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 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 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 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 Fx
input to Fy
+
+
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 a3 in Part 22 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.
and
for the longitudinal and lateral forces. Here they are, in isolation:
for which 
for which 
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 
, 
, 
,
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 
. These are
pure numbers, so they're commensurable. They are unity when 
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:
, should have its maximum all the way
around the traction circle, that is, whenever 
.
and 
.
, 
, 
as the input, with the appropriate algebraic signs, satisfies
requirements 1. Multiplying the outputs by the ratio of s to 
showing the traction circle very clearly:
