The goal in this and the next instalment of the Physics of Racing is
to combine the magic formulae of parts 21 and 22, so that we have a model of
tyre forces when turning and braking or turning and accelerating at the same
time. In this part, we figure out combination slip, and in the next
instalment, we figure out combination grip. Roughly speaking, slip is the input
and grip is the output to our model. Slip comes from control inputs on brakes,
throttle and wheel, grip comes from reaction forces of the ground on the tyres.
The regular magic formulae apply only to a tyre generating longitudinal or
lateral forces in isolation, that is, to a tyre accelerating or braking and not
turning, or a tyre turning but not accelerating or braking. In part 7, we
approximated the response under combination slip by noting that it follows the
circle of traction. A tyre cannot deliver maximal longitudinal grip when it's
delivering lateral grip at the same time, and vice versa. According to my
sources, modelling of combination slip and grip is an area of active research,
which means we are on our own, once again, in the original, risk-taking spirit
of the Physics of Racing series. In other words, we're going way out on a
limb and this could all be totally wrong, but I promise you lots of fun physics
on the journey.
From part 21, recall our definition for the longitudinal slip,
, the
input to the longitudinal magic formula
where V is the forward speed of the hub w.r.t. ground, Re
is a constant, the effective radius, for a given tyre, and
Re
is the backward speed of the CP w.r.t. the hub. Therefore,
Re- V
is the backward speed of the CP w.r.t. EARTH. A slick technique for proving
this, and, in fact, for figuring out any combinations of relative
velocities (see part 19) is as follows. Write V = HUB - GD,
meaning speed of the HUB relative to the ground (GD). Now write Re= -(CP - HUB)
meaning the backward speed of the CP relative to the HUB; the overall minus sign
outside reminds us that we want Re positive when
the CP moves backwards w.r.t. the hub. Now, just do arithmetic:
voila, backward speed of the CP w.r.t. the ground. This realization
gives us intuition into the sign of
:
if and only if the CP moves backwards
faster than the hub moves forward; the car accelerates forward-visualize that in
your head; in that case,
Re
is greater than V;
Re- V
is greater than zero; and is positive.
It turns out that we developed this formula only for the case when V
is positive, that is, the car is moving forward. And, in fact, the formula only
works in that case. To generalize it to cars moving in reverse, we'd best
analyse it in excruciating detail. A moment's reflection reveals that there are
eight cases: two signs for V, two signs for
Re,
and two cases for whether the absolute value of V is greater than the
absolute value of
Re,
yielding eight = (2 2
2)
possibilities, which have the following physical interpretations:
Car (hub) moving forward, CP moving slowly forward w.r.t. ground,
resisting car motion. This models driving slowly in reverse gear while
moving forward. Slowly, here, means slowly relative to V,
or, precisely, that
, where
is the
absolute value or unsigned magnitude of
Re and
is the absolute value or unsigned magnitude of V.
Same as above, just with CP moving quickly forward, that is
with .
Car moving forward, CP moving slowly backwards, just not quickly enough to
accelerate the car. This is braking or engine braking in forward gear. Wheel
lockup while moving forward falls in this case, too.
Car moving forward, CP moving quickly backwards, accelerating the car
forward.
Car moving backward, CP moving slowly forwards, just not quickly enough to
accelerate the car backward. This is braking or engine braking in reverse,
and wheel lockup in reverse falls in this case.
Car moving backward, CP moving quickly forward, accelerating in reverse.
Car moving backward, CP moving slowly backward, resisting motion.
Car moving backward, CP moving quickly backward, resisting motion.
We've caught all this in the following diagram, in which we have drawn V and
Re
as arrows, pointing in the actual direction that the hub moves w.r.t. the ground
and the CP moves w.r.t. the hub, respectively. Algebraically, V and
Re
have opposing sign conventions, so
Re
is negative when its arrow points up. In looking at this table, note that the longitudinal force
Fx has the same sign as
.
When Fx and
are positive, the
car is being forced forward by the ground's reacting to the tyres. When they're
negative, the car is being forced backwards. So, to figure out which way the car
is being forced, just look at the sign of
.
Inspection of this table reveals that the following new formula works
in all cases:
Where the numerator, Re
- V is the signed difference
of the two speeds and the denominator is unsigned. It is perhaps
surprising that there is so much richness in such a little formula. However, it
is precisely this richness that we must maintain as we add steering, that is,
lateral slip angle at the same time. The best way to do that is to vectorize the
formula so that the algebraic signs of the vector components take the place of
the signed quantities V and
Re.
The approach here parallels the approaches of parts 16 and 19. We want the signed component Vx
to take the place of the old, signed V, the signed component Lx of
the slip velocity L to take the place of the old
Re,
and V now to denote the unsigned magnitude of the vector V, that
is .
The next table summarizes these changes:
Quantity
old notation
new notation
vector
signed, forward speed of hub w.r.t. EARTH
V
Vx
V
signed, backward speed of CP w.r.t. hub
-Wx
W
unsigned magnitude of hub speed
V
or V
signed, backward speed of CP w.r.t. EARTH
-Lx
L = V + W
signed, longitudinal slip
?
Slip velocity, L [Lx, Ly,
Lz] = V + W is the plain-old
vector velocity of the CP w.r.t. EARTH with no secret sign convention to confuse
things. As an aside, we note that when the car sticks to the ground on flat
road, we may assume Lz = 0. W is CP velocity
w.r.t. hub. In the TYRE system, W has only a (signed) x-component, that
is, WTYRE = [Wx, 0, 0].
These definitions hold whether the car is moving forward or backward,
accelerating or braking.
The big question mark in the table indicates that we do not have a vector for
combination slip because we measure its longitudinal and lateral components
differently, as a ratio and as an angle, respectively. Note that, since lateral
slip is
the angle made by V in the TYRE system, it is
.
Since L = V + W, it's easy to see that
,
which is a most convenient expression, though some attention must be paid to
the quadrant in which the angle falls. We resolve this in the next two
instalments of PhORS as we stitch together the two magic formulae to make
Combination Grip.
But first, let's update the big diagram, showing all eight cases with a
little slip angle thrown into the mix, and the vector sum, L = V + W,
replacing the ad hoc, signed quantities of the old notation. The sign of
the slip angle
does
not introduce new cases so long as
because the
right-hand and left-hand cases are precisely symmetrical. The nice thing, here,
is that we can treat all eight cases the same way-the nature of vector math
takes care of it because the magnitude of a vector is always unsigned. Using
signed, scalar quantities, we had to dissect the system and introduce absolute
value to get everything to work. Absolute value has always struck me as a
kind of crock or kludge to use when the math is just not sufficiently
expressive. The main contribution of this instalment is to fix that problem.
, the
input to the longitudinal magic formula
Re
is the backward speed of the CP w.r.t. the hub. Therefore,
2
, where
is the
absolute value or unsigned magnitude of
is the absolute value or unsigned magnitude of V.
.
.
The next table summarizes these changes:
or V
[Lx, Ly,
Lz] = V + W is the plain-old
vector velocity of the CP w.r.t. EARTH with no secret sign convention to confuse
things. As an aside, we note that when the car sticks to the ground on flat
road, we may assume Lz = 0. W is CP velocity
w.r.t. hub. In the TYRE system, W has only a (signed) x-component, that
is, WTYRE = [Wx, 0, 0].
These definitions hold whether the car is moving forward or backward,
accelerating or braking.
is
the angle made by V in the TYRE system, it is
.
Since L = V + W, it's easy to see that
,
because the
right-hand and left-hand cases are precisely symmetrical. The nice thing, here,
is that we can treat all eight cases the same way-the nature of vector math
takes care of it because the magnitude of a vector is always unsigned. Using
signed, scalar quantities, we had to dissect the system and introduce absolute
value to get everything to work. Absolute value has always struck me as a
kind of crock or kludge to use when the math is just not sufficiently
expressive. The main contribution of this instalment is to fix that problem.