The vehicle has basically only lateral and yaw dynamics (often simply called lateral dynamics), described by the following differential equations (cf. (3.64))
may=Y=Y1+Y2
Jzr˙=N=Y1a1−Y2a2
(6.4) while
max=mvr=X=X1+X2−1
2ρSCxu2 (6.5)
is now an algebraic equation, the unknown being(X1+X2).
2The left and right wheels of the same axle are normally equipped with the same kind of brake.
Therefore, the braking torque is pretty much the same under ordinary operating conditions, and, again, (6.1) holds true. However, there are important exceptions. The left and right braking forces can be different if: (a) the grip is different and one wheel is locked; (b) the friction coefficients inside the two brakes is different (for instance, because of different temperatures, which is often the case in racing cars); (c) some electronic stability system, like ESP or ABS, has been activated.
6.2 Fundamental Equations of Vehicle Handling 133 With an open differential, it is easy to solve (6.4) with respect to the front and rear lateral forces
Y1=ma2
l ay+Jz
l r˙ma2
l ay Y2=ma1
l ay−Jz
l r˙ma1
l ay
(6.6)
where we took into account that|Jzr˙| |mayai|, since in a carJz< ma1a2 and
|˙rai| |ay|. In a two-axle vehicle with open differential the lateral forces are linear functions of the lateral accelerationay. This is a very peculiar and important result, which greatly impacts on the whole vehicle model, as will be shown.
According to (3.114) and (3.115), the lateral load transfers are linear functions ofY1andY2. Employing (6.6) we obtain the following simplified equations for load transfers in vehicles with open differential
ΔZ1maykφ1kφ2
t1kφ
h−q kφ2 +a2q1
lksφ
1
+a2q1
lkφs
2
+a2q1+a1q2
lkpφ
2
=mayη1
ΔZ2may
kφ1kφ2 t2kφ
h−q kφ1 +a1q2
lksφ
1
+a1q2 lkφs
2
+a2q1+a1q2 lkpφ
1
=mayη2
(6.7)
The two constantsη1andη2 depend, in a peculiar way, on the roll stiffnesses, on the heights of the no-roll centers3and on the longitudinal position of the center of gravity.
Similarly, the suspension roll angles (3.110) can be set as functions of the lateral acceleration only4
φ1s=may 1 ksφ
1
kφ1kφ2 kφ
h−q kφ2 +a2q1
lkφp
1
−a1q2
lkφp
2
=mayρ1s
φ2s=may
1 ksφ
2
kφ1kφ2 kφ
h−q kφ1 +a1q2
lkφp
1
−a2q1 lkφp
1
=mayρ2s
(6.8)
The same applies to tire roll anglesφip φp1 =may 1
kφp
1
kφ1kφ2
kφ
h−q kφ2 +a2q1
lkφs
1
+a2q1
lksφ
2
+a2q1+a1q2
lkφp
2
=mayρ1p
φp2 =may 1 kφp
2
kφ1kφ2 kφ
h−q kφ1 +a1q2
lkφs
1
+a1q2 lksφ
2
+a2q1+a1q2 lkφp
1
=mayρ2p (6.9)
If, for simplicity, the tires are supposed to be perfectly rigid, that iskpφ
i → ∞,
we haveρ1p=ρ2p=0,ρ1s=ρ2s=(h−q)/ kφand the expressions of the lateral load
3We call no-roll center what is commonly called roll center. This aspect is discussed in Sect.3.8.8.
4In this model the roll inertial effects are totally disregarded.
transfers become simpler ΔZ1may1
t1
kφ1(h−q) kφ +a2q1
l
=mayη1
ΔZ2may
1 t2
kφ2(h−q) kφ +a1q2
l
=mayη2
(6.10)
as in (3.118).
The total vertical loads (3.79) on each tire can also be simplified by discarding the longitudinal load transfer. Moreover, cars with an open differential are not so sporty to have significant aerodynamic vertical loads. Therefore, combining (3.79) and (6.7), we get
Z11=Fz11=mga2
2l −mayη1=Z10
2 −ΔZ1(ay) Z12=Fz12=mga2
2l +mayη1=Z10
2 +ΔZ1(ay) Z21=Fz21=mga1
2l −mayη2=Z20
2 −ΔZ2(ay) Z22=Fz22=mga1
2l +mayη2=Z20
2 +ΔZ2(ay)
(6.11)
which shows that all vertical loads are (linear) function of the lateral acceleration.
According to (3.123) and taking into account (6.8), we get the following expres- sion for the steering angles of the wheels
δij=δij0 +δvτij+Υijφis(ay)
=δij0 +δvτij+Υijρisay
=δij(δv, ay) (6.12)
which are functions ofδvand, again, of the lateral accelerationay. More precisely, the termδvτijis the steer angle due to the steering wheel rotationδv, the termδij0 is the toe-in/out angle , and the termΥijφis(ay)is the roll steer angle.
Under the assumed operating conditions (6.3), the tire lateral slips (3.49) become σy11=(v+ra1)−uδ11
u−rt1/2 v+ra1
u −δ11
σy12=(v+ra1)−uδ12
u+rt1/2 v+ra1
u −δ12 (6.13)
6.2 Fundamental Equations of Vehicle Handling 135 σy21=(v−ra2)−uδ21
u−rt2/2 v−ra2
u −δ21 σy22=(v−ra2)−uδ22
u+rt2/2 v−ra2
u −δ22
sinceu |rti/2|as discussed in (3.4). The lateral slips can be conveniently rewrit- ten taking (6.12) into account
σy11=v+ra1
u −δvτ11−δ011−Υ11ρ1say σy12=v+ra1
u −δvτ12−δ012−Υ12ρ1say σy21=v−ra2
u −δvτ21−δ021−Υ21ρ2say σy22=v−ra2
u −δvτ22−δ022−Υ22ρ2say
(6.14)
or, more compactly5
σyij =σyij(v, r, u, δv)=σyij(β, ρ, ay, δv) (6.15) Let, γi10 = −γi02=γi0 be the camber angles under static conditions, and let Δγi1=Δγi2=Δγibe the camber variations. The camber angles of the two wheels of the same axle are thus given by
γi1=γi0+Δγi, γi2= −γi0+Δγi (6.16) where the camber variationΔγi, according to (3.83), (6.8) and (6.9), depends on the lateral accelerationay
Δγimay
qi−bi
bi
ρis−ρip
=mayχi (6.17)
since the track variation termΔti/(2bi)is usually negligible.
The lateral forces exerted by the tires on the vehicle depend on many quantities, as shown in the second equation in (2.72). For sure, there is a strong dependence on the vertical loadsZij and on the lateral slipsσyij, while, in this model, we can neglect the longitudinal slipsσxij. The camber angles γij need to be considered, since they are quite influential, even if small. According to (3.125), the spin slipsϕij are directly related toγij. Therefore, the simplified model for each lateral force is
Fyij =Fyij
Z0i/2−ΔZi(ay), γi0+Δγi(ay), σyij(v, r, u, δv) cos
δij(δv) (6.18)
5Here we are abusing the notation: different functions bear the same name. However, the meaning should be sufficiently clear and unambiguous.
The lateral forceYi for each axle is obtained by adding the lateral forces of the left tire and of the right tire (cf. (3.58))
Yi=Fyi1cos δi1(δv)
+Fyi2cos δi2(δv)
(6.19) or, more explicitly, taking also (6.15) into account
Y1=Fy11
Z01/2−ΔZ1(ay), γ10+Δγ1(ay), σy11(v, r, u, δv) cos
δ11(δv) +Fy12
Z10/2+ΔZ1(ay),−γ10+Δγ1(ay), σy12(v, r, u, δv) cos
δ12(δv)
=Y1(v, r, u, δv)=Y1(β, ρ, ay, δv) Y2=Fy21
Z02/2−ΔZ2(ay), γ20+Δγ2(ay), σy21(v, r, u, δv) cos
δ21(δv) +Fy22
Z20/2+ΔZ2(ay),−γ20+Δγ2(ay), σy22(v, r, u, δv) cos
δ22(δv)
=Y2(v, r, u, δv)=Y2(β, ρ, ay, δv)
(6.20)
The effects ofay on the steering anglesδij can be neglected in the cosine terms because they are very small. On the other hand, these effects are very influential on the congruence equations (6.14).
It must be clearly understood that the functions in (6.20) are known functions.