Previously, the vehicle dynamic characteristics have been studied with the assumption that the lateral force is proportional to the side-slip angles of the tires. It is important now to try to un- derstand how the vehicle dynamics are affected when the lateral force is not proportional to the side-slip angles (e.g., at large tire slip angles).
The close relation between tire’s lateral force,Y, and side-slip angles,b, has been discussed in Sections 2.3.1 and 2.4.2. For simplicity, takingKas the cornering stiffness atbẳ0 and a friction force,mW, as a saturated lateral force that can be approximated as a second-order polynomial of b, the following is obtained:
YẳKb K2
4mWb2 (3.98)
The relation is shown inFigure 3.32.
When a vehicle, with weightmgis making a circular motion with lateral acceleration€y, as in Eqn (3.48), the lateral forces acting at the front and rear wheels are as follows:
2Yfðbfị ẳlrmg
l y€ẳ2Kfbf Kf2
mlrmgl b2f (3.99)
2Yrðbrị ẳlfmg
l y€ẳ2Krbr Kr2 mlfmgl
b2r (3.100)
wherebfandbrare the front and rear wheel side-slip angles.
Using these equations, the equivalent cornering stiffness values,vYf/vbfandvYr/vbr, are the following:
vYf
vbf ẳKf 1 Kf
mlrmgl bf
!
ẳKf ffiffiffiffiffiffiffiffiffiffiffiffi 1€y m s
(3.101)
FIGURE 3.32
Approximation of tire nonlinear characteristics.
vYr
vbrẳKr 1 Kr
mlfmgl br
!
ẳKr
ffiffiffiffiffiffiffiffiffiffiffiffi 1€y m s
(3.102)
These are the gradients of the lateral forces to side-slip angle at the equilibrium point of circular motion with€y. Ify€=m1, then the following results:
vYf vbf ẳKf
1 €y
2m
(3.103) vYr
vbrẳKr
1 €y
2m
(3.104) The cornering stiffness of the vehicle during circular motion decreases with the lateral acceleration when the lateral acceleration approaches the limit or the friction coefficient between the road and tire decreases abruptly. In the region where€yis small compared tom, the cornering stiffness could be treated as decreasing linearly. The previous condition is shown inFigure 3.33.
Next, the characteristics of the vehicle motion in the region where the tire exhibits its nonlinear characteristics will be looked at. Consider the very small motion of the vehicle in response to a very small steering input from the initial condition of circular motion with the lateral accelerationy. Equations of motion at that time are expressed as follows:€
m
gy€þV db
dtþr
ẳ2Yf
bfþdblfr V
þ2Yr
brbþlrr V
(3.105) Idr
dtẳ2lfYf
bfþdblfr V
2lrYr
brbþlrr V
(3.106) Becaused,b, andrare very small, the following results:
Yf
bfþdblfr V
yYfðbfị ỵvYf vbf
dblfr V
FIGURE 3.33
Change of equivalent cornering stiffness due to lateral acceleration.
Yr
brbþlrr V
yYrðbrị ỵvYr
vbr
bþlrr V
And, from the equilibrium conditions, we have the following:
mgy€ẳ2Yfðbfị ỵ2Yrðbrị 2lfYfðbfị 2lrYrðbrị ẳ0
Substituting these equations intoEqns (3.105) and (3.106)and rearranging them gives the following final equations:
mVdb dtþ2
vYf
vbfþvYr
vbr
bþ 8>
><
>>
: mVþ
2
lfvYvbf
flrvYvbr
r
V
9>
>=
>>
; rẳ2vYf
vbfd (3.107)
2
lfvYf
vbflrvYr
vbr
bþIdr dtþ
2
l2fvYvbf
fþl2rvYvbr
r
V rẳ2lfvYf
vbfd (3.108)
These are the linearized equations of motion for the region where tire characteristics are nonlinear, based on the theory of small perturbation. In the region where tire characteristics are nonlinear, the tire cornering stiffness values ofKfand Krare now replaced by the equivalent cornering stiffness ofvYf/vbfandvYr/vbrinEqns (3.101) and (3.102)orEqns (3.103) and (3.104).
Here, when€y=m1, expressing the equivalent cornering stiffness as the following:
vYf
vbfẳKfẳKf
1 €y 2m
vYr
vbrẳKrẳKr
1 €y 2m
then several parameters that show the vehicle dynamic characteristics are obtained through the following equations. First of all, the stability factor is shown:
Aẳ m 2l2
lrKrlfKf KfKr ẳ m
2l2
lrKrlfKf KfKr
1þ €y
2m
ẳA
1þ y€ 2m
(3.109) And, the natural frequency now becomes the following:
unẳ2 ffiffiffiffiffiffiffiffiffiffiffiffi KfKr q
l mk
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þAV2 p
V
ẳ2 ffiffiffiffiffiffiffiffiffiffi KfKr
p l
mk
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þAV2 p
V
1
1þ 1
1þAV2 y€
4m
(3.110)
ẳun
1
1þ 1
1þAV2 y€
4m
Furthermore, the approximated response time of yaw rate given byEqn (3.85)is as follows:
teẳ 1 u2nTr
ẳ 1 u2nTr
1þ 1
1þAV2
€ y 2m
(3.111)
ẳte
1þ 1
1þAV2
€ y 2m
These show the change of vehicle dynamic characteristics with respect to lateral acceleration,
€
y, in the region where the tire characteristic is nonlinear.
Figure 3.34is an example of the vehicle yaw rate and lateral acceleration frequency response.
It shows the vehicle dynamic characteristics during circular motion at different lateral acceler- ation values. This gives an idea of the effect of tire nonlinear characteristics. The dynamic characteristics change substantially with the lateral acceleration due to the saturation property of tire characteristic to side-slip angle.
Freqency (Hz)
Gain (1/s)
Freqency (Hz)
Gain (m/s 2/deg) G4.0:
G 6 . 0 :
G y:0.0
:
y: y: y:
G 2 . 0 :
V=120(km/h) V=120(km/h)
Lateral acceleration Yaw rate
Freqency (Hz)
)ged( elgna esahP
Freqency (Hz)
)ged( elgna esahP
( ) ( )( ) ( )
FIGURE 3.34
Effect of tire nonlinear characteristics on frequency response.
PROBLEMS
3.1 Referring toFigure 3.4(b), confirm that it is acceptable to regard the side-slip angles of right and left wheels as almost identical, and by using the bicycle vehicle model that it is reasonable when the vehicle speed is higher than 40 km/h, the yaw rate is less than 0.1 rad/s and the vehicle track is 1.4 m.
3.2 DeriveEqns (3.29) and (3.30)fromEqns (3.26)–(3.28).
3.3 Geometrically, show that the side-slip angle during steady-state cornering at low speed is described by the third equation inEqn (3.33).
3.4 Give the geometric proof ofEqn (3.34).
3.5 UsingEqn (3.39), find the vehicle speed at which steady-state yaw rate reaches the peak value when the vehicle is understeer. This speed is called characteristic speed. Show that the peak value is half of the yaw rate value of the neutralsteer vehicle at the characteristic speed.
3.6 Find the vehicle speed at which the steady-state side-slip angle is equal to zero using Eqn (3.40), and calculate the value undermẳ1500 kg, lfẳ1.1 m,lrẳ1.6 m, and Krẳ60 kN/rad.
3.7 Calculate the stability factor usingEqn (3.43)undermẳ1500 kg,lfẳ1.1 m,lrẳ1.6 m, Kfẳ55 kN/rad, andKrẳ60 kN/rad.
3.8 Calculate the static margin usingEqn (3.45)for the same vehicle parameters as used in Problem 3.7.
3.9 Calculate the critical vehicle speed for the OS vehicle with the parameters used in Example 3.7.
3.10 Confirm that for the vehicle with a static margin, SM, of almost zero, the inverse of the vehicle natural frequency, 1/un, is nearly equal to the vehicle response time expressed by Eqn (3.74).
3.11 UsingEqn (3.110), estimate what percent of the vehicle’s natural frequency is reduced due to circular turning with the lateral acceleration, 2.0 m/s2, on a dry road surface,mẳ1.0.
3.12 Execute the vehicle response simulation to a single, 0.5 Hz, sine-wave steering input with an amplitude of 0.04 rad at vehicle speeds of 60, 100, and 140 km/h, respectively, using the Matlab-Simulink simulation software. Use the same vehicle parameters as in Example 3.6.
3.13 Find the steady-state side-slip angle caused by disturbance yaw rate,DrC, at a steering angle equal to zero by usingEqn (3.24).
3.14 UsingEqn (3.25), find the steady-state yaw rate,DrR, caused by the restoring yaw moment that is produced by the side-slip angle calculated in Problem 3.13; assume the steering angle is equal to zero.
3.15 From Problem 3.14, it is possible to obtain the ratioDrR/DrC. A ratio larger than 1.0 means that the result is larger than the cause, and the result causes larger next results, and so on. The vehicle, eventually, becomes unstable. Find the vehicle speed that satisfies DrR/DrCẳ1.0, and confirm that the speed found is identical to the critical vehicle speed obtained byEqn (3.42).
REFERENCES
[1] Whitcomb DW, Milliken WF. Design implication of a general theory of automotive stability and control. Proc IMechE (AD) 1956. 367–91.
[2] Ellis JR. Vehicle Dynamics. London: London Business Book Ltd; 1969.