There are four different types of external forces acting on a road vehicle:
(1) weight (gravitational force);
(2) aerodynamic force;
(3) road-tire friction forces;
(4) road-tire vertical forces.
60 3 Vehicle Model for Handling and Performance
3.5.1 Weight
The weight W is simply given by
W= −Wk= −mgk (3.54)
wheregis the gravitational acceleration. As well known, the weight force is applied inG. Therefore, it does not contribute to MG.
3.5.2 Aerodynamic Force
The aerodynamic force
Fa= −Xai+Yaj+Zak (3.55) depends essentially on the vehicle shape and size, and on the relative speed Va
between the vehicle and the air. An in-depth discussion on vehicle aerodynamics is beyond the scope of the present work. Here it may suffice to state without proof that
Xa=1
2ρaVa2CxSa, Ya=1
2ρaVa2CySa, Za=1
2ρaVa2CzSa (3.56) whereρais the air density,Va= |Va|,Sais the area of the vehicle frontal projection (frontal area) andCx, Cy, Cz are shape coefficients. Traditionally Cx>0, which explains the minus sign in (3.55). If Va is directed like the vehicle axis i, that is Va= −Vai, the coefficientCy=0 and henceYa=0.
In a modern car, the frontal areaSais about 1.8 m2and the drag coefficientCx ranges between 0.30 and 0.35. A Formula One car has a frontal area of about 1.3 m2 and a drag coefficient which ranges between 0.7 and 1. It is quite usual to provide directly the product SaCx as a more effective way to compare the aerodynamic efficiency of cars. For instance, a Formula One car hasSaCxof about 1.2 m2, while a commercial one may have it below 0.6 m2.
Formula 1 cars haveCz with very high negative values to achieve a very high aerodynamic downforce. Typically,SaCz −5.2 m2.
In general, the aerodynamic force Fain not applied atG(why should it be?) and therefore it contributes to MGwith an aerodynamic moment Ma=Maxi+Mayj+ Mazk, the biggest component beingMay (pitch moment).
It is common practice to do like in Fig.3.6, thus defining the front and rear aerodynamic vertical forces (positive upward) according to
Za1=1
l[Zaa2−May+Xah] =1
2ρaVa2Cz1Sa
Za2=1
l[Zaa1+May−Xah] =1
2ρaVa2Cz2Sa
(3.57)
Fig. 3.6 Aerodynamic forces
whereCz1andCz2 have been introduced. In other words, in straight running, the aerodynamic force Fa is given as two vertical loadsZ1a andZa2 acting directly on the front and rear tires, respectively, plus the aerodynamic dragXa acting at road level.
3.5.3 Road-Tire Friction Forces
The road-tire friction forces Ftij are the resultant of the tangential stress in each footprint, as shown in (2.15). Typically, for each tire, the tangential force Ftij is split into a longitudinal componentFxij and a lateral componentFyij, as shown in Fig.3.7. It is very important to note that these two components refer to the wheel reference system shown in Fig.2.2, not to the vehicle frame.
Ifδij is the steering angle of a wheel, the components of the tangential force in the vehicle frameSare given by
Ftij =Xiji+Yijj
whereXij=Fxijcos(δij)−Fyijsin(δij)
Yij=Fxijsin(δij)+Fyijcos(δij) (3.58) with obvious simplifications ifδij is very small.
To deal with shorter expressions, it is convenient to define X1=X11+X12, X2=X21+X22
Y1=Y11+Y12, Y2=Y21+Y22 ΔX1=X12−X11
2 , ΔX2=X22−X21
2 ΔY1=Y12−Y11
2 , ΔY2=Y22−Y21 2
(3.59)
62 3 Vehicle Model for Handling and Performance Fig. 3.7 Road-tire friction
forces
Even for not so small steering angles, simpler expressions can be obtained by observing that small errors in the values of the steering anglesδij have marginal influence on the global equilibrium.2More precisely, in the equilibrium equations we can “force” the steering angles of the front wheelsδ11andδ12both to be equal toδ1=(δ11+δ12)/2. Similarly, the rear wheels can be set to have the same (often zero) steering, that isδ2=(δ21+δ22)/2.
It should be clearly understood that often, in real vehicles, the two wheels of the same axle are intentionally slightly nonparallel. Assuming the two wheels to be parallel is harmless for the global equilibrium of the vehicle, whereas it would be quite influential on the tire behavior.
Strictly speaking, the tangential forcesFyij are not applied at the center of the contact patch. In general, there are also vertical momentsMzij. However, these mo- ments have negligible effects on the dynamics of the vehicle as a whole. Indeed, takingMzij into account would mean displacing by only a few centimeters the ac- tion lines of Ftij.
On the other hand, vertical moments do affect quite a bit the steering system. In particular, they must be included in vehicle models with compliant steering system.
2But not on the tire slips.
3.5.4 Road-Tire Vertical Forces
The road-tire vertical forcesFzijk are the resultant of the normal pressure in each footprint, as in (2.13).
As discussed in Sect.2.6.3, the displacement with respect to the center of the footprint of the line of action of the vertical forces is the main cause of rolling re- sistance. This phenomenon can be neglected when studying, e.g., extreme braking or handling, whereas it is of paramount importance for the estimation of fuel con- sumption or of power losses in general.
It is customary to add the vertical forces of the same axle
Z1=Fz11+Fz12 and Z2=Fz21+Fz22 (3.60) and to define the differences
ΔZ1=Fz12−Fz11
2 and ΔZ2=Fz22−Fz21
2 (3.61)
usually called lateral load transfers.
Inverting these equations yields for the vertical load on each wheel Fz11=Z1
2 −ΔZ1=Z11, Fz12=Z1
2 +ΔZ1=Z12
Fz21=Z2
2 −ΔZ2=Z21, Fz22=Z2
2 +ΔZ2=Z22
(3.62)