Part 25: Combination Grip

©Copyright February 2001

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 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

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 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

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

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
*F*must_{x}*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*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._{y} - 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*with increasing , we must_{x}*decrease*the output of*F*. Likewise, while we increase the input to_{x}*F*with increasing , we must decrease the output. This requirement is a bit of a balancing act because often there_{y}*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 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. |