The Physics of Racing,
Part 7: The Traction Budget
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 tyre in contact with the
ground. Figure 1 shows how the bottom surface of a tyre 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 tyre. For the rest of
the discussion, we will always imagine that we view the tyre this way. From this point of
view, "up" on the diagram corresponds to forward forces and motion of the tyre
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 tyre viewed from the top as
though with "X-ray vision."
The figure shows a shaded, elliptical region, where the tyre presses against the
ground. All the interaction between the tyre and the ground takes place in this contact
patch: that part of the tyre that touches the ground. As the tyre rolls, one bunch of
tyre 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 tyre and the car as a whole. We can use this fact to develop a simplified view of the
interaction between tyre and ground. This simplified view lets us quickly and easily do
approximate calculations good within a few percent. (A full-blown, mathematical analysis
requires tyre coordinates that roll with the tyre, 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 tyre-ground
interaction involves a great deal more complexity.)
You will recall that forces on the tyre 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
tyre 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 tyre and to ignore the forces on the ground, which point the opposite way.
You will also recall that a tyre has a limited ability to stick to the ground. Apply a
force that is too large, and the tyre slides. The maximum force that a tyre can take
depends on the weight applied to the tyre: F  µW
where F is the force on the tyre, µ is the coefficient of
adhesion (and depends on tyre compound, ground characteristics, temperature, humidity,
phase of the moon, etc.), and W is the weight or load on the tyre.
By Newton's second law, the weight on the tyre depends on the fraction of the car's
mass that the tyre must support and the acceleration of gravity, g = 32.1 ft / sec2.
The fraction of the car's mass that the tyre must support depends on geometrical factors
such as the wheelbase and the height of the centre 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 g braking manoeuvre, 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 g 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 W = f(a)mg
where f(a) is the fraction of the car's mass the tyre must support and also
accounts for weight transfer, m is the car's mass, and g is
the acceleration of gravity.
Finally, by Newton's second law again, the acceleration of the tyre due to the force F
applied to it is a = F / f(a)m We can now combine
the expressions above to discover a fascinating fact:
a = F / f(a)m amax
The maximum acceleration a tyre can take is µg, a constant, independent
of the mass of the car! While the maximum force a tyre can take depends very much
on the current vertical load or weight on the tyre, the acceleration of that tyre does not
depend on the current weight. If a tyre can take one g 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 amax
being constant is only approximately true, because µ changes slightly as
tyre 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 tyre
independently of details of weight transfer. The tyre 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 tyre that will give you one g, 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 ay, combines with the
left-right component, ax, 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 tyre.
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,
tyre loads do cause slight, dynamic variations. Car characteristics also give rise to
variations. Imagine a car with slippery tyres in the back and sticky tyres 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 tyre 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 tyres and a small circle for
the slippery rear tyres. 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 tyres very effectively.
A traction budget diagram for a poorly handling car.
The traction budget is a versatile and simple technique for analysing 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. |
