5.3.1 EFFECTS OF STEERING SYSTEM CHARACTERISTICS ON VEHICLE MOTION WITH FIXED STEERING ANGLE
The previous section showed the steering system could be treated as a mechanical system with two-degree-of-freedom,aandd. This section will study the effect of steering system char- acteristics on the vehicle motion by treating the steering angle,a, as fixed and not a state of motion of the mechanical system. In reality, this is when the steering wheel angle is maintained purposely at a fixed value regardless of the vehicle motion or when the steering wheel angle is set with a fixed pattern regardless of steering wheel inertia, damping force, or restoring force.
As mentioned earlier, in the previous chapters, the front wheel steering angle is given freely and fixed. When the front wheel steering angle or the steering wheel angle is fixed, this is called fixed control.
Here,ais no longer treated as a state of motion,Eqn (5.5)becomes meaningless, and the steering system motion could be expressed byEqn (5.6).
Isd2d dt2þCsdd
dtỵKsðdaị ẳ2xKf
bþlfr
V d
ð5:6ị
And,a, here, is not a variable but something known as an input. In relation toa, the front wheel steering angle,d, is known fromEqn (5.6). The vehicle motion in relation to steering angle, d, had been derived in Chapter 3.
mVdb
dtỵ2ðKfỵKrịbỵ
mVþ2
VðlfKflrKrị
rẳ2Kfd ð3:12ị
2ðlfKflrKrịbỵIdr dtþ2
l2fKfþl2rKr
V rẳ2lfKfd ð3:13ị
These cross-coupledEqns (5.6), (3.12), and (3.13) are the equations of the vehicle motion due to a steering angle,a, when taking the steering system into account.
To study the effect of static characteristics of the steering system, either the steering angle is fixed or rapid operation of the steering wheel is omitted, i.e.,d2d/dt2anddd/dtare small.IsandCs
are small too; thus,Is(d2d/dt2) andCs(dd/dt) can be neglected. The front wheel steering angle is determined by the following equation:
Ksðdaị ẳ2xKf
bþlfr
V d
(5.7) dis derived as follows:
dẳ Ks
Ksþ2xKfaþ
1 Ks
Ksþ2xKf
bþlfr
V
By taking the following,
eẳ Ks
Ksỵ2xKf ẳ 1
1þ2xKKsf (5.8)
dẳeaỵ ð1eị
bþlf
Vr
(5.9) and puttingdinto Eqns (3.12) and (3.13) gives the following:
mVdb
dtỵ2ðeKfỵKrịbỵ
mVþ2
VðlfeKflrKrị
rẳ2eKfa (5.10) 2ðlfeKflrKrịbỵIdr
dtỵ2ðl2feKfỵl2rKrị
V rẳ2lfeKfa (5.11)
Equations (5.10) and (5.11) express the vehicle motion to steering angle, a, when considering the static characteristic of the steering system. Comparing these equations with Eqns (3.12) and (3.13) shows that the front wheel cornering stiffness,Kf, is replaced byeKfin Eqns (5.10) and (5.11). In other words, the vehicle response to steering wheel angle,a, has the characteristics of the vehicle response to front wheel steering angle withKfreplaced byeKf. FromEqn (5.8), the value ofeis smaller than one, and the equivalent front wheel cornering stiffness is decreased. Larger restoring moment coefficients and smaller steering system stiffness values increase this effect. If the front wheel cornering stiffness becomes smaller, the vehicle steer characteristics change toward US, and the vehicle exhibits a larger tendency to US. It is true to say that a larger restoring moment coefficient and smaller steering system stiffness gives stronger US characteristics and better directional stability of the vehicle.
It is impractical to assume that the front wheel steering angle is possible to be fixed to a prior given value. Even if the vehicle shows some OS tendency from the theoretical study as in Chapter 3, the vehicle might actually have a US tendency when the steering system restoring moment coefficient and stiffness are considered. In other words, the vehicle’s actual US and OS steer characteristics not only depend on the tire characteristics and longitudinal positions of front and rear wheels, but they also greatly depend on the steering system stiffness. The front wheel steering angle,ad, produced by the steering system restoring moment,Ts, is called the steering system compliance steer.
Example 5.1
Calculate the effect of the compliance steer on the equivalent reduction of the cornering stiffness when the cornering stiffness of the front tire itself,Kf, pneumatic trailþcaster trail,x, and rigidity of the steering system,Ks, are equal to 80 kN/rad, 0.035 m, and 10.0 kNm/rad, respectively. Also confirm the effect of the equivalent reduction of the cornering stiffness on the steer characteristics of the vehicle withmẳ1500 kg,lfẳ1.1 m,lrẳ1.6 m, andKrẳ60.0 kN/rad.
Solution
UsingEqn (5.8), the following is obtained:
eẳ 1
1ỵ2xKKsfẳ 1 1þ2:00:03580000
10000
ẳ0:64
So, the equivalent cornering stiffness of the front wheel is 0.6480.0ẳ51.2 kN/rad.
Using Eqn (3.43), the stability factor of the vehicle with the compliance steer taken into consid- eration is as follows:
Aẳ m 2l2
lfKflrKr
KfKr ẳ 1500 22:72
1:1510001:660000
5100060000 ẳ0:00134 and, the stability factor of the vehicle with no consideration of the compliance steer is as follows:
Aẳ m 2l2
lfKflrKr
KfKr ẳ 1500 22:72
1:1800001:660000
8000060000 ẳ0:00017
As shown inExample 5.1, it is very important to understand that the compliance steer has a great effect on reducing the equivalent cornering stiffness of the front wheel and eventually the vehicle steer characteristics. A normal value of the coefficienteis around 0.5 to 0.7, which means that once the tire is put on the front axle of the vehicle, the cornering stiffness of the front wheel is 50–70% of that of the original tire itself. It should be understood throughout this book that the symbolKf, used for the cornering stiffness of the tire, includes the effect of the compliance steer unless otherwise stated.
Previously, the steering angle,a, was either fixed or prevented from rapid changes so that Is(d2d/dt2) andCs(dd/dt) inEqn (5.6)could be neglected. This allowed study of the effect of static characteristics of the steering system. Strictly speaking, if the steering wheel can be operated more quickly, this should be considered. The equations of motion for the vehicle motion to steering angle,a, can be obtained by slightly modifyingEqns (5.6), (3.12), and (3.13).
mVdb
dtỵ2ðKfỵKrịbỵ
mVþ2
VðlfKflrKrị
r2Kfdẳ0 (5.12) 2ðlfKf lrKrịbỵIdr
dtỵ2ðl2fKfỵl2rKrị
V r2lfKfdẳ0 (5.13)
2xKfb2lfxKf
V rþIsd2d dt2þCsdd
dtỵ ðKsỵ2xKfịdẳKsa (5.14) Based on these equations, it is expected that the front wheel inertia and damping friction around the kingpin will cause a delay in the response of the front wheel steering angle,d, to steering angle,a. The delay is also in the vehicle response to steering angle. The smaller the restoring moment coefficient and the stiffness of the steering system are, the greater this effect is.
Figure 5.4is an example of the response of vehicle yaw rate,r, to periodical steering angle,a, with different steering system stiffness values[1]. As expected fromEqns (5.12)–(5.14), with the decrease of the steering system stiffness, the vehicle response lag to steering angle becomes larger.
The previous discussion showed that the smaller the steering system stiffness is, the more liable the vehicle is to a US characteristic. Chapter 3 described how a vehicle with a US char- acteristic has less delay in the response to actual steering angle. However, the US vehicle with low steering system stiffness has a larger delay to steering angle, particularly when the front wheel inertia and damping friction cannot be neglected. Therefore, excessively reducing the steering system stiffness is not something that is desired in the vehicle transient response to somewhat higher speed steering input. In the case of the restoring moment coefficient, the larger the coefficient, the more likely the vehicle will show US characteristics.Equations (5.12)–(5.14)
show a larger restoring moment coefficient will not cause any delay in the vehicle response to steering angle. Therefore, the restoring moment coefficient, 2xKf, is desired to be as large as possible provided the steering force is not too heavy.
Example 5.2
Section 3.4.4 showed that it is possible to analyze the vehicle dynamics in the tire’s nonlinear region by making the equivalent cornering stiffness decrease with increasing the lateral acceleration. In this sec- tion, however, only the saturation property of the tire to side-slip angle is dealt with, and nothing is taken into consideration for the fact that the pneumatic trail decreases with increasing side-slip angle. As the pneumatic trail decreases with the side-slip angle, namely with the lateral acceleration, the aligning torque decreases, and the compliance steer is decreased, which has an effect on the loss of the cornering stiffness due to the compliance steer.
Referring to Sections 2.3.3 and 3.4.4, the pneumatic trail,x0, decreases proportionally to the side-slip angle and tends to be zero at the saturation point of the lateral force. The following equation is obtained:
x0ẳx 0
@1 Kf
mllrWbf
1
A (E5.1)
Taking the compliance steer in the steering system into account, find the equivalent cornering stiffness of the front tire at the large tire side-slip region.
Solution
Equation (3.101) suggests the following description for the pneumatic trail,x0:
x0ẳx 0
@1 Kf
mllrWbf 1 Aẳx
ffiffiffiffiffiffiffiffiffiffiffiffi 1y€ m s
(E5.2) normal steering system stiffness
reduced steering system stiffness yaw rate gain (deg/sec/deg)phase (deg)
FIGURE 5.4
Effect of steering system stiffness.
Also the cornering stiffness of the front wheel at a large side-slip angle,Kf0, is described as follows:
K0fẳKf
ffiffiffiffiffiffiffiffiffiffiffiffi 1€y m s
(E5.3)
So, referring toEqn (5.8), the cornering stiffness reduction coefficient at a large side-slip angle,e0, is the following:
e0ẳ 1
1ỵ2xK0Ks0fẳ 1 1þ2xKKsf
1€ymẳ e
1 ð1eịm€y (E5.4)
From the previous equation, the equivalent cornering stiffness at a large side-slip angle, considering the compliance steer, is as follows:
e0Kf0ẳ e 1 ð1eịm€yKf
ffiffiffiffiffiffiffiffiffiffiffiffi 1y€ m s
zeKf
1
e1
2 y€
m
(E5.5)
It is interesting inExample 5.2that the reduction rate of the cornering stiffness due to the side-slip angle, namely to the lateral acceleration, inEqn (E5.5)decreases compared with Eqns (3.101) and (3.102) or (3.103) and (3.104). This means that the equivalent cornering stiffness reduction due to the side-slip angle at the front wheel is smaller than that at the rear wheel, especially under large tire side-slip angle. It is interesting to see that though the steering system compliance steer makes the vehicle US, there is a possibility to weaken this aspect at larger side slips and make the vehicle tend to OS. This is achieved by reducing the effects of compliance steer on the effective cornering stiffness of the front tire at large side slips.