In a previous section on steady cornering, the assumption was made that the relation between tire side force and the slip angle remained in the linear
region. Now, steady cornering in the nonlinear range will be studied. We will describe, at least qualitatively, what happens to a car as it approaches the limiting speed in a corner. By limit speed, we mean the highest speed that the tires will allow the car to have while maintaining a given curve radius.
It is almost obvious that a tire with a fixed normal force can only produce a limited lateral force before it begins to slide. Roughly speaking, the maximum lateral force corresponds to the normal force times a coefficient of friction for rubber on pavement. The Coulomb friction model suggests that once slipping starts, the friction force does not change. As was pointed out inChapter 4,it is not quite that simple because tests on real tires show that the lateral force vs. slip angle curve typically reaches a maximum lateral force at moderate slip angles, and then begins to decrease at very large slip angles. At any rate, the maximum side forces possible from the tires limit the maximum speed in a given corner because they limit the maximum centripetal acceleration of the center of mass, U2/R.
It may be less obvious that the limitation may come either at the front or at the rear. In order for the car to remain in a steady turn, the moments about the center of mass from the front and rear axles must sum to zero. (Otherwise, constant angular momentum and angular velocity cannot be maintained.) Thus, if either end of the car has reached its maximum force, the extra force capability at the other end cannot be used because the
moment would not remain at zero. The result is that attempts to increase the cornering speed beyond the limit will result in either the front end or the rear end beginning to slide out.
The analysis to be presented below will still use some small angle approximations. This is reasonable because the slip angle at which the maximum lateral tire force occurs is often only about 10j or 15j. Even racecars approaching limit speeds in the middle of a corner usually do not exhibit large attitude angles. If the speed limit in a corner is exceeded, the car will either slide out of the corner or spin, at which time the small angle approximation will no longer apply.
Finally, the ‘‘bicycle model’’ will continue to be used. When we talk about forces and slip angles at the front and rear, we will mean the total force at the front and rear axles from both wheels. We also assume that the slip angles are essentially equal for the two wheels at the front and also for the two wheels at the rear. This means that the force-slip angle relation- ships are not attributable only to the tires themselves, but to the design of the suspensions at the front and rear.
There are many factors that can influence the axle characteristics (Bundorf, 1967b). For example, consider the camber angle, i.e., the angle the tires make with the normal to the road surface. When a car is cornering, it tends to roll toward the outside of the turn. Different suspension designs result in different camber angle changes when the car body rolls and a camber change affects lateral tire force generation.
Also, in a steady turn, the normal forces for the two outside tires must be larger than the normal forces on the two inside tires. However, the distribution of the changes in normal forces at the between the tires at the front and rear is affected by the suspension stiffness and the presence or absence of antisway bars at the front or at the rear. Because of the nonlinear dependence of lateral tire forces on the normal forces, the distribution of total roll stiffness between the front and rear axles affects how the front and rear axle forces are related to the corresponding slip angles. This important means of adjusting the handling characteristics of vehicles was discussed briefly in Chapter 4 with respect to the tire characteristics plotted inFig. 4.2(see Problem 6.8).
Finally, it should be mentioned that even with identical tires on all wheels, handling engineers can affect the axle characteristics by specifying different tire pressures at the front and rear.
The number of factors affecting understeer and oversteer (or handling properties in general) is too large to discuss in any detail here,
but one should just keep in mind that the axle force-slip angle curves are determined largely, but not entirely, by the tire characteristics. As will be seen, what are important are always differences between front and rear forces that are often similar in magnitude. Thus sometimes small changes have large effects, particularly in the limit.
A. Steady Cornering with Linear Tire Models
Before discussing cornering with a nonlinear tire model, which is necessary when considering limit cornering behavior, it is worthwhile to review some results for the case in which it is assumed that the tires remain in the linear region. In the section on steady cornering, the relation between steer angle and the slip angle at the front and rear was derived, see Eq. (6.52). This relation remains true regardless of the relationship between lateral force and slip angle. If a linear relationship is assumed, the slip angles are proportional to the lateral acceleration, U2/R, and a constant understeer coefficient K2was defined [see Eq. (6.69)].
The steer angle relationship then takes the form of Eq. (6.66). In this form, K2 indicates how d varies when the lateral acceleration in ‘‘g’s’’ changes. If K2>0, d increases as the lateral acceleration increases and this is an understeer situation. As can be seen inFig. 6.8in the section on steady cornering, when d is positive, the slip angles are negative. Understeer really means that (af) is larger in magnitude than (ar). For this reason, it is sometimes convenient to redefine the slip angles so that only positive quantities appear in the steer angle equation.
a1u af; a2u ar ð6:102Þ
With these definitions, Eq. (6.52) becomes
d¼ ða1 a2Þ þ ða þ bÞ=R ð6:103Þ
With these new definitions of the slip angles, for the linear case,
K2¼ ða1 a2Þ=ðU2=RgÞ ð6:104Þ
and K2is positive (understeer) when a1>a2and negative (oversteer) when a1<a2.
B. Steady Cornering with Nonlinear Tire Models
For the nonlinear case, there is no constant understeer coefficient; howev- er, one can define a variable coefficient that expresses how the slip angle difference or the steer angle changes as the lateral acceleration changes.
K2u
dða1 a2Þ dðU2=RgÞ¼
dd
dðU2=RgÞ ð6:105Þ
The variable understeer coefficient can be found graphically from plots of the axle force characteristics Yf(a1) and Yr(a2). It proves to be convenient to normalize the axle lateral forces by the weight forces. As was shown in the previous section on steady cornering, in the absence of aerodynamic forces, the lateral forces are proportional to the weight forces, Eqs. (6.60) and (6.61), and in fact, the ratios of the side forces to the weight forces is always equal to the lateral acceleration in g’s.
Yf Wf ¼ Yr Wr ¼U2 Rg ð6:106Þ
From plots of the axle forces normalized by the weight forces, one can determine the slip angles given the lateral acceleration, as shown in Fig.
6.16.
Combining the two plots in Fig. 6.16, one can see how (a1a2) changes as U2/Rg varies, as shown inFig. 6.17.
In the case shown in Fig. 6.17, a1is always greater than a2, and the difference (a1a2) increases as U2/Rg is increased, until a value is reached at which the front slides out and a steady turn cannot be achieved.
FIG. 6.16 Front and rear slip angles determined from normalized force vs. slip angle curves.
Finally, a single plot relating the acceleration to the slip angle difference can be constructed from the combined plot of Fig. 6.17, as shown in Fig. 6.18.
In the example of Fig. 6.18, the variable understeer coefficient is always positive and increases continuously as the lateral acceleration is increased from zero to the maximum value possible before the front axle slides and a steady turn is no longer possible. However, there are many other possibilities, as shown inFig. 6.19.Because the understeer coefficient has to do with the difference in front and rear slip angles, occasionally small effects that come into play at certain values of lateral acceleration can make significant changes in the steering behavior of the vehicle.
FIG. 6.17 Determination of slip angle difference from lateral acceleration.
FIG. 6.19 Plots equivalent to Figs. 6.17 and 6.18 for a number of particular cases.
From the plots inFig. 6.19,one can see that the understeer coefficient can vary with the lateral acceleration in a steady turn. A car that under- steers at low acceleration can begin to oversteer at higher acceleration for example. Furthermore, the limiting lateral acceleration may be achieved when either the front axle or the rear axle finally slides out.
It is also possible to imagine a car for which both slip angle curves are identical. The understeer coefficient would then always be zero and both ends of the car would reach their maximum lateral for simultaneously. One can think of this as the situation sometimes described as a ‘‘four-wheel drift.’’ This is the origin of the notion that a neutral steer car is a sort of optimum even when the concept is extended into the nonlinear regime.