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Page 7
TIRE LOADINGS

Or: Where The Rubber Meets The Road

A complete understanding of vehicle dynamics requires a knowledge which encompasses aerodynamics, tire characteristics, suspension design, and can even include the psychological factors involved in the car/driver interface. With such complexities, it is tempting to make simplifications to ease the problem of analysis. Such simplifications, unfortunately, often limit the application to very restricted examples. There is, however, one simplification which is both universal in its application and stark in its simplicity.

I doubt if a publisher would accept it, but suppose I were to write a children's book about "Mister Road." Mister Road is the road surface upon which every vehicle travels. Mister Road clings to each vehicle as the vehicle tavels through his turns and starts and stops at his intersections. (Yeah, this is getting silly, but stick with me on this. There is a point to it.) But, poor Mister Road is blind. The only knowledge he has of those cars overhead are their four (more or less) tire patches which make contact with him. Whether it's Aunt Rose's 1953 Kaiser or Mark Martin's Cup car, Mister Road knows only of those four tire patches.

I hope that bit of silliness has driven home the importance of those four relatively small tire patches when analyzing the dynamics of a race car. We can talk all we want about slip angles and tire pressure and Panhard bar heights and instant centers, but it all comes down to how well those four little patches of rubber cling to the track surface.

It is at this point that an automotive engineering textbook would probably begin with a chapter on tire characteristics and testing methods. This would then develop into a consideration of understeer and oversteer and then into various dynamic stability parameters. But, again, I believe that, since those involved in motorsports are mostly concerned with the limiting behavior of the car, another simplification is warranted. As a result of that belief, I'll devote some space to those four tire patches and how they affect performance at the limits of tire/road surface adhesion.

Coefficient of Friction

When your high school physics teacher spoke of the "coefficient of friction," He probably introduced it with a picture of a block atop a level surface, explaining that the necessary force to move the block along the surface equalled the product of the coefficient of friction and the weight of the block. He probably went on to explain that, for most material combinations (materials of block and surface), the force necessary to initiate motion is slightly more than that required to maintain motion. In other words, the value of the static coefficient of friction is slightly higher than the value of the sliding coefficient of friction. He then pointed out that the measured values of the coefficient are independent of the area at the interface. To demonstrate a means of measuring the coefficient, he might have placed a block on a surface, which surface he then tilted until the block began to move. The tangent of the surface's angle of tilt is equal to the friction coefficient. If the materials are properly chosen for such a demonstration, it can be shown that the angle tangent can easily exceed unity, meaning the coefficient of friction can easily exceed a value of "1." (I have often used a plastic drafting triangle and a drafting eraser for such a demonstration.)

Well, when it comes to tires and most road surfaces, we'll have to modify...very slightly...some of the things you were taught. In the first place, the interface area obviously DOES affect the value of the friction coefficient. I'm not going to attempt to explain the exact mechanical nature by which the rubber "grips" the track surface, but it cannot be denied that there is an effect. Another parameter, interacting with the tire patch area, is the temperature of the rubber. A limitation to an increase of tire patch area seems to be the ability to keep the rubber warm enough to efficiently use the increased area. The weaving of race cars behind the pace car is, of course, indicative of their desire to keep the tire rubber at temperatures higher than ambient.

Also, tire tests seem to indicate that a slight bit of relative motion...between tire patch and track surface...might actually raise the value of the friction coefficient. This sliding is difficult to quantify during cornering, but, in traction, we can say that the friction coefficient reaches a peak value when the wheel speed is about 10% greater than that corresponding to the "non-slipping" speed.

The Friction Circle

Going back to your high school physics and the block on the flat horizontal surface: Your teacher didn't specify a particular direction for the displacement of the block on the surface. While direction isn't important in describing the characteristics of friction, a force inherently has direction. So, it would take an infinite number of force vectors to describe the possibilities. The "tail" of each vector would conveniently be positioned beneath the block, meaning a line connecting the "heads" of all the vectors would constitute a circle. This circle is commonly called the "friction circle." The heavier the block, the larger the diameter of the circle.

Well, we're not interested in pushing blocks around on a table top, but this same friction circle concept can be applied to the tire patch. The greater the wheel loading, the larger the friction circle, or, in other words, the greater the wheel loading, the more force required for the tire to lose grip with the track surface. And, since the force vector can be in any direction in the plane of the track surface, it is not limited to either a traction force or a cornering force, but can be any vector sum of the two. The magnitude of the vector sum cannot exceed the limitation imposed by the friction circle.

By adding the force vectors at all four tires, we can consider a force vector and a friction circle for the entire car. Or, it might be advantageous to consider separate vector sums and friction circles for the rear tire pair and for the front tire pair.

(At this point, if you're a dragracer, you might be wondering what all this has to do with your problems, since you're not all that concerned with cornering forces. While this is true, I would urge you to continue reading, for there is a principle involved which most certainly does affect the performance of every car at the dragstrip.)

If the friction circle "grew" linearly with increasing load, the weight of your oval or roadrace car would be of little significance. If you doubled the weight, the inertial forces would be doubled, but the friction circle would also be doubled and maximum cornering speeds would be unaffected. As you're well aware, however, that isn't what happens! Tires are certainly non-linear in their behavior.

An Important Principle

The result leads to a very significant principle, generally well understood by the oval and roadrace crowd, but often totally ignored by the dragracers: The maximum grip from a tire pair is achieved when they are equally loaded. For the front tire pair of a car in a corner, the maximum vector sum of the two tires would be achieved when the two tires are equally loaded. This is impossible, however, since the outside tire is always loaded more than the inside. So, the constructor minimizes this loading disparity by seeking the lowest possible center of gravity height.

Suppose that, while traveling at the maximum speed on a constant radius curve, the rear of the car tends to lose traction first. To avoid this, we might want to reduce the grip of the front tire patches. This could be done by increasing the load difference between the two front tires. This is perhaps the first thing the oval or road course racer learns.

Also, note that, for a constant radius turn, the maximum cornering speed for a car with identical tires on all four wheel will be achieved when half the car's weight is on the front tires. There might, however, be other factors which would cause the constructor to use a different weight distribution.

Many...if not most...dragrace cars are rear wheel drive (RWD) with a beam rear axle. The driveshaft torque reaction is received through the engine and transmission mounts, but only a portion of it is delivered back to the rear axle, meaning that the right rear tire tends to lose loading while the left rear gains. This load imbalance means that the total tire traction force is less than the maximum possible. The cure, obviously, is to achieve equal rear tire loading.

Factors That Influence Tire Patch Effectiveness

There are other factors...in addition to unequal loading of the four tires...which can affect the tire patch performance. Within each tire patch, there might exist differences in loading. This can be due to differences in wheel width, stiffness of the tire sidewall, or tire air pressure or, of course, any combination of these differences.

Load variations might also occur as a function of time. These would be greatly affected by the wheel rate, car mass, and the shock absorber coefficient.

So, while "Mister Road" might not be able to tell the difference between Aunt Rose's 1953 Kaiser and Mark Martin's Cup car, there are many factors which would affect the driving impressions of Aunt Rose or Mark Martin.

The Car's Total Friction Circle

The force vectors from the four tire patches can be combined into a single force vector, with the possible directions described by a single Friction Circle. It would be necessary that the center of this circle, in the plan (overhead) view, coincide with the center of gravity of the car. If it did not, a moment balance would indicate that the car was in the process of "spinning." For that matter, it is necessary that the combined force vectors for all four tires pass through the CG at all times, whether the car is operating at the limits of adhesion or not. How is this achieved? Well, that's what the steering wheel is for. As the tires patches were discussed, no mention has been made of the "direction" of the tires, or, in other words, their alignment with the path of the car. That's because I wanted to consider the limiting performance of the tire and...at that limit...tire orientation and steering wheel angle lose their significance. "Handling" becomes simply a matter of patch loading and tire friction coefficient.

This does not mean, however, that the tire alignment with the path (slip angle) and the other tire performance characteristics are not important. They're extremely important! But, for a racecar, they're important only in that they must be properly used to force the origin of the total tire force vector...while the car is operating at less than maximum...to coincide with the origin of the total friction circle as the car reaches the maximum performance which has been in view during the discussion of this page.

An example might be useful: Suppose a car has the preponderance of its weight on the rear wheels and suppose, to compensate, higher tire air pressure is used in the rear tires. With this condition, the car handles quite well at lateral accelerations less than those associated with breakaway. But, at breakaway, the tire air pressure is no longer a significant factor and the origin of the total tire force vector suddenly shifts. Obviously, this is not a desirable condition in a racecar.

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