In this instalment of the Physics of Racing, we complete the
program begun last time to combine the magic formulae of parts 21 and 22, so that we have
a model of tire forces when turning and braking or turning and accelerating at the same
time. Parts 21 and 22 introduced the magic formulae. The first one takes longitudinal slip
as input and produces longitudinal grip as output. The other one takes lateral slip as
input and produces lateral grip. Slip depends primarily on driver inputs, grip is force
generated at the ground. Longitudinal means in the straightahead direction. Lateral
means sideways, as in the forces for turning. Since the magic formulae work only in
isolation, we have work to do to model turning and braking at the same time and turning
and accelerating at the same time. Last time, we vectorized slip  the input  to come
up with combination slip, captured in the vector slip velocity.
That vector measures the velocity of the contact patch with respect to (w.r.t.) the ground
in one, handy definition. This time, we first boil down combination slip to new inputs for
the old magic formulae. In the old magic formulae, we measure longitudinal slip as a
percentage of unity, that is, as a percentage of breakaway sliding; and we measure lateral
slip as an angle in degrees. These are not commensurable, meaning that we do
not use the same units of measurement for both kinds of slip. That's why there was a big,
fat question mark in the vector slot for combination slip in one of the tables in part 24.
Once we make them commensurable, then we stitch the magic formulae together to get one
vector gripping force as a function of one vector slip. This finally allows us to compute
the forces delivered by a tire under combination control inputs.
Once again, we are in uncharted territory, so take it all in the forfun spirit of this
whole series of articles. I don't represent anything I do here as authoritative racing
practice. I only claim to be bringing the fresh perspective of a stubbornly naïve
physicist to the problems of racing cars as an amateur. The standard practice of the
professional racing engineering community may be completely different. This is the Physics
of Racing, not the Engineering of Racing. I'm after the fundamental principles
behind the game. I use techniques that may be foreign to the engineers that build and race
cars professionally. My results may not be precise enough for final application. I may
take approximations that simplify away things that are actually critically important. On
purpose, I'm figuring things out on my own. Often, this helps me understand published
engineering information better. Just as often, it helps me debunk and debug the
conventional wisdom. If you find mistakes, gaffs, or laughable dumb stuff, or if you know
better ways to do things, I encourage you to fire up debate, publish rebuttals, or write
to me directly. I've done my best to track down the latest and greatest information, but
I've found lots of errors, ambiguities, and inexplicabilities in the open literature. I
also suspect a conspiracy, meaning that I'd bet that the tyre manufacturers and pro racing
teams don't publish their best informationI certainly wouldn't if I were they.
Disclaimers out of the way, we now have enough tools on the table to combine the two
magic formulae. Recall the formulae from parts 21 and 22: and for the
longitudinal and lateral forces. Here they are, in isolation:
There are a lot of ways we could stitch them together. This is not the kind of
situation where there is one right answer. Instead, in the absence of hard theory or
experimental data, we have the freedom to be creative, with the inevitable risk of being
wrong. We pick a method that satisfies some simple, intuitive, physical requirements.
First, we must put the inputs on the same footing. Ask "what is the value of for which has its
maximum, and what is the value of for which has its maximum?" Call these two values and . They are
constants for given F_{x} and : characteristics of a particular tyre and car and
surface. So, we can finesse the notation and just write and . The
maxima identify points on the rim or edge of the 'traction circle'. The grip decreases
when
exceeds and when exceeds . Let's illustrate with , = 0, and the
constants from Genta's alleged Ferrari. Once we substitute all that in (and we'll let you
check our arithmetic from the data in prior articles), we get
We evaluate these equations for = 0, = 0, getting , , and
showing a small lateral force (about 16 lbs) due to conicity and ply steer. The source of
that problem is the constant offset in S, which results from a_{9}
and a_{10}'s being nonzero. We just set them to zero for now. Let's plot , slip
on the horizontal axis and grip on the vertical:
The maximum positive grip occurs, just by eyeball, around = 0.08. To the
left of the maximum, adding more slip  more throttle  generates more grip. To the right
of the maximum, adding more slip generates less grip. That's where we've lost
traction. We can find the maximum precisely by plotting the slope of this curve,
since the slope is zero right at the maximum:
Using secret physicist methods, I've found that this curve crosses the horizontal axis
 that is, goes to zero  at precisely = 0.0796. This was so much fun that we'll just
do it again for . First, the curve proper:
Notice the same kind of stability situation as we saw before. To the left of the
maximum, more slip  more steering  means more grip. To the right of the maximum, more
slip means less grip. Here's the slope:
We find that the maximum of the original curve, the zerocrossing of the slope, occurs
at = 3.273°
Once we find the maxima, we can create new, nondimensional quantities by scaling and by these
values, namely . These are pure numbers, so they're commensurable. They are
unity when and have the values of maximum traction in isolation of one another. We can then
write new functions and which have their maxima at s = 1
and a = 1. We seek a vectorvalued function of s and a
whose longitudinal x component expresses the longitudinal force component and whose
lateral y component expresses the lateral force component under combination slip.
Build this up from and so that it satisfies the following requirements:
 The magnitude of , that is, , should have its maximum all the way around the
traction circle, that is, whenever .
 The individual components should agree completely with the old magic formulae whenever
there is pure longitudinal or pure lateral slip. Mathematically, this means that and .
 For a fixed, positive value of (throttle), as (steering) increases, the input to F_{x}
must increase. Say what? Here's the idea. Suppose you're on the limit of
longitudinal grip. When steering increases, the forward grip limit must be exceeded, and a
great way to model that is just to shove the input over the cliff to larger . We want
the same behaviour the other way, namely, for a fixed value of (steering), as (throttle)
increases, the input to F_{y} increases to model the fact that at maximum
steering adding throttle exceeds the limit. We model the three other cases entailing
negative values of and below.
 Below the limits, we do not want dramatic increases in forward grip when steering
increases, and vice versa. So, although we must increase the input to F_{x}
with increasing , we must decrease the output of F_{x}. Likewise, while
we increase the input to F_{y} with increasing , we must decrease the
output. This requirement is a bit of a balancing act because often there is an
increase of steering grip with braking, as we see in the technique of trail braking.
However, there is usually no increase in steering grip with increased throttle in a
frontwheeldrive car, even below the limits. In the modelling of combined effects like
this, it's necessary to include weight transfer with the combination grip formula. That
simply means that until we have a full model of the car up and running, we won't be able
to evaluate fully the quality of this combination magic grip formula.
The following table fleshes out requirement 3 for the cases of braking ( < 0 ) or
turning left ( < 0 ). The essential idea is that if the magnitude of either parameter
increases, then the magnitudes of the inputs to the old magic formulae must increase, but
honouring the algebraic signs. If a parameter is positive, it should get more positive as
the magnitude of the other parameter increases. Similarly, if a parameter is negative, it
should get more negative as the magnitude of the other parameter increases.
sgn( ) 
sgn( ) 
Trend 
Trend 
input to F_{x} 
input to F_{y} 
+ 
+ 
increasing 
fixed 
increasing 
increasing 
+ 
+ 
fixed 
increasing 
increasing 
increasing 
+ 
 
increasing 
fixed 
increasing 
decreasing 
+ 
 
fixed 
decreasing 
increasing 
decreasing 
 
+ 
decreasing 
fixed 
decreasing 
increasing 
 
+ 
fixed 
increasing 
decreasing 
increasing 
 
 
decreasing 
fixed 
decreasing 
decreasing 
 
 
fixed 
decreasing 
decreasing 
decreasing 
Without further ado, here's our proposal for the combination magic grip
formula:
Using as the input, with the appropriate algebraic signs, satisfies
requirements 1. Multiplying the outputs by the ratio of s to and a to magically
satisfies requirements 2, 3, and 4. There is, in fact, plenty of freedom in the choice of
the outer multiplier: strictly speaking, any power of the ratios would do for requirements
2 and 4, and some care will be required to get the signs right for requirement 3. Until we
have a good reason to change it, we'll just go with the ratio straight up,
especially since it automatically gets the signs right. We close this instalment with a
plot of the magnitude showing the traction circle very clearly:
The stability criteria are visually obvious, here. If the current, commensurable slip
values, s and a, are inside the central "cup" region, then
increasing either component of slip increases grip. If they're outside, then increasing
slip leads to decreasing grip and the driver is in the "deep kimchee" region of
the plot.
ERRATA: The Physics of Racing series has been fairly errorfree over the years,
but I caught three small errors in part 22 whilst going over it for this instalment. The
good news is that they did not affect any final results. I defined the WHEEL frame at the
wheel hub but later I implied that it is centred at the contact patch (CP). In fact, the
frame at the CP is the important one, and we call it TYRE from now on, avoiding the
ambiguous "WHEEL". We never actually used the improperly defined WHEEL frame,
so, again, final results were not affected. Also, the dimensions for a_{3}
in Part 22 should be N/Degree, not just N, because a_{3} furnishes the
dimensions for B, which always appears in the combination SB, and has
dimensions of degrees. Finally, the dimensions for a_{6} are 1/KN, not KN. 
