3.3 VEHICLE STEADY-STATE CORNERING
3.3.3 STEADY-STATE CORNERING AND TIRE NONLINEAR
Section 3.3.2studied the characteristics of vehicle steady-state cornering to give a basic un- derstanding of the vehicle dynamics. It also looked at the vehicle steer characteristics (US, NS, and OS) as an important concept for describing a basic vehicle’s dynamic characteristics. Pre- viously, it was assumed that the lateral forces acting on the front and rear wheels are the lateral forces that are proportional to the side-slip angles. However, as seen in Chapter 2, the lateral force may not necessarily be proportional to the side-slip angle. In the case of larger side-slip angles and under certain other conditions, lateral forces may show nonlinearity in their characteristics.
This subsection will examine the effect of nonlinear tire characteristics on the vehicle steady- state cornering characteristics.
Asmglr/landmglf/lare the vertical loads of the front and rear wheels, taking the ratio of the lateral force obtained byEqn (3.49)to the vertical load at the front and rear wheels asmfandmr, respectively, the following are obtained:
mf ẳ j2Yfj
mglr=lẳ€y; mrẳ j2Yrj
mglf=lẳy€ (3.52)
This means, during steady-state cornering, the lateral force at the front and rear wheels divided by their respective vertical loads is always equal to the lateral acceleration of the center of gravity,€y.
Assuming the lateral force produced by the tire side-slip angles is the only lateral force, then mfandmrdepend only onbfandbr. If the tire cornering characteristics of the front and rear wheels (bfmfandbrmr) are given and the lateral acceleration of the vehicle center of gravity,€y, is known, the front and rear wheel side-slip angles,bfandbr, at that instant can be found.
When the tire cornering characteristic is not linear, the front and rear tire cornering char- acteristics,bfmfandbrmr, are given, for example, as inFigure 3.18. As seen previously, if the vehicle is in steady-state cornering, the lateral acceleration€yis equal tomfandmr. Thus, by knowing the lateral accelerationy, the front and rear wheel side-slip angles,€ bfandbr, at that instant are known, andbfbrcan be determined.
For vehicle steady-state cornering, regardless of whether lateral force acting at the front and rear wheels is proportional to the side-slip angle or not, the geometric relation ofEqn (3.46)is always satisfied. FromEqn (3.46), if the constant radius of the circular motion isr0, then the following results:
dẳ l
r0þbfbr (3.46)0
Table 3.1 Steer Characteristics and Steady-State Turning
A[ Lm 2l2
lfKfLlrKr
KfKr
Relative Position of NSP and C.G.
Effects of Vehicle Speed on Steady-State Turning
r (d[Constant)
d
(r[Constant) Yaw Rater
Side- Slip
Angles Critical Speed
US >0 Increase with
speed
Increase with speed
Increase with speed to some extent and decrease
bf>br Non-critical
NS ẳ0 Constantrẳdl Constantdẳrl Proportional
increase with speed
bfẳbr Non-critical
OS <0 Decrease with
speed
Decrease with speed
Rapid increase with speed
bf<br
Vcẳ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2K
fKrl2 mðlfKflrKrị
q
CHAPTER3FUNDAMENTALSOFVEHICLEDYNAMICS
By knowing€yand determiningbfbrfrom Figure 3.18, the relation between lateral ac- celeration,€y, and steering angle,d, can be plotted usingEqn (3.46)0for constant radius circular motion, as inFigure 3.19.
Ifmf<mrwith regard to tire side-slip angle, the vehicle shows US characteristics because bf>br. Ifmfẳmr, thenbfẳbr, and the vehicle characteristic is, therefore, NS. Whereas, if mf>mr, thenbf<br, and the vehicle characteristic becomes OS.
For the case when the vehicle characteristic is US, as inFigure 3.18, if€yis small and the side- slip angles are small, thenbfbris positive, and its value increases almost proportionally to€y.
Here,y€could also be assumed to be proportional tod. When€yis greater than a certain value, bfbr is no longer proportional toy, and it increases rapidly with€ y. Consequently,€ d also
linear tire characteristics
y
side-slip angle
normalized lateral force
. .
FIGURE 3.18
Nonlinear tire characteristics of front and rear.
lateral acceleration
stable unstable
for linear tire characteristics
steering angle
yC
. . FIGURE 3.19
Steering angle to lateral acceleration under constant turning radius.
increases rapidly with€y, and the vehicle shows a strong US characteristic. Whereas, for the case when the vehicle characteristic is OS, when€yis small, thenbfbris negative, and the value increases almost proportionally toy. Thus,€ dalso decreases proportionally toy. When€ y€is greater than a certain value, bfbr increases rapidly with y, and the vehicle shows a strong OS€ characteristic.
For the case of OS,ddecreases rapidly with large€y. At€yẳy€c,dẳ0. This point defines the limit where under steady-state cornering, with constant radiusr0, it is not possible to increase the lateral acceleration by raising the velocity. This€yis considerably smaller than the case where the tire cornering characteristic is linear. The tire nonlinear cornering characteristic further re- duces the critical velocity for the OS vehicle. Chapter 2 showed that, in practice, a lateral force saturates after a certain side-slip angle. It is important to note that the vehicle with OS charac- teristics becomes unstable at lower speeds than that with a linear characteristic. This velocity limit for circular motion is dependent on the radius of the circular motion, which is different than that for a linear tire characteristic.
Consider the case when the vehicle is making a circular motion with a constant speed of VẳV0. Here,r0ẳV02=g€y, and the relation between lateral acceleration,€y, and required steering angle,d, fromEqn (3.46), is as follows:
dẳ l
V02gy€þbfbr (3.53)
The value ofbfbrcan again be determined fromFigure 3.18by knowingy. Using€ Eqn (3.53), the relationship between steering angle, d, and lateral acceleration, €y, during constant-speed circular motion is shown inFigure 3.20. This figure shows that a US vehicle with€ysmall has dincreasing almost proportionally toy. After€ y€reaches a certain value,dincreases rapidly withy,€ and the vehicle reveals a strong US characteristic.
Whereas, for the case of OS, when€yis small,dincreases almost proportionally toy, but after€
€
yreaches a certain value, the increase ofdwith€ybecomes weaker. Finally, when€yẳ€yc,dreaches its peak. Wheny€is greater than€yc,ddecreases. This reduction ofddecreases the turning radius at a constant speed. This means that when the vehicle is about to move to the right with regard to its
lateral acceleration unstable
for linear tyre characteristics
steering angle
stable
y. .C
FIGURE 3.20
Steering angle to lateral acceleration under constant vehicle speed.
current traveling direction, the steering should be turned to the left. This is impossible and has no physical meaning in practice. This point shows that circular motion with a radius that causes
€
yẳy€cand the motion with smaller radius is not possible. For an OS vehicle, there is always a lower limit to the cornering radius possible, regardless of vehicle speed. Furthermore, y€c at constant turning radius, as shown inFigure 3.19, has the same meaning asy€cfor the constant speed shown inFigure 3.20.
The front and rear wheel tire cornering characteristics shown inFigure 3.18are always either mf>mrormf<mrfor the whole range of side-slip angles. This is not always the case and depends on circumstances.
Consider the front and rear wheel tire cornering characteristicsbfmfandbrmr, as shown inFigure 3.21. (A) is the case where ifbfandbrare smaller thanbp, thenmf<mr, and ifbfandbr
are larger thanbp, thenmf>mr. In contrast, (B) is the case where ifbfandbrare smaller thanbp, thenmf>mr, and ifbfandbrare larger thanbp, thenmf<mr.
If the vehicle is making a circular motion of constant radius,r0, Eqn (3.46)0is formed as previously. By knowing the lateral acceleration,y,€ bfbrcan be determined fromFigure 3.21.
UsingEqn (3.46)0, the relationship between steering angle,d, and lateral acceleration,€y, during constant radius circular motion is obtained as shown inFigure 3.22.
In the case of (A), when€yis small,bfbrincreases with€y;dalso increases, and the vehicle shows a US characteristic. After€yreaches a certain value, with the increase ofy,€ bfbrde- creases, and at€yẳ€yp, the value becomes zero. As€y>€yp,bfbrdecreases rapidly with€y, and the vehicle reveals an OS characteristic. This tendency continues to increase with€y, and at€yẳ€yc, dẳ0. This point, as inFigure 3.19, shows that it is impossible to have circular motion with higher lateral acceleration with radiusr0.
In the case of (B), wheny€is small,ddecreases with€y, and the vehicle shows OS charac- teristic. Aftery€reaches a certain value,dincreases withy, and at€ €yẳy€p, the value returns tol/r0. At€y>€yp,dincreases rapidly, and the vehicle reveals a strong US characteristic.
If the vehicle is making a circular motion with a constant speed ofVẳV0,Eqn (3.53)is formed. Therefore, by knowing the lateral acceleration €y, bfbr can be determined from
linear tire characteristics
side-slip angle
normalized lateral force
yP
. .
y. .
FIGURE 3.21
Nonlinear tire characteristics of front and rear.
Figure 3.21and usingEqn (3.53), the relationship between steering angle,d, and lateral ac- celeration,y, during constant radius circular motion is shown in€ Figure 3.23.
In the case of (A), at smally, the vehicle shows a US characteristic, but at large€ €y, the vehicle characteristic changes to OS. Circular motion with a radius that causes€yS€ycbecomes impos- sible. (B) is opposite to (A), whereas€ybecomes larger, and the vehicle shows a US characteristic.
As seen in case (A), the vehicle has a US characteristic at small lateral acceleration,€y, but when€ygets large, the steer characteristic changes to OS due to the nonlinear tire characteristic, and circular motion becomes impossible after a certain velocity. While there is no critical velocity based on the concept of linear tire, the stable vehicle might fall into a statically unstable condition during circular motion at larger lateral accelerations. As shown byFigure 3.22andFigure 3.23, the change of steer characteristic due to€yis called the reverse-steer.
linear tire characteristics
lateral acceleration
steering angle
P C
y. . y. .
stable unstable
FIGURE 3.22
Steering angle to lateral acceleration under constant turning radius.
linear tire characteristics
lateral acceleration
steering angle
unstable stable
C
yP
. . y. .
FIGURE 3.23
Steering angle to lateral acceleration under constant vehicle speed.
Figures 3.22 and 3.23also show that in case (B), when the lateral acceleration gets larger, the vehicle exhibits a strong US characteristic, and no matter how large the steering angle is, circular motion with lateral acceleration beyond a certain point is impossible. This condition is called vehicle plow. When the front and rear wheels reach the upper limit of the lateral force at the same time, it is called vehicle drift. In the case of (A), the condition where the vehicle shows a strong OS characteristic with the rear wheels reaching the limit earlier than the front wheels and the vehicle becomes statically unstable is called vehicle spin.