Brian Beckman
physicist and member of
No Bucks Racing Club
P.O. Box 662
Burbank, CA 91503
©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: 
where 
is the force on the tire, 
is the coefficient of adhesion (and depends on tire
compound, ground characteristics, temperature, humidity, phase of the moon, etc.),
and 
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, 
. 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 
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
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 
where 
is the fraction of the car's mass the tire must support
and also accounts for weight transfer, 
is the car's mass, and is the acceleration of gravity.
Finally, by Newton's second law again, the acceleration of the tire due to the force applied to it is 
We can now combine the expressions above to discover
a fascinating fact: 
The maximum acceleration a tire can
take is 
, 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 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 
being
constant is only approximately true, because 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 , 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 
, combines with the left-right
component, 
, not by adding, but
by the Pythagorean formula: 
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.
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.
