3.4.1 VEHICLE TRANSIENT RESPONSE TO STEERING INPUT [1]
3.4.1.1 Transient response and directional stability
The basic equations that describe the vehicle motion are defined byEqns (3.12) and (3.13)in Section 3.2as follows:
mVdb
dtỵ2ðKfỵKrịbỵ
mVþ2
VðlfKflrKrị
rẳ2Kfd (3.12)
2ðlfKflrKrịbỵIdr
dtþ2 l2fKfþl2rKr
V rẳ2lfKfd (3.13)
Once the equations of motion of a dynamic system, such as (3.12) and (3.13), are given, the vehicle response todcan be obtained by solving the equations of motion under suitable conditions. If the system is linear, the transient behavior of the dynamic system can be understood by solving the equations of motion directly or investigating the eigenvalues of the characteristic equation.
The characteristic equation of the dynamic system that is our subject of study is given by Eqn (3.14):
s2þ 2
42m l2fKf þl2rKr
ỵ2IðKfỵKrị mIV
3 5sþ
4KfKrl2
mIV2 2ðlfKflrKrị I
ẳ0 (3.54) Or, it can be written as follows:
s2ỵ2DsỵP2ẳ0 (3.55)
where
2Dẳ2m l2fKfỵl2rKr
ỵ2IðKfỵKrị
mIV (3.56)
P2ẳ4KfKrl2
mIV2 2ðlfKflrKrị
I (3.57)
And, the vehicle yaw inertia moment could be written as follows:
Iẳmk2 (3.58)
Here,kis called the vehicle yaw moment radius. SubstitutingEqn (3.58)intoEqn (3.56), 2D could be written as the following:
2Dẳ 2 mV
ðKfỵKrị
1þk2 lflr
k2=lflr
þ 1
k2ðlflrịðlfKflrKrị
(3.56)0 And, iflfzlrandKfzKr, then the following is derived:
2Dẳ2ðKfỵKrị mV
1þk2 lflr k2=lflr
(3.56)00 By substitutingEqn (3.58)intoEqn (3.57),P2could be written as the following:
P2ẳ4KfKrl2 m2k2V2
1 m
2l2
lfKflrKr
KfKr V2
(3.57)0 Now, the response of the system with the characteristic equation given byEqn (3.55)is expressed byC1el1tþC2el2twithl1andl2as the roots of the characteristic equation:
l1;2ẳ D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2P2 p
(3.59) The transient response characteristics and stability of the system is dependent on whetherl1 andl2are real numbers or complex numbers and on the sign ofl1andl2if they are integers or the sign of the real part ofl1andl2if they are complex numbers. Based onEqn (3.59), the value ofl1andl2is dependent onDandP. FromEqn (3.56), it is apparent that D>0, and the transient response characteristic and motion stability of the vehicle can be classified into the following categories based onDandP:
1. WhenD2P20 andP2>0,l1andl2are negative real numbers, and motion converges without oscillation (stable).
2. WhenD2P2<0,l1andl2are complex numbers with the negative real part, and motion converges with oscillation (stable).
3. WhenP20,l1andl2are positive and negative real numbers, and motion diverges without oscillation (unstable).
This is under the premise that the steering angle of the vehicle is predetermined and not changeable in response to the vehicle’s behavior. It should be noted that
the vehicle does not always show this behavior, and the driver plays a key role in the vehicle stability. This situation not only applies to the vehicle motion, but for ships and aircrafts alike. The control of the motion by the driver (in some cases, by control actuators) on board of the moving body itself is an important matter in studying the motion of the moving bodies (refer to Chapters 8–10 for a detailed approach).
Next, the response of the vehicle to steering input, which can be divided into three catego- ries as previously described, is investigated. Also, the motion stability will be studied in more detail, and the type of vehicle and of situation that gives motion for cases 1, 2, and 3 is investigated.
First, case 3 is examined. FromEqn (3.57), the first term ofP2is always positive; thus, for P20, it is only the second term that can be negative, in other words,lfKflrKr>0. TakingVc
as the velocity whereP2ẳ0, the following is obtained fromEqn (3.57)0: 1 m
2l2
lfKflrKr KfKr
Vc2ẳ0 (3.60)
hence,
Vcẳ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KfKr
mðlfKflrKrị s
lẳ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2lKfKr
mðKfỵKrị
1 SM s
(3.61) Then, for all velocities greater thanVc,P20. This is the condition for static instability of the mechanical system as described previously inSection 3.3.2.
When the vehicle shows OS characteristics, the vehicle’s lateral motion becomes unstable at velocityVc, and it diverges without oscillation in response to a fixed steering input. As can be seen clearly fromEqn (3.61), this stability limit is greatly dependent on the SM.Figure 3.24 shows the vehicle stability limit by SM and velocity; and, it shows that the smaller the absolute value of SM and the larger the total cornering stiffness of the front and rear wheels is, the larger the stability limit velocity.
30 40 50 60 70 80 90 100
-0.4 -0.3 -0.2 -0.1 0
V (km/h)
SM (-)
2(Kf+Kr )= 138 (kN/rad) 2(Kf+Kr )= 184 (kN/rad) 2(Kf+Kr )= 230 (kN/rad)
FIGURE 3.24
Relation of critical vehicle speed to SM.
Example 3.5
It is obvious that the stability condition of the vehicle is described by the following:
1m 2l2
lfKflrKr
KfKr
V20
Show that there are upper and lower limits of the front and rear cornering stiffness,KfandKr, respectively, for the vehicle to be stable, and draw a schematic diagram of the limits with respect to the vehicle speed.
Solution
The previous inequality is rewritten as follows:
2l2 mV2
lf
Kr
þlr
Kf
0
It turns out to be a form of upper limit of the front cornering stiffness:
KflrKr
lf
V2 V22lml2Kr
f
(E3.7) And, as the lower limit of the rear cornering stiffness, the following inequality is obtained:
KrlfKf
lr
V2
V2þ2lml2Krf (E3.8)
The schematic diagram of the upper and lower limits of the cornering stiffness with respect to vehicle speed is shown inFigure E3.5.
Stable
Stable Unstable
Unstable
V2 V2
Kf Kr
r f
r K
l l
f r
f K
l l
0 0
f r
ml K l2 2
FIGURE E3.5
Next, iflfKflrKr<0 when the vehicle shows US characteristics or if the vehicle reveals OS characteristic butV<Vc, thenP2is always greater than zero, and the vehicle motion is stable.
This corresponds to cases 1 and 2. Now, usingEqns (3.56) and (3.57),D2P2is calculated as follows:
D2P2ẳ2ðlfKf lrKrị
I þ
2 4 8<
:
m l2fKfþl2rKr
ỵIðKf ỵKrị
mI
9=
;
2
4KfKrl2 mI
3
5 1
V2 (3.62) Transforming the coefficient of 1/V2gives the following:
8<
:
m l2fKfþl2rKr
IðKfỵKrị
mI
9=
;
2
ỵ4ðlfKflrKrị2
mI >0 (3.63)
Because the coefficient of 1/V2ofEqn (3.62)is always positive, if the first termlKflrKr
is positive or zero, then,D2P2is also positive or zero. Hence, if the vehicle steer char- acteristic is OS or NS, the vehicle transient steering response will always be without oscil- lation, stable or not. WhenlKflrKris negative, the value ofD2P2is dependent onV, where above a certain value,D2P2changes from positive to negative. In other words, when the vehicle shows US characteristics, the vehicle transient steering response is without oscillation at a vehicle speed lower than a certain value, but beyond that the response becomes oscillatory.
UsingEqns (3.45) and (3.58)and assuming thatlfzlrandKfzKr, thenD2P2can be calculated by transformingEqn (3.62):
D2P2ẳ2ðlfKflrKrị mk2 þ
"
ðKfỵKrị2 m2
1k2 lflr k2=lflr
2
ỵ4ðlfKflrKrị2 m2k2
# 1 V2
zðKfỵKrị2 m2
"
lflr
k2 8m
lðKfỵKrịSMỵ
(1k2 lflr
k2=lflr
2
þ16lflr
k2 SM2 )
1 V2
#
(3.62)0 From this equation, if SM>0, i.e., the vehicle has a US characteristic, and the velocity,Vs, where vehicle transient steering response becomes oscillatory is as follows:
Vsẳ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2lðKfỵKrị
m (
1 16
ð1k2=lflrị2 k2=lflr
1 SMþSM v )
uu
t (3.64)
In this equation,Vsis affected byk2/lflr. FromEqn (3.64), whenk2/lflrẳ1,Vsis minimum and ifk2/lflris greater or smaller than this,Vsalways becomes larger. It is interesting to see that when the vehicle yaw moment inertia is larger or smaller than a certain value, the speed where vehicle transient steering response becomes oscillatory always becomes larger. Also, an inter- esting thing is thatVsbecomes a minimum at the following:
SMẳ1k2 lflr 4 ffiffiffiffiffiffiffiffiffiffiffiffiffi
k2=lflr
p
This analysis has shown that the characteristics of a vehicle’s transient response to steering are particularly affected by the vehicle traveling speed and steer characteristics. This is shown inTable 3.2. The stability problem of the vehicle motion, as shown in the table, is called the vehicle directional stability. The image of the vehicle response to a pulse steering input is shown in Figure 3.25. The motion response characteristic corresponding to 1, 2 and 3 in Table 3.2is clear.