In this instalment, we review the other side of the magic formula: the
one that computes lateral or cornering forces from slip angles (or grip angles). This
formula is sufficiently similar to the longitudinal version of Part 21 that we can skip
many preliminaries. But it's sufficiently different as to require careful exposition,
leading us to define coordinate frames that will serve us throughout the rest of the Physics
of Racing series. This instalment will be one to keep on hand for future reference. Diving
right in, just like its longitudinal sibling, this formula requires some magical
constants, fifteen of them this time. Again, from Genta's possibleFerrari data sheet:
a_{0} 
1.799 
dimensionless 
a_{1} 
0 
1/MN 
a_{2} 
1688 
1/Kilo 
a_{3} 
4140 
N 
a_{4} 
6.026 
KN 
a_{5} 
0 
1/Degree 
a_{6} 
0.3589 
KN 

a_{7} 
1 
dimensionless 
a_{8} 
0 
dimensionless 
a_{9} 
6.111/1000 
Degree/KN 
a_{10} 
3.224/100 
Degree 
a_{11} 
0 
1/MN  Degree 
a_{12} 
0 
1/Kilo Degree 
a_{13} 
0 
1/Kilo 
a_{14} 
0 
N 

where N is Newton, KN is KiloNewton, and MN is MegaNewton. As with the
longitudinal magic formula, there are lots of zeros in this particular sample case, but
let us not confuse particulars with generalities. The formula can account for much more
general cases.
The first helper is the peak, lateral friction coefficient µ_{yp} = a_{1}F_{z} + a_{2},
measured in inverse Kilos if F_{z} is in KN. Next is D = µ_{yp}F_{z},
which is a factor with the form of the Newtonian model: normal force times
coefficient of friction. In our sample, a_{1} is zero, so µ_{yp }acts
exactly like a Newtonian friction coefficient. In all cases, we should expect a_{1}F_{z
}to be much smaller than a_{2} so that it will be, at most, a
small correction to the Newtonian behaviour.
To get the final force, we correct with the following empirical factor:
This has exactly the same form as the empirical correction factor in the longitudinal
version, but the component pieces, S, B, and E are different, here.
where is the slip angle and is the camber angle of the wheel. In practice, we must
carefully account for the algebraic signs of the camber angles so that the forces make
sense at all four wheels. The usual negative camber, by the 'shop' definition, as measured
on the wheelalignment machine, will generate forces in the positive Ydirection on the
righthand side of the car and in the negative Ydirection on the lefthand side of the
car. This comment makes much more sense after we've covered coordinate frames, below.
As before, we get B from a product, albeit one of greatly different form
where is the absolute value of the camber angle, that is, a positive number no
matter what the sign of . This gives
Almost done; include and sneak in an additive correction for ply steer and conicity,
which we'll leave undefined in this article:
To arrive at the final formula
This form is almost identicalin formto the longitudinal version of the magic formula.
The individual subcomponents are different in detail, however.
The most important input is the slip angle, . This is the difference
between the actual path line of the car and the angle of the wheel. To be precise, we must
define coordinate systems. We'll stay close to the conventions of the Society of
Automotive Engineers (SAE), as published by the Millikens in Race Car Vehicle Dynamics.
Note that this may differ from some frames we've used in the past, but we're going to
stick with this set. There's a lot of intense verbiage in the following, but it's
necessary to define precisely what we mean by wheel orientation in all generality. Only
then can we measure slip angle as the difference between the path heading of the car and
the wheel orientation.
First, is the EARTH frame, whose axes we write as {X, Y, Z}.
The Z axis is aligned with Earth's gravitation and points downward. The
origin of EARTH is fixed w.r.t. the Earth and the X and Y axes point in
arbitrary, but fixed, directions. A convenient choice at a typical track might be the
centre of start/finish with X pointing along the direction of travel of the cars up
the main straight. All other coordinate frames ultimately relate back to EARTH, meaning
that the location and orientation of every other frame must be given w.r.t. EARTH,
directly or indirectly. The next coordinate frame is CAR, whose axes we
write as {x, y, z}. This frame is fixed w.r.t. the sprung mass of the
car, that is the body, with x running from tail to nose, y to driver's
right, and z downward, roof through seat. Its instantaneous orientation w.r.t.
EARTH is the heading, . Precisely, consider the line formed by the intersection of EARTH's XY
plane with CAR's xz plane. The angle of the that line w.r.t. EARTH's X axis
is the instantaneous heading of the car. It becomes undefined only when the car it points
directly upstanding on its tailor directly downstanding on its nose. To emphasize, heading
is measured in the EARTH frame.
The next coordinate frame is PATH. The velocity vector of the car traces
out a curve in 3dimensional space such that it is tangent to the curve at every instance.
The Xdirection of PATH points along the velocity vector. The Zdirection of PATH is at
right angles to the X direction and in the plane formed by the velocity vector and the
Zdirection of EARTH. The Y direction of PATH completes the frame such that XYZ form an
orthogonal, righthanded triad. The path of the car lies instantaneously in the XY plane
of PATH. PATH ceases to exist when the car stops moving. Path heading is the
angle of the projection of the velocity vector on EARTH's XY w.r.t. the Xaxis of EARTH.
Milliken calls this course angle, (Greek upsilon). Path heading, just like heading, is measured in
the EARTH frame. The sideslip angle of the entire vehicle is the path
heading minus the car heading, . This is positive when the right side of the car
slips in the direction of travel.
The next set of coordinate frames is ROAD_{i}, where i
varies from 1 to 4; there are four frames representing the road under each wheel, numbered
as 1=Left Front, 2=Right Front, 3=Left Rear, 4=Right Rear. Each ROAD_{i} is
located at the force centre of its corresponding contact patch at the point R_{i} = (R_{i}^{x},
R_{i}^{y}, R_{i}^{z}) w.r.t. EARTH. This point moves
with the vehicle, so, more pedantically, the origin of ROAD_{i} is R_{i}(t)
written as a function of time. To get the X and Y axes of ROAD_{i}, we begin with
a temporary, flat, coordinate system called TA_{i} aligned with
EARTH and centred at R_{i}, then elevate by an angle 90° < < 90°,
to get temporary frame TB_{i}, and bank by an angle 90° < < 90°, in
that order, as illustrated below:
Consider any point P in space with coordinates P = (P^{x}, P^{y},
P^{z}) w.r.t. EARTH. A little reflection reveals that its location w.r.t. TA_{i}
is P_{Ai }P  R_{i}, just subtracting coordinates
componentbycomponent. To get coordinates in TB_{i}, we multiply by the orthogonal
matrix (once again, see http://www.britannica.com/
for brushup) that does not change the Y components, but increases the Z and decreases the
X components of points in the first quadrant for small, positive angles, namely:
We pick this matrix by inspection of the figure above or by application of the
righthandrule (yup, see Britannica) Finally, to
bank the system, we need the orthogonal matrix that does not change the X components, but
increases the Y and decreases the Z components of firstquadrant points for small,
positive angles, namely:
In case you missed it, we snuck in a reliable, seatofthepants method for getting the
signs of orthogonal matrices right. In any event, given P and R_{i},
we compute the coordinates, P_{R}i, of the point P in ROAD_{i}
as follows:
If the angles are small, , and the matrix can be simplified to
Even at 20 degrees, the errors are only about 6% in the cosine and 2% in the sin,
resulting in a maximum error of 12% in the lower right of the matrix. This matrix
approximation is suitable for the majority of applications. One feature of orthogonal
matrices is that their inverse is their transpose, that is, the matrix
derived by flipping everything about the main diagonal running from upper left to lower
right. In the smallangle approximation, we get
The righthand side is very close to the unit matrix because the squares of small
angles are smaller, yet. With the inverse matrix we can convert from coordinates in ROAD_{i}
to coordinates in EARTH:
The last set of coordinate frames is WHEEL_{i}. As with ROAD_{i},
there is one instance per wheel. WHEEL_{i} is centred at the wheel hub.
Under normal rolling, the coordinates of its origin in ROAD_{i} are W_{Ri}
= (0, 0, R_{i}), where R_{i} is the loaded radius of the
tyrewheel combination. Pedantically, R_{i }should be corrected for
elevation and banking, but such corrections would be small for ordinary angleson the
order of plus it seems not to be standard practice (I can find no reference to
it in my sources). More important is the orientation of WHEEL_{i}. Consider
the plane occupied by the wheel itself. This plane intersects ROAD_{i }in a
line that defines the X direction of WHEEL_{i}, with the positive direction
being as close to that of travel as possible. The Y direction points to driver's right.
The wheel plane is tilted by a camber angle, , about the Xaxis of the WHEEL coordinate system. To
emphasize: WHEEL_{i }does not include wheel camber, and it differs
from ROAD only by a rotation about ROAD's Z axis that accounts for the pointing direction
of the wheel.
At this point, you should create a mental picture of these coordinate frames under
typical racing conditions. Picture a CAR frame yawed at some heading w.r.t. EARTHand
perhaps pitched and rolled a bit; a PATH frame aligned at some slightly different path
heading; and individual ROAD and WHEEL frames under each tyre contact patch, where the
ROAD frames are perhaps tilted a bit w.r.t. EARTH and the WHEEL frames are aligned with
the wheel planes but coplanar with the ROAD frames. For a car travelling on a flat road at
a stable, flat attitude, the XY planes of CAR, PATH, and EARTH would all coincide and
would differ from one another only in the yaw angles and . When some tilting is engaged, and are still defined by the precise
projection mechanisms explained above.
Now, imagine the Xaxis of CAR projected on the XY plane of each WHEEL frame and
translatedwithout changing its directionto the origin of WHEEL. The angle of WHEEL's X
axis, which is the same as the plane containing the wheel, w.r.t. the projection of CAR's
X axis, defines the steering angle, , of that wheel. Finally, imagine PATH's X axis
projected onto the XY plane of WHEEL in exactly the same way. Its angle w.r.t. to the X
axis of WHEEL, in all generality, defines the slip angle. Since WHEEL is lilted w.r.t.
gravitational down, the load, F_{x}, on the contact patch, which we
need for the magic formula, must be computed in WHEEL. It will be smaller than the total
weight, W_{i}, by factors of and , which are
obviously unity under the smallangle approximation.
At last, we can plot the magic formula:
The horizontal axis measures slip angle, in degrees. The vertical axis measures
lateral, cornering force, in Newtons. The deep axis measures vertical load on the contact
patch, in KiloNewtons. We can see that these tyres have a peak at about 4 degrees of slip
and that cornering force goes down as slip goes up on either side of the peak. On
the high side of the peak, we have dynamic understeer, where turning the wheel more makes
the situation worse. This is a form of instability in the control system of car and
driver.
As a final comment, let me say that I am somewhat dismayed that the magic formula does not
account for any variation of the lateral force with speed. Intuitively, the forces
generated at high speeds must be greater than the forces at low speed with the same slip
angles. However, the literaturesometimes explicitly, and sometimes by sin of
omissionstates that the magic formula doesn't deal with it. One of the reasons is that,
experimentally, effects of speed are extremely difficult to separate from effects of
temperature. A fastmoving tyre becomes a hot tyre very quickly on a test rig. Another
reason is that theoretical data is usually closely guarded and is not likely to make it
into a consensus approximation like the magic formula. This is a fact of life that we hope
will not affect our analyses too adversely from this point on. 