3.3 VEHICLE STEADY-STATE CORNERING
3.3.1 DESCRIPTION OF STEADY-STATE CORNERING
The equations of motion derived inSection 3.2using vehicle fixed coordinates will be used to describe steady-state cornering. During steady-state cornering, there will be no changes in the side-slip angle and the yaw velocity. Here the steady-state conditions,db/dtẳ0 anddr/dtẳ0, can be substituted intoEqns (3.12) and (3.13), giving the following:
2ðKfỵKrịbỵ
mVþ2
VðlfKflrKrị
rẳ2Kfd (3.24)
FIGURE E3.2
2ðlfKflrKrịbỵ2 l2fKfỵl2rKr
V rẳ2lfKfd (3.25)
The solution forbandris shown next:
bẳ
2Kf mVþ2
VðlfKflrKrị 2lfKf
2 l2fKfþl2rKr
V
d
D (3.26)
rẳ
2ðKfỵKrị 2Kf 2ðlfKflrKrị 2lfKf
d
D (3.27)
whereby,
Dẳ
2ðKfỵKrị mVỵ2
VðlfKflrKrị 2ðlfKflrKrị 2 l2fKfỵl2rKr
V
(3.28)
ExpandingEqns (3.26)–(3.28)and rearranging them, givesbandras follows:
bẳ 0
@ 12lmllf
rKrV2 12lm2
lfKflrKr
KfKr V2 1 A lr
ld (3.29)
rẳ 0
@ 1
12lm2
lfKflrKr
KfKr V2 1
A V
ld (3.30)
If the vehicle is traveling with a constant speed,V, and the yaw velocity isr, the radius of the steady-state cornering,r, is formulated as follows:
rẳV r ẳ
1 m
2l2
lfKflrKr
KfKr V2
l
d (3.31)
Equations (3.29)–(3.31) describe the vehicle’s steady-state cornering with steering angle,d, and constant traveling speed,V. They show physically how the side-slip angle,b, yaw velocity, r, and circular radius, r, respond to steering angle at different traveling speeds.
Assuming that the vehicle is traveling at a very low speed (Vz0), thenV2can be neglected inEqns (3.29)–(3.31).b,r, andrcan now be described as follows:
bðVz0ịẳbsẳlr
ld rðVz0ịẳrsẳV
ld rðVz0ịẳrsẳl
d
(3.32)
Equations (3.29)–(3.31)can express the vehicle steady-state cornering as follows:
b
bsẳ 1m2lllf
rKrV2 12lm2
lfKflrKr
KfKr V2 (3.29)0
r rs
ẳ 1
12lm2
lfKflrKr
KfKr V2 (3.30)0
r
rsẳ1 m 2l2
lfKflrKr
KfKr V2 (3.31)0
The previous equations show how the conditions for vehicle steady-state cornering change with the traveling speed, V. The state of vehicle circular motion at very low speeds,Vz0, can be used as the reference.
3.3.1.2 Description by geometry
The study of vehicle steady-state cornering gave the response ofb,r, andrto steering angle,d, by simply putting steady-state conditions into the vehicle equations of motion derived fromSection 3.2. Further derivations of theseEqns (3.29)–(3.31)or(3.29)0–(3.31)0show how the steady state cornerings change with the traveling speed,V.
Next, the vehicle steady-state cornering will be studied geometrically to understand the vehicle motion more intuitively or in a more direct sense. Using geometry, though the steering angle and yaw velocity are positive in the anticlockwise direction in the equations of motions, they are taken as positive in the direction shown byFigure 3.7(a).
First, let us consider the steady-state cornering of the vehicle at very low speeds,Vz0. In this circumstance, a centrifugal force does not act on the vehicle, lateral forces are not needed, and no side-slip angle is produced as both front and rear wheels travel in the heading direction of the wheels, respectively, and make a circular motion aroundOs, as shown inFigure 3.7(a). From this figure, the geometric relations are formulated as follows:
rsẳl d rsẳV
rsẳV ld bsẳ lr
rsẳlr
ld
(3.33)
whereby 0<d1 andlr. As seen fromFigure 3.7(a), the actual steering angle for the left and right front wheels is notd, but it is a little smaller for the left wheel and a little larger for the right wheel. In practice, this is achieved by a steering link mechanism, but ifd1 andrs[d, then the difference is very small, and the left and right wheels can be considered as having the same steering angle,d.Equation (3.33)agrees with the steady-state cornering ofEqn (3.32)that is obtained from the equations of motion with Vz0. This geometrical relation is called the Ackermann steering geometry, anddẳl/rsis called the Ackermann angle.
When the circular motion of the vehicle is considered at larger speeds, the centrifugal force becomes significant. The cornering forces at the front and rear wheels are needed to
balance this centrifugal force, and the side-slip angles are produced. When the centrifugal force acts at the vehicle center of gravity, the circular motion condition shown inFigure 3.7(a) is no more accurate, andEqn (3.33)is not accurate either. Figure 3.7(b)shows the circular motion when the front and rear wheel side-slip angles,bfandbr, are produced by the cen- trifugal force.
The center of the circular motion is the intersecting point of the two straight lines perpen- dicular to the front and rear wheel’s traveling direction, denoted as O. The geometric relations are formulated as follows:
rẳ l
dbfþbr (3.34)
assuming that 0<d1, 0<bf,br1, andrs[l,d.
FIGURE 3.7
(a) Steady-state turning at low speed. (b) Steady-state turning with centrifugal force.
Here,rẳV/r. So, the following is formulated:
rẳVðdbfỵbrị
l (3.35)
Furthermore, fromFigure 3.8, the following is derived:
bỵbrẳlr r Hence,
bẳlr
rbrẳlr
ldlrbfþlfbr
l (3.36)
Equations (3.34)–(3.36)express the vehicle circular motion and are derived from the steady-state cornering geometric relation. The front and rear wheel side-slip angles, bf and br, can be found from the magnitude of the centrifugal force acting at the vehicle center of gravity. The centrifugal force is dependent on the vehicle speed,V. So,r,r, andbinEqns (3.34)–(3.36)also change withV.
The steady-state cornering conditions with traveling speed have been determined by derivingEqns (3.29)–(3.31). From the previous discussion, it is known that this occurs because the centrifugal force changes with speed. This causes the front and rear side-slip angles to change; which, in turn, changes the circular motion geometry and conditions of steady-state cornering.
Equations (3.34) and (3.36), which express the vehicle steady-state cornering, are derived from a geometric relation. They are not influenced by the relationship between lateral forces and side-slip angles,bfandbr, or any lateral forces acting on the front and rear wheels. As long as jdj 1 and r[l, d, the equations are valid under any conditions. Contrary to this, it is important to note thatEqns (3.29)–(3.31) and (3.29)0–(3.31)0are only valid when the lateral force acting on the front and rear wheels is the lateral force that is proportional to the side-slip anglesbfandbr.
FIGURE 3.8
Side-slip angle during steady-state cornering.
It is possible to introduceEqns (3.29)–(3.31)from the geometric descriptions of the steady- state turning. The centrifugal force is expressed bymV2/rwhile the lateral forces exerted on the front and rear tires are proportional to the side-slip angle and expressed by 2Kfbf and2Krbr, respectively. If the vehicle is in steady-state turning, the following equilibrium equations arise:
mV2
r2Kfbf2Krbrẳ0 2lfKfbfỵ2lrKrbrẳ0 From the previous two equations, the side-slip angles are obtained:
bf ẳmV2lr
2lKf
1
r; brẳmV2lf 2lKr
1 r
PuttingbfandbrintoEqns (3.34)–(3.36)givesEqns (3.29), (3.30), and (3.31), respectively.