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    The Physics of Racing, Part 7: The Traction Budget

    Math and Physics

    Myopic Rhino
    physicist and member of
    No Bucks Racing Club

    P.O. Box 662
    Burbank, CA 91503

    (C)Copyright 1991

    This month, we introduce the traction budget. This is a way of thinking about the traction available for car control under various conditions. It can help you make decisions about driving style, the right line around a course, and diagnosing handling problems. We introduce a diagramming technique for visualizing the traction budget and combine this with a well-known visualization tool, the "circle of traction," also known as the circle of friction. So this month's article is about tools, conceptual and visual, for thinking about some aspects of the physics of racing.

    To introduce the traction budget, we first need to visualize a tire in contact with the ground. Figure 1 shows how the bottom surface of a tire might look if we could see that surface by looking down from above. In other words, this figure shows an imaginary "X-ray" view of the bottom surface of a tire. For the rest of the discussion, we will always imagine that we view the tire this way. From this point of view, "up" on the diagram corresponds to forward forces and motion of the tire and the car, "down" corresponds to backward forces and motion, "left" corresponds to leftward forces and motion, and "right" on the diagram corresponds to rightward forces and motion.

    The bottom surface of a tire viewed from the top as though with "X-ray vision."


    The figure shows a shaded, elliptical region, where the tire presses against the ground. All the interaction between the tire and the ground takes place in this contact patch: that part of the tire that touches the ground. As the tire rolls, one bunch of tire molecules after another move into the contact patch. But the patch itself more-or-less keeps the same shape, size, and position relative to the axis of rotation of the tire and the car as a whole. We can use this fact to develop a simplified view of the interaction between tire and ground. This simplified view lets us quickly and easily do approximate calculations good within a few percent. (A full-blown, mathematical analysis requires tire coordinates that roll with the tire, ground coordinates fixed on the ground, car coordinates fixed to the car, and many complicated equations relating these coordinate systems; the last few percent of accuracy in a mathematical model of tire-ground interaction involves a great deal more complexity.)

    You will recall that forces on the tire from the ground are required to make a car change either its speed of motion or its direction of motion. Thinking of the X-ray vision picture, forces pointing up are required to make the car accelerate, forces pointing down are required to make it brake, and forces pointing right and left are required to make the car turn. Consider forward acceleration, for a moment. The engine applies a torque to the axle. This torque becomes a force, pointing backwards (down, on the diagram), that the tire applies to the ground. By Newton's third law, the ground applies an equal and opposite force, therefore pointing forward (up), on the contact patch. This force is transmitted back to the car, accelerating it forward. It is easy to get confused with all this backward and forward action and reaction. Remember to think only about the forces on the tire and to ignore the forces on the ground, which point the opposite way.

    You will also recall that a tire has a limited ability to stick to the ground. Apply a force that is too large, and the tire slides. The maximum force that a tire can take depends on the weight applied to the tire: _8602_tex2html_wrap72.gif where _8602_tex2html_wrap84.gif is the force on the tire, _8602_tex2html_wrap86.gif is the coefficient of adhesion (and depends on tire compound, ground characteristics, temperature, humidity, phase of the moon, etc.), and _8602_tex2html_wrap88.gif is the weight or load on the tire.

    By Newton's second law, the weight on the tire depends on the fraction of the car's mass that the tire must support and the acceleration of gravity, _8602_tex2html_wrap90.gif. The fraction of the car's mass that the tire must support depends on geometrical factors such as the wheelbase and the height of the center of gravity. It also depends on the acceleration of the car, which completely accounts for weight transfer.

    It is critical to separate the geometrical, or kinematic, aspects of weight transfer from the mass of the car. Imagine two cars with the same geometry but different masses (weights). In a one _8602_tex2html_wrap94.gif braking maneuver, the same fraction of each car's total weight will be transferred to the front. In the example of Part 1 of this series, we calculated a 20%weight transfer during one _8602_tex2html_wrap94.gif braking because the height of the CG was 20%of the wheelbase. This weight transfer will be the same 20%in a 3500 pound, stock Corvette as in a 2200 pound, tube-frame, Trans-Am Corvette so long as the geometry (wheelbase, CG height, etc.) of the two cars is the same. Although the actual weight, in pounds, will be different in the two cases, the fractions of the cars' total weight will be equal.

    Separating kinematics from mass, then, we have for the weight _8602_tex2html_wrap73.gif where _8602_tex2html_wrap96.gif is the fraction of the car's mass the tire must support and also accounts for weight transfer, _8602_tex2html_wrap98.gif is the car's mass, and _8602_tex2html_wrap94.gif is the acceleration of gravity.

    Finally, by Newton's second law again, the acceleration of the tire due to the force _8602_tex2html_wrap84.gif applied to it is _8602_tex2html_wrap74.gif We can now combine the expressions above to discover a fascinating fact: _8602_tex2html_wrap75.gif_8602_tex2html_wrap76.gif The maximum acceleration a tire can take is _8602_tex2html_wrap104.gif, a constant, independent of the mass of the car! While the maximum force a tire can take depends very much on the current vertical load or weight on the tire, the acceleration of that tire does not depend on the current weight. If a tire can take one _8602_tex2html_wrap94.gif before sliding, it can take it on a lightweight car as well as on a heavy car, and it can take it under load as well as when lightly loaded. We hinted at this fact in Part 2, but the analysis above hopefully gives some deeper insight into it. We note that _8602_tex2html_wrap108.gif being constant is only approximately true, because _8602_tex2html_wrap86.gif changes slightly as tire load varies, but this is a second-order effect (covered in a later article).

    So, in an approximate way, we can consider the available acceleration from a tire independently of details of weight transfer. The tire will give you so many gees and that's that. This is the essential idea of the traction budget. What you do with your budget is your affair. If you have a tire that will give you one _8602_tex2html_wrap94.gif, you can use it for accelerating, braking, cornering, or some combination, but you cannot use more than your budget or you will slide. The front-back component of the budget measures accelerating and braking, and the right-left component measures cornering acceleration. The front-back component, call it _8602_tex2html_wrap114.gif, combines with the left-right component, _8602_tex2html_wrap116.gif, not by adding, but by the Pythagorean formula: _8602_tex2html_wrap77.gif

    Rather than trying to deal with this formula, there is a convenient, visual representation of the traction budget in the circle of traction. Figure 2 shows the circle. It is oriented in the same way as the X-ray view of the contact patch, Figure 1 , so that up is forward and right is rightward. The circular boundary represents the limits of the traction budget, and every point inside the circle represents a particular choice of how you spend your budget. A point near the top of the circle represents pure, forward acceleration, a point near the bottom represents pure braking. A point near the right boundary, with no up or down component, represents pure rightward cornering acceleration. Other points represent Pythagorean combinations of cornering and forward or backward acceleration.

    The beauty of this representation is that the effects of weight transfer are factored out. So the circle remains approximately the same no matter what the load on a tire.

    The Circle of Traction.


    In racing, of course, we try to spend our budget so as to stay as close to the limit, i.e. , the circular boundary, as possible. In street driving, we try to stay well inside the limit so that we have lots of traction available to react to unforeseen circumstances.

    I have emphasized that the circle is only an approximate representation of the truth. It is probably close enough to make a computer driving simulation that feels right (I'm pretty sure that "Hard Drivin' " and other such games use it). As mentioned, tire loads do cause slight, dynamic variations. Car characteristics also give rise to variations. Imagine a car with slippery tires in the back and sticky tires in the front. Such a car will tend to oversteer by sliding. Its traction budget will not look like a circle. Figure 3 gives an indication of what the traction budget for the whole car might look like (we have been discussing the budget of a single tire up to this point, but the same notions apply to the whole car). In Figure 3 , there is a large traction circle for the sticky front tires and a small circle for the slippery rear tires. Under acceleration, the slippery rears dominate the combined traction budget because of weight transfer. Under braking, the sticky fronts dominate. The combined traction budget looks something like an egg, flattened at top and wide in the middle. Under braking, the traction available for cornering is considerably greater than the traction available during acceleration because the sticky fronts are working. So, although this poorly handling car tends to oversteer by sliding the rear, it also tends to understeer during acceleration because the slippery rears will not follow the steering front tires very effectively.

    A traction budget diagram for a poorly handling car.


    The traction budget is a versatile and simple technique for analyzing and visualizing car handling. The same technique can be applied to developing driver's skills, planning the line around a course, and diagnosing handling problems.

    converted by: rck@home.net
    Thu Sep 29 14:11:06 PDT 1994

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