In the study of the stability of pneumatic tired vehicles, it is of primary importance to describe the generation of lateral forces by the tires. Obviously, these are the forces that allow a car to be steered or allow any wheeled vehicle to negotiate a turn. It is also obvious that no lateral force is necessary for a vehicle traveling on a straight and level road when it is not under the influence of any external loads. On the other hand, if one considers a small perturbation from a straight-line path, it is the lateral forces from the tires that will determine whether the vehicle will return to the original state of straight-line motion or will deviate ever more from it, i.e., whether the vehicle is stable or unstable.
A tire mounted on a wheel subjected to no lateral force at all will generally roll along a surface in the direction its wheel is pointed unless the tire is significantly unsymmetrical. Assuming that the wheel supports a load, there is a normal force between the road surface and the contact patch of the tire. If now a small lateral force is applied to the wheel, a corresponding lateral force will arise in the contact patch between the tire and the road surface due to friction. Although the lateral force may be too small to cause the tire to slide sideways over the surface, it will no longer be the case that the wheel continues to move in the direction it is pointed. Rather, it will acquire a sideways or lateral velocity in addition to its forward velocity. Thus the total velocity vector of the wheel will point in a different direction than the direction of the center plane of the wheel. This angle between the wheel and the direction the wheel moves when a lateral force is applied is called the slip angle, a. The slip angle is shown in Fig. 4.1. For small lateral forces, the slip angle is small and it is due to the distortion of the tire. The most obvious aspect of this distortion is the
deflection of the tire sidewall. The wheel appears to drift sideways as parts of the tire surface entering the contact patch are pushed to the side before they finally become locked to the road surface by the friction. (Parts of the tire exiting the rear of the contact move back toward their undistorted position as they lose the influence of the lateral friction force.) If the lateral force is increased sufficiently, there will come a point at which the entire contact patch begins to slide sideways. At this point, the slip angle can approach 90j.
In the analysis of the stability of vehicles using pneumatic tires, it is common to use a linearized model of the relation between the lateral force generated by the tire, Fy, and the slip angle,a. Over a certain range, the lateral force and the slip angle are found to be nearly proportional to each other. The linear tire force model will apply approximately for small values of the lateral force and for small values of the slip angle. As long as the lateral force remains small enough that the tire does not begin to skid, the slip angle remains quite small. For automobile tires, the slip angle typically remains less than about 10j even for quite vigorous cornering.
For small slip angles, a proportionality constant, the cornering coefficient, Ca, can be defined as the slope of the line relating lateral force to slip angle. At large slip angles, the tire begins to skid and the force vs. slip angle relation is significantly nonlinear. At a certain slip angle, the lateral force reaches a maximum, and for even larger values, the force tends to decline somewhat.
Although the slip angle is a main determinant of the side force, it is certainly not the only one. For example, if the wheel midplane is not at right angles to the ground plane but is tilted over at a so-called camber angle, this angle does influence side force generation. The effect is some- times called camber thrust, but 1j of camber angle produces much less lateral force than 1j of slip angle. Thus changes in camber angle can often be neglected in a first analysis of vehicle stability. More importantly, if the wheel has a braking or driving torque, the lateral force generation can be significantly affected as will be demonstrated below.
A. Effect of Normal Force
Perhaps the most obvious influences on the side force generation for tires are the nature of the surface on which the tire rolls and the normal force, Fz, supported by the tire. It is common to assume that the side force at a given slip angle is roughly proportional to the normal force. This is consistent with the Coulomb friction model assumption that when the tire skids, the
skidding force is approximately equal to a coefficient of friction times the normal force. The coefficient of friction depends on the road surface as well as on the tire material. Unfortunately, friction is a complicated phenom- enon and cannot always be represented accurately using the coefficient of friction as a simple constant parameter as often is assumed in elementary treatments of friction.
For pneumatic tires, the coefficient of friction idea is often general- ized to be the ratio of the lateral force to the normal force. This ratio then becomes a function of the slip angle. As the tire begins to skid, this ratio assumes a maximum value, but for very large slip angles, the ratio actually begins to drop off. Fig. 4.2 shows how a typical passenger tire generates a side force as a function of slip angle. Note that the condition of the roadway surface can have a major effect on the possible magnitude of the lateral force as anyone who has driven on an icy road can testify.
The generalized lateral coefficient of friction, ly, plotted in the graph is simply defined as the ratio of the side force to the normal force, Fy/Fz. Thus the lateral force can be written as
Fy¼ lyFz ð4:1Þ
The graph implies that the side force is strictly proportional to the normal force at a given slip angle, but this is only approximately true for a range of normal forces. When the normal force becomes very large, the ratio of side force to normal force at a given slip angle actually decreases. This
FIG. 4.2 The ratio of side force to normal force for a tire on several surfaces as a function of slip angle.
phenomenon is not represented in graphs such as the ones shown inFig.
4.2.
The graphs in Fig. 4.2 give a good idea of the basic form of the relation between side force and slip angle. As the graphs show, there is no strictly linear portion of the curves, but up to about 5j, a linear approx- imation is reasonable. For many stability analyses, the tire is assumed to be operating in the vicinity of zero slip angle so a linear approximation to the relationship between the lateral force and the slip angle can be used. The cornering coefficient, Ca, is defined as the coefficient in the linearized version of the law relating the lateral force to the slip angle
Fy¼ Caa ð4:2Þ
The cornering coefficient Cais the slope of a graph of Fyas a function ofa at the origin (witha in radians), assuming a constant normal force.
Using the assumption implied by Fig. 4.2 that the lateral force is proportional to the normal force, Eq. (4.1), one can find the cornering coefficient from the slope of the graphs inFig. 4.1near the origin, dly/da. In the linear region, the friction coefficient is approximately (dly/da)a, and thus Eq. (4.1) becomes
Fy¼ ðdly=daÞaFz¼ fðdly=daÞFzga ð4:3Þ
A comparison of Eqs. (4.2) and (4.3) shows that
Ca¼ fðdly=daÞFzg ð4:4Þ
Obviously, Cadepends on the road surface as can be seen in Fig. 4.2, but as mentioned above, the dependence on the normal force implied by the graphs in Fig. 4.2 and given in Eqs. (4.1), (4.3), and (4.4) is not strictly correct. As a result, the cornering coefficient is not strictly proportional to the normal force. As the normal force increases, the lateral force at a given slip angle begins to increase less than strictly proportionally at high normal force levels. Thus Fig. 4.2 and the relation, Eq. (4.4), derived from it should be regarded only as useful approximations for a restricted range of normal forces. SeeFig. 4.3for a plot of tire data showing the effect of changes in normal force.
Fig. 4.3 shows an important effect associated with changes in normal force. For a constant slip angle, there is a range of normal forces for which the lateral force is essentially proportional to the normal force as is assumed in Eqs. (4.1)–(4.4) and in Fig. 4.2. However, as the normal force is increased to ever-higher levels, the lateral force for a constant slip angle
does not continue to increase at the same rate. In fact, at some point, the lateral force ceases to increase at all with increasing normal force and even begins to decrease somewhat in some cases. This effect is particularly noticeable in Fig. 4.3 for slip angles from about 4j to 10j.
The curvature of the relation between lateral and normal force for constant slip angle will have implications for the vehicle models to be studied inChapters 5and6.In many cases, it is convenient to consider a single equivalent wheel to represent the lateral forces generated by two wheels on a single axle. If the slip angles for both wheels are the same and if they both carry the same load, then it is easy to see that the total lateral force is just twice the lateral force for a single wheel. Furthermore, the axle cornering coefficient is twice the cornering coefficient for a single wheel as calculated, for example, by Eq. (4.4).
When a vehicle is negotiating a corner, there will be a transfer of load with the outside wheel taking more of the load and the inside wheel taking less. (Just how much load transfer takes place depends on a number of factors including the height of the vehicle center of mass, the distance
FIG. 4.3 Composite plot of lateral force, Fy, vs. slip angle,a, and normal force,
between the wheels on the axle, and the details of the suspension system.) If the cornering coefficient was truly proportional to the normal force, the cornering coefficient for the axle could still be calculated as twice the co- efficient for one wheel. Assuming that the total load supported by the axle is constant whether the vehicle is cornering or not, the increase in load on the outside wheel would be matched by the decrease in load carried by the inside wheel. The change in the cornering coefficient on one wheel would compensate the change in cornering coefficient on the other.
As can be seen inFig. 4.3,the curvature of the lateral vs. normal force relation at a given slip angle means that the sum of the lateral forces for two wheels with equal normal forces is more than the sum of the lateral forces when one wheel has its normal force incremented by a certain amount and the other has its normal force decremented by the same amount. In simple terms, when load transfer occurs at an axle, the lateral force and the cornering coefficient at a given slip angle are less than they would be in the absence of load transfer. This effect is often used by suspension designers to adjust the handling properties of cars by varying the amount of load transfer taken by the front and rear axles through the use of antisway bars, the stiffness of the suspension springs, etc. What they are doing is influencing the effective cornering coefficients at the front and rear axles. How the cornering coefficients influence the stability of the car will be discussed in Chapter 6.
For the tire law sketched in Fig. 4.2, the maximum lateral force on dry concrete is a little more that 0.8 times the normal force. If the normal force were due solely to the weight supported by the tire, this would imply that the maximum lateral acceleration would be 0.8 times the acceleration of gravity.
Most passenger cars can barely achieve this level of lateral acceler- ation because of a number of effects including the fact that not all tires achieve their maximum levels of side force simultaneously in a typical case. Race cars may achieve much larger levels of lateral acceleration if they use tires using rubber compounds with higher maximum friction coefficients. Very high levels of lateral acceleration are possible particularly if cars have bodies that generate aerodynamic down-force so that the normal force on the tires is larger than the weight of the car. Maximum tire friction coefficients greater than unity are possible (usually at the expense of decreased tire life) and the aerodynamic down-force allows the tires to generate larger side forces without any increase in weight or mass, and thus lateral accelerations greater than one ‘‘g’’ are common for certain types of race cars.