The suggested approach, which explains why steady-state data are also relevant for the transient behavior, is applied here to the single track model. The goal is to clarify the matter by an example.
10Actually, the real critical speed can be lower than the value predicted by (6.124), as shown in [5, pp. 216–219]. Basically, (6.124) may not predict the right value because in real vehicles we control the longitudinal force, not directly the forward speed. Therefore, a real vehicle is a system with three state variables, not just two. This additional degree-of-freedom does affect the critical speed, unless the vehicle is going straight.
6.15 The Single Track Model Revisited 181 For simplicity, we assumeu=ua andu˙=0 and hence start with the linearized equations of motion (6.98). In the single track model (with open differential), the stability derivatives (6.83) can be given a more explicit form (cf. (6.59)) taking into account the axle characteristics
Yβ=∂Y1
∂α1
∂α1
∂β +∂Y2
∂α2
∂α2
∂β = −∂Y1
∂α1−∂Y2
∂α2 = −Φ1−Φ2
Yρ=∂Y1
∂α1
∂α1
∂ρ +∂Y2
∂α2
∂α2
∂ρ = −a1∂Y1
∂α1+a2∂Y2
∂α2 = −a1Φ1+a2Φ2
(6.127)
and Nβ=a1
∂Y1
∂α1
∂α1
∂β −a2
∂Y2
∂α2
∂α2
∂β = −a1
∂Y1
∂α1+a2
∂Y2
∂α2 = −a1Φ1+a2Φ2
Nρ=a1∂Y1
∂α1
∂α1
∂ρ −a2∂Y2
∂α2
∂α2
∂ρ = −a12∂Y1
∂α1−a22∂Y2
∂α2= −a21Φ1−a22Φ2
(6.128)
where
Φ1=∂Y1
∂α1 and Φ2=∂Y2
∂α2 (6.129)
are the slopes of the axle characteristics at the equilibrium point. Obviously,Φi>0 in the monotone increasing part of the axle characteristics. These slopes are simple to be defined, but not so simple to be measured directly. It is also worth noting that
Yρ=Nβ (6.130)
To proceed further, let
δ1=τ1δv and δ2=τ2δv=χ τ1δv (6.131) thus linking the rear steering to the front steering. To have front steering only it suffices to setχ=0. We can now obtain also the more explicit expressions of the control derivatives
Yδ=(Φ1+χ Φ2)τ1, Nδ=(Φ1a1−χ Φ2a2)τ1 (6.132) Therefore, the linearized equations of motions (6.98) are as follows, whereY and Ndo not depend onu
m
uaβ˙t+u2aρt
= −(Φ1+Φ2)βt−(Φ1a1−Φ2a2)ρt+(Φ1+χ Φ2)τ1δvt
Jzuaρ˙t= −(Φ1a1−Φ2a2)βt−
Φ1a12+Φ2a22
ρt+(Φ1a1−χ Φ2a2)τ1δvt (6.133) Formulổ (6.89) become, in this case
2ζ ωn= −tr(A)= 1 ua
Φ1+Φ2
m +Φ1a21+Φ2a22 Jz
(6.134)
and
ω2n=det(A)= 1 u2amJz
Φ1Φ2(a1+a2)2−mu2a(Φ1a1−Φ2a2)
(6.135) and hence
ζ = (Φ1+Φ2)Jz+(Φ1a21+Φ2a22)m 2√
Jzm
Φ1Φ2l2−mu2(Φ1a1−Φ2a2) (6.136) These parameters characterize the handling behavior in the neighborhood of an equi- librium point. Actually, the fundamental “bricks” on which everything is built are the six design parameters
Φ1
m, Φ2
m, a1, a2, Jz
m, χ (6.137)
in addition to the control parametersuandδv(t ), with constantu(τ1 has no rele- vance).
At steady-state, the linearized equations of motion become a linear algebraic system of equations
ma˜y=mu2aρp= −(Φ1+Φ2)βp−(Φ1a1−Φ2a2)ρp+(Φ1+χ Φ2)τ1δva 0= −(Φ1a1−Φ2a2)βp−
Φ1a12+Φ2a22
ρp+(Φ1a1−χ Φ2a2)τ1δva
(6.138) which, when solved, provides the (linear approximation of the) handling maps in the neighborhood of the equilibrium point (cf. (6.80))
βp=βp(a˜y, δva)=vp ua =
a2+a1χ l
τ1δva−m l2
Φ1a21+Φ2a22 Φ1Φ2
˜ ay ρp=ρp(a˜y, δva)= rp
ua = 1−χ
l
τ1δva−m l2
Φ2a2−Φ1a1
Φ1Φ2
˜ ay
(6.139)
We remark that this is a local linear approximation of the handling maps. In the suggested approach, these two maps fully describe the vehicle handling features at steady state.
The components of the gradients gradβp and gradρp (defined in (6.66)) are therefore given by
βy= −m l2
Φ1a12+Φ2a22 Φ1Φ2
= −Kβy, βδ=τ1
a2+χ a1
l
= −Kβδ
ρy= −m l2
Φ2a2−Φ1a1 Φ1Φ2
= −Kρy, ρδ=τ1
1−χ l
= −Kρδ
(6.140)
As already stated, all these components can be obtained experimentally from stan- dard steady-state tests, without having to bother about Ackermann steer angle and the like.
6.15 The Single Track Model Revisited 183 From the results displayed in the first column in (6.140) we get
Φ1
m = a2
l(Kβy +a1Kρy), Φ2
m = a1
l(Kβy−a2Kρy) (6.141) which show that there may exist different vehicles with exactly the same values of Kβy= −βyandKρy= −ρy. Therefore, more than two parameters are necessary.
Summing up, for the single track model, the six coefficientssi in (6.119) are s1=βy= − m
(a1+a2)2
Φ2a22+Φ1a12 Φ1Φ2
= −Kβy
s2=ρy= − m (a1+a2)2
Φ2a2−Φ1a1
Φ1Φ2
= −Kρy s3=βδ=τ1a2+χ a1
a1+a2
s4=ρδ=τ1 1−χ a1+a2
s5=Nδ
Jz =τ1
Φ1a1−χ Φ2a2
Jz
s6=Yδ m =τ1
C1+χ Φ2 m
(6.142)
Of course, they depend on the six design parameters listed in (6.137):Φ1/m,Φ2/m, a1,a2,Jz/m,χ.
Assumingχas given, we obtain from the first four equations in (6.142) Φ1
m = ρδ(βδ−χ τ1)
τ1(χ−1)[βyρδ−ρy(βδ−τ1)]= s4(s3−χ τ1) τ1(χ−1)[s1s4−s2(s3−τ1)] Φ2
m = ρδ(τ1−βδ)
τ1(χ−1)[βyρδ−ρy(βδ−χ τ1)]= s4(τ1−s3)
τ1(χ−1)[s1s4−s2(s3−χ τ1)] (6.143) a1=τ1−βδ
ρδ =τ1−s3
s4
a2=βδ−χ τ1
ρδ =s3−χ τ1
s4
Then, also considering the fifth equation Jz
m = 1 s5
(βyρδ−ρyβδ)(βδ−χ τ1)(βδ−τ1)
[βyρδ−ρy(βδ−τ1)][βyρδ−ρy(βδ−χ τ1)] (6.144) In the single track model there are not enough design parameters to fulfill all six equations. Therefore, the value ofs6depends on the other five parameters si,i= 1, . . . ,5.
Table 6.1 Features of vehicles with different amounts of rear steeringχ, but with almost identical transient handling behavior. The old understeer gradientKconveys misleading information
χ C1
[N/rad]
C2 [N/rad]
a1 [m]
a2 [m]
Jz [kg m2]
m [kg]
K [deg/g]
Kρy [deg/g]
−0.10 76 629 93 559 0.91 1.93 3 169 1 365 4.16 1.46
−0.05 74 900 91 452 0.91 1.80 2 759 1 365 3.97 1.46
0.00 73 000 90 000 0.91 1.67 2 400 1 365 3.78 1.46
+0.05 70 899 89 144 0.91 1.54 2 084 1 365 3.59 1.46
+0.10 68 565 88 851 0.91 1.41 1 803 1 365 3.40 1.46
6.15.1 Different Vehicles with Almost Identical Handling
It is kind of interesting to employ Eqs. (6.143) and (6.144) to obtainΦ1/m,Φ2/m, a1,a2andJz/mfor givens1–s5, but different values ofχ, that is with a different amount of rear steering. This way, it is possible to create vehicles that look very different, but which ultimately have almost exactly the same handling behavior. The little difference being due to the terms6that cannot be set to the same value for all vehicles, due to the lack of parameters in the single track model.
A vehicle with front steering only hasχ=0, while, e.g.,χ = −0.05 means a rear steering angleδ2= −0.05δ1, and so on.
But, let us do some numerical examples. Let us consider a vehicle with only front steering (i.e.,χ=0), with the following features:
• m=1 365 kg;
• Jz=2 400 kg m2;
• a1=0.912 m;
• a2=1.668 m;
• Φ1=C1=73 000 N/rad;
• Φ2=C2=90 000 N/rad.
From (6.142) we can compute allsifor this vehicle.
Then we can set a non-zero value forχ and, employing the very same values of s1,s2,s3,s4ands5, compute the corresponding physical quantitiesC1,C2,a1,a2, Jz andm, according to (6.143) and (6.144). The results for some values of χ are shown in Table6.1. The five vehicles there reported are strikingly different, yet they have (almost) the same handling behavior, and not limited to steady state. For the driver, they behave quite the same way even under transient conditions, like under a step steering input.
The amazing similarity of the handling dynamics between these five vehicles can be appreciated looking at Figs.6.34,6.35,6.36and6.37, where the time-histories of some variables are shown and compared. All figures refer to a step steerδ1=2.2◦, withu=30 m/s, starting from a straight trajectory.
But perhaps the most astonishing result is that these vehicle, although with almost identical handling, do not have the same understeer gradientK. Just have a look at
6.15 The Single Track Model Revisited 185 Fig. 6.34 Vehicle slip angle
β(t )after a step steering input
Fig. 6.35 Yaw rater(t )after a step steering input
Fig. 6.36 Derivativev(t )˙ of the lateral speed after a step steering input
Fig. 6.37 Lateral accelerationay(t )= ˙v+ur after a step steering input
the next to last column in Table6.1. In other words, they would have been classified as very different if evaluated in terms of their understeer gradientK.
The conclusion is that the classical understeer gradient is not a good parameter and should be abandoned. It should be replaced by the gradient components pro- posed in (6.66) and discussed in Sect.6.13, which have proven to really provide a measure of the dynamic features of a vehicle. In particular, the gradient com- ponentKρy, shown in the last column in Table6.1, is the real measure of under- steer/oversteer.