In this instalment of the Physics of Racing, we complete the program begun last time to combine the magic formulae of

The Magic Formula: Longitudinal Version and

The
Magic Formula: Lateral Version, so that we have a model of tire forces when turning and braking or turning and accelerating at the
same time.

The Magic Formula: Longitudinal Versionand

The
Magic Formula: Lateral Version 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 straight-ahead 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

Combination Slip. 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 for-fun 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 information-I 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

The Magic Formula: Longitudinal Versionand

The
Magic Formula: Lateral Version:

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 Fx 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 a9 and a10's being non-zero. 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 zero-crossing of the slope, occurs at

= 3.273°

Once we find the maxima, we can create new, non-dimensional 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 vector-valued 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 Fx 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 Fy 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 Fx with increasing , we must decrease the output of Fx.
Likewise, while we increase the input to Fy 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 front-wheel-drive 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 error-free over the years, but I caught three small errors in

The Magic Formula: Lateral Version 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 a3 in

The Magic Formula: Lateral Version
should be N/Degree, not just N, because a3 furnishes the dimensions for B, which always appears in the combination SB, and has dimensions
of degrees. Finally, the dimensions for a6 are 1/KN, not KN.