Driving a car is a classic problem in control. Here, we mean control in the technical sense of control theory, an established branch
of engineering science (once again, I find
http://www.britannica.com to have a very nice, brushup article on that term). In a
moreorless continuous fashion, the driver compares desired direction, speed, and acceleration with actual direction, speed, and
acceleration. The driver uses visual input to sense actual direction and speed; and uses visceral, inertial feedbackthe butt
sensorfor actual acceleration. When the actual differs too much from the desired, the driver applies throttle, brake, steering,
and gear selection to change the actual. These inputs cause the tyres to react with the ground, which pushes back against the tyres,
and through the suspension, pushes the body of the car and driver. Drivers in highspeed circumstances can also generate desired
aerodynamic forces, as in slipstreaming, in the "slingshot pass," and in the Earnhardt TIP manoeuvre, where the driver "takes the
air off" the spoiler of the car in front of him.
Tyres generate forces by sticking and sliding and everything in between. They transmit these forces to the wheels by elastic
deformation. The elastic deformation is extremely complex and theoretical computation requires numerical solution of finiteelement
equations. However, despite fierce trade secrecy, industry and academia have reached apparent consensus in recent years on a formula
that summarizes experimental and theoretical data. This socalled magic formula is not a solution to equations of motiona solution
in such a form is not feasible. It's just a convenient fitting of commonplace mathematical functions to data. It allows one to
compute forces at a higher precision than something like RARS (see
RARS,
A Simple Racing Simulator and
Space, Time, and Rubber of the Physics of
Racing [PhOR]), but without integrating equations. Therefore, forces can be computed within a reasonable time, say in a realtime
simulation program.
To understand the magic formula, we need first to define its inputs, which include slip. Slip is an indirect measure of the fraction
of the contact patch that is sticking. It is frequently asserted in the literature that a tyre with no slip at all cannot create
forces. It has taken me a very long time to accept this assertion. Why can I steer a tintoy car with metal tyres on a hard surface
like Formica? If there is any slip in such tyres, it is microscopic, yet there are sufficient forces to brake and steer, even if just
a little. I finally caved in when I realized that the forces are minute, also. If there is any friction between the tyre and the
surface, there MUST be slip, as it is defined below. Though to a very small degree, the Formica and the tiny contact patches of the
tin tyres actually twist and stretch each other. The only way to eliminate slip completely is to eliminate GRIP completely. Any
grip, and you will have slip.
There are two, slightly different flavours of the magic formula. The longitudinal one is the subject of this entire instalment of
PhOR, and we cover the lateral one in the next instalment. Longitudinal slip is along the mean plane of the wheel and might also be
called circumferential or tangential. It creates braking and accelerating forces. Lateral slip is our old friend
grip
angle, and it generates cornering forces.
We write longitudinal slip as
. It's defined by the actual angular velocity,
,
of a wheel plus tyre under braking or acceleration,
compared to the corresponding angular velocity of the same wheel plus tyre when rolling freely. We write the freerolling angular
velocity as
, where V is the current, instantaneous velocity of the hub
centreline of the wheel with respect to the ground, and Re is the effective radius, a constant defined below. Since the dimensions of
V are length/time, and the dimensions of any radius are length, the ratio,
,
has dimensions of inverse time. In fact, it should be viewed as measuring radians per unit time, radians being the natural,
dimensionless measure of angular rotation. There are
radians in one
rotation or one circumference of a circle, just as the length of the circumference is
times the radius.
Let's begin the discussion of longitudinal slip with a question. Consider a wheeltyre combination with 13inch radius or 26inch
diameter, say a 25550/16 tyre on a 16inch rim. The "50" in the tyre specification is the ratio of the sidewall height to the tread
width, which is also written into the specification as 255, millimetres understood. We get a sidewall height of 50 percent of 255 mm,
which is 5.02 inch. Therefore, the total, unloaded radius, half of the treadtotread diameter, is about 5 + 16/2 = 13 Inch. Now
consider a rigid tyre of the same radius, made, say, of steel or of wood with an iron tread like old Western wagon wheels. The
question is whether, given a certain constant hub velocity, pneumatic tyres spin faster than, slower than, or at the same speed as
equivalent rigid tyres?
At first glance, one might say, "Well, faster, obviously. Since the pneumatic tyre compresses radially under the weight of the car,
its radius is actually smaller than the unloaded radius at the point of contact, where it sticks and acquires linear velocity equal
in magnitude and opposite in direction to the hub velocity. Since smaller wheels spin faster than larger ones at the same speed, the
pneumatic tyre spins faster than the equivalent rigid tyre of the same unloaded radius. Let the unloaded, natural radius of the
pneumatic tyre be R, also the radius of the equivalent solid tyre. If the hub has velocity V, the solid tyre spins with angular
velocity
. Since the loaded radius, of the pneumatic tyre, Rl, is smaller
than R, V/Rl, the angular velocity of the loaded pneumatic tyre, must be larger than V/R."
This is partly correct. The pneumatic tyrewheel combination does spin faster than a rigid wheel of the same unloaded radius, but it
does not spin as fast as a rigid wheel of the same loaded radius, which is the height of the hub centre off the ground under load.
The reason is that the tyre also compresses circumferentially or tangentially, setting up complex longitudinal twisting in the
sidewall. The tangential speed of a particle of tread varies as the particle goes around the circumference of the tyre.
Let's mentally follow a piece of tread around as the wheel, not necessarily the tyre, turns at a constant radial velocity,
.
Imagine a plug of yellow rubber embedded in the tread, so that you could visually track it or photograph it with a movie camera or
strobe system as it moves around the circumference. The rubber of the tread does not travel at constant speed, even though the wheel
supporting the tyre does. At the top of the tyre, the radius is almost exactly R, the unloaded radius, so the tread moves with
tangential velocity
. As the yellow plug rolls around and approaches the
contact patch from the front, it slows down in the bunched up area at the leading edge of the contact patchjust forward of it. There
is a bunchedup area, because the tyre is made up of elastic material that gets squeezed and stretched out of the contact patch and
piles up ahead of the contact patch as it rolls into it from the direction of the leading edge. Eventually, the plug enters the patch,
in the centre of which it must move at speed
relative to the hub centre,
that is, backwards at a speed dictated by the loaded radius and the wheel velocity. We've assumed that the plug is not slipping on the
ground at the point where it has speed
with respect to the hub. This
means that it has speed zero with respect to the ground at that point.
The average of the tangential velocities around the wheel defines the effective radius, Re, as follows. Let
measure the angular position, from 0 to
, around the wheel. Suppose we
knew the tangential velocity with respect to the hub centre, V(
), at every .
We could easily measure this with our strobe light and cameras. V(
) gives us
the radius at every angular position via the equation V(
)/ = R(
),
where is the constant angular velocity of the wheel. The average would be computed by the following integral:
Let's run some numbers. 10 mph is 14
feet/second or 176 inches/second.
With an unloaded circumference of
inch/revolution, we get 176/
= 2.154 revs per second, or 129 RPM for each 10 MPH. Under ordinary circumstances, the effective radius will be no more than a few
percent less than then the unloaded radius, and the RPMs should be, then, a few percent more than 129 RPM per 10 MPH. At 100 MPH, the
tyre is under considerable stress and spins at something over 1,300 RPM.
Now we're in a position to define longitudinal slip, written
. We want a
quantity that vanishes when the wheel rolls at constant speed, increases when the wheel accelerates the car by pulling the contact
patch backwards, and decreases below zero when then wheel brakes the car by pushing the contact patch forward. Under acceleration, the
wheel and tyre combination will tend to spin a little faster than it would do while free rolling. We already know that, for a given V,
the freerolling angular velocity is
, by definition. The actual angular
velocity,
, then, is higher under acceleration. So, if we know V,
,
and the constant Re, then we can define the longitudinal slip as the ratio, minus 1, so that it's zero under freerolling
conditions:
Just looking at this formula, a freerolling wheel has
,
= 0
a lockedup wheel under braking has
= 0,
= 1
and an accelerating wheel has a positive of any value.
The magic formula yields the longitudinal force, in Newtons, given some constants and dynamic inputs. The formula takes eleven
empirical numbers that characterize a particular tyre {b0, b1 ...b10}. The dynamic parameters are Fz, or weight, in KiloNewtons on
the tyre, and the instantaneous slip,
. The eleven numbers are measured for
each tyre. We borrow an example from Motor Vehicle Dynamics by Giancarlo Genta. On page 528, he offers the following numbers for a
car that appears to be a Ferrari 308 or 328, to which I have added dimensions:
b_{0} 
1.65 
dimensionless 
b_{1} 
0 
1/MegaNewton 
b_{2} 
1688 
1/Kilo 
b_{3} 
0 
1/MegaNewton 
b_{4} 
229 
1/Kilo 
b_{5} 
0 
1/KiloNewton 

b_{6} 
0 
1/(KiloNewton)^{2} 
b_{7} 
0 
1/KiloNewton 
b_{8} 
10 
dimensionless 
b_{9} 
0 
1/KiloNewton 
b_{10} 
0 
dimensionless 

Though the majority of these values are zero for the tyres on this car, it is by no means always the case. In fact, the
'largesaloon' example just before the (alleged) Ferrari in Genta's book has no zeros.
We build up the magic formula in stages. The first helper quantity is µp = b1 Fz + b2. This is an estimate of the peak, longitudinal
coefficient of friction, fitted as a linear function of weight (see
The Traction Budget).
From this definition, we begin to see what's going with the dimensions. A typical, streetable sports car might weigh in at 3,000 lbs,
which is about 3,000 / 2.2 = 1,500 * 0.9 = 1,350 kg, which is about 1,350 * 9.8 = 13,200 Newtons, or 13.2 KiloNewtons (look, ma, no
calculator!). Let's assume each tyre gets a quarter of that to start off with, or 3.3 KN. b1 multiplies that number to give us
something with dimensions of KiloNewton/MegaNewton, which we write simply as 1/Kilo (inventing units onthefly, one Mega = 1 Kilo
squared). b2 has the same dimensions, so it's kosher to add it in, yielding µp = 1688 / Kilo in this case. The next step is the helper
D = µpFz, which will be in Newtons. We now see the reason for the 1/Kilo unit. In our case, we get about D = (1700  12) * 3.3 = 5610
 40 = 5570 N. The important point is that D is linear in Fz, so µp acts, mathematically, like a coefficient of friction, as promised.
b2 is a pretty direct measurement of stickiness, times 1,000 for convenience. This model tyre has a coefficient of friction of almost
1.7! Not my data, man.
The next step is to compute the product of a new helper, B, times b0 and the aforecomputed D. The magicians who created the formula
tell us that Bb0D = (b3 Fz2 + b4Fz) exp(b5Fz). This slurps up a few more of the magical eleven empirical numbers, and a pattern
emerges. These bi numbers serve as coefficients in polynomial expressions over Fz. So, b5Fz is dimensionless, as must be the argument
of the exponential function. b3Fz2 + b4Fz has dimensions of Newtons, as does the entire product. Therefore, B must be dimensionless.
We need B in the next step, so let's solve for it now:
,
Where we've been able symbolically to divide out one factor of Fz, convenient especially for numerical computation, where overflow is
an everpresent hazard. Continuing with our numerical sample, b3Fz + b4 = 229 / Kilo, the exponential is unity, and the numerator
is
yielding B = 229 / 2786 = 0.0822. Most importantly, B depends only weakly on Fz. In the sample case, not at all, because b1 = b3 = b5
= 0, but there are lots of other ways to characterize the algebraic dependence of B on Fz.
The next step is to account for the longitudinal slip with another helper, S = (100
+ b9Fz + b10); in our sample case, this reduces
to just S = 100
.
Only one more helper is needed, and that's E = ( b6Fz2 + b7Fz + b8), very straightforward. The final formula is
Once again, don't try to find any physics in here: it's just a convenient formula that fits the data reasonably well. Plugging in
numbers for
= 0, because that's an easy sanity check to do in our heads, we
see immediately the result is zero. Let's try S = 10, ten percent slip. SB = 0.822, tan1(0.822) = 0.688, E = 10, so the argument
of the outer arctangent is SB  10 * (0.266) = SB + 2.66 = 3.48, tan1(3.48) = 1.29, 1.29 b0 = 2.13, sin(2.13) = 0.848, and,
finally, D * 0.848 = 4720 Newtons. Lots of longitudinal force for a 3,300 N vertical load!
Let's plot the whole formula:
The horizontal axis measures S = 100
, which is really just slip in percent. The deep axis, going into the page, measures Fz from 5
KN, nearest us, to zero in the back. The vertical axis measures the result of applying the formula to our model tyre, so it's
longitudinal forceforce of launching or braking. Notice that for a load of 5 KN, the model tyre can generate almost 8 KN of force.
Very sticky tyre, as we've already noticed! Also notice that the generated force peaks at around
= 0.08, or 8 percent. The peak would be something one could definitely feel in the driver's seat. Overcooking the throttle or brakes would produce a palpable reduction in gforces as the tyres start letting go. Worse than that, increasing braking or throttle beyond the peak leads to reduced grip. This is an instability area, where increasing slip leads to decreasing grip.
Finally, note that the function behaves roughly linearly with Fz, showing that it acts like a Newtonian coefficient of friction,
albeit a different one for each value of slip.