W/O Relaxation
0.5
Lat. Accel/Steer (Gs/deg)
0
0.8
0
0
0
0.8
With Tire Relaxation
Without Tire Relaxation
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
0
-0.5
0
0.5
Time (sec)
1
Figure 7. Understeering Vehicle Response to Step Steer Input for
Increasing Velocities (Arrows)
Phase (deg)
Lateral Acceleration (Gs)
Sideslip Angle (deg)
5
0
5
1
Understeer Angle (deg)
Yaw rate (deg/s/deg)
10
With Tire Relaxation
Without Tire Relaxation
10
0
Phase (deg)
∆ 2 (V → 0 ) = λ f λr J z m  s
Yawrate/Steer (deg/s/deg)
equation (23) with tire relaxation as the velocity approaches
zero reduces to:
0
-90
As can be seen from the response, the model with
relaxation exhibits more oscillation at low speeds compared to
the model without relaxation. Interestingly, this oscillation
causes the understeer angle to become negative for the speed of
30 kph, as the yaw rate overshoots its steady-state value.
Oversteering Vehicle. The oversteering vehicle, as
defined by the parameters in Table 1, was analyzed similarly.
The yaw rate and lateral acceleration transfer functions for a
few selected velocities (30, 60, 90, & 120 kph) are shown in
Figures 8 & 9, respectively.
Figure 9. Lateral Acceleration Transfer Function of Oversteering Vehicle
with and without Tire Relaxation for Increasing Velocity (Arrows)
The yaw rate and lateral acceleration response looks similar
to the previous case with the exception of steady-state
performance. The yaw rate and lateral acceleration gain
increase greatly as the velocity is increased as expected for an
oversteering vehicle [1, 4, 5].
The understeer angle transfer function for the oversteering
vehicle is shown in Figure 10. The steady-state understeer
angle is 180 degrees out-of-phase with respect to the steering
input. This is consistent with the expected negative value of
understeer angle for an oversteering vehicle.
6
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1
0.5
0
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
90
0
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
Figure 10. Understeer Angle Transfer Function of Oversteering Vehicle
with and without Tire Relaxation for Increasing Velocity (Arrows)
It is also noteworthy that the asymptotes of magnitude and
phase of the oversteering vehicle are identical to that of the
understeering vehicle in Figure 5.
Therefore, for the
oversteering and understeering vehicles the understeer
performance for increasing frequencies of excitation converge
to pure understeer as seen by an understeer angle of one degree
per degree of steering.
The loci of the poles for the oversteering vehicle over a
range of increasing velocities are shown in Figure 11.
Model with Tire Relaxation
25
25
20
20
15
15
10
10
5
5
Imaginary
Imaginary
Model without Tire Relaxation
0
-5
0
-5
-10
-10
-15
-15
-20
-20
-25
-40
-30
-20 -10
Real
0
10
-25
-40
-30
-20 -10
Real
0
10
Figure 11. Loci of Poles for Oversteering Vehicle with and without Tire
Relaxation for Increasing Velocity (Arrows)
The model without relaxation, including two negative real
poles, reaches instability when one of the poles crosses the
imaginary axis. The loci of the poles for the model with
relaxation progressively move from four imaginary poles to
four real poles for increasing velocity as shown by the arrows in
Lateral Acceleration (Gs)
Phase (deg)
180
30
Sideslip Angle (deg)
1.5
Figure 11. Accordingly, at low vehicle velocities there exist two
distinct frequencies of oscillation. Similarly to the model
without relaxation, one of these poles crosses the imaginary axis
at a critical velocity inducing instability. Note that at high
velocities, the two models converge to each other meaning that
tire relaxation does not dramatically reduce the critical velocity.
The step steer response for the oversteering vehicle,
showing the yaw rate, side slip angle, lateral acceleration, and
understeer angle is shown in Figure 12.
20
10
0
0
0.5
Time (sec)
1
1.5
Understeer Angle (deg)
With Tire Relaxation
Without Tire Relaxation
Yaw rate (deg/s/deg)
Understeer Angle/Steer (deg/deg)
2
1
0.5
0
0
0.5
Time (sec)
1
2
0
-2
-4
0
1
0.5
Time (sec)
1
W/ Relaxation
W/O Relaxation
0.5
0
-0.5
-1
0
0.5
Time (sec)
1
Figure 12. Oversteering Vehicle Response to Step Steer Input for
Increasing Velocities (Arrows)
The model with relaxation length again shows oscillation at
lower vehicle speeds, particularly on the understeer angle. The
usefulness of the understeer angle as a response is shown by
how it reveals the most dramatic differences in performance,
especially at lower vehicle speeds.
CONCLUSIONS
In this work, transient understeer performance of a vehicle
was analyzed using a frequency domain approach. The effects
of tire relaxation on transient handling performance is shown
through an analysis of an understeering and oversteering vehicle
using transfer functions and complex plane locus of the system
poles for the models. The following observations were made
from the analysis:
• Relaxation length produces larger magnitude ratios for
the understeer angle transfer function for both vehicles.
• For increasing velocity, a locus of the poles analysis
shows that the poles of the model with tire relaxation
converge to poles of the model without relaxation. In
other words, the most dramatic effects of tire
relaxation are apparent at low velocities, where highly
oscillatory poles are introduced by tire relaxation for
both oversteering and understeering vehicles.
• Relaxation length does not significantly affect the
stability or critical velocity magnitude of the studied
7
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Copyright © 2009 by ASME
oversteering vehicle, due to the reduced effect of
relaxation at such a velocity (145 kph).
• For increasing excitation frequencies, an understeering
and oversteering vehicle converge to pure understeer
as measured by a one degree understeer angle per
degree of steering.
• As expected, the steady-state performance of a model
with and without tire relaxation are identical, i.e, in
steady-state tire relaxation has no effect on the
response of the vehicle.
It is also proposed in this paper that the understeer angle
transfer function makes for an easy to interpret parameter for
such on-center analyses.
8.
9.
10.
11.
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