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.
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. |