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
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 µyp = a1Fz + a2, measured in inverse Kilos if Fz is in KN. Next is D =
µypFz, which is a factor with the form of the Newtonian model: normal force times coefficient of friction. In our sample, a1 is zero,
so µyp acts exactly like a Newtonian friction coefficient. In all cases, we should expect a1Fz to be much smaller than a2 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 ROADi, 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 ROADi is located at the force centre of its corresponding
contact patch at the point Ri = (Rix, Riy, Riz) w.r.t. EARTH. This point moves with the vehicle, so, more pedantically, the origin
of ROADi is Ri(t) written as a function of time. To get the X and Y axes of ROADi, we begin with a temporary, flat, coordinate system
called TAi aligned with EARTH and centred at Ri, then elevate by an angle 90° <
< 90°, to get temporary frame TBi, and bank by an
angle 90° <
< 90°, in that order, as illustrated below:
Consider any point P in space with coordinates P = (Px, Py, Pz) w.r.t. EARTH. A little reflection reveals that its location w.r.t.
TAi is PAi
P  Ri, just subtracting coordinates componentbycomponent. To get coordinates in TBi, we multiply by the orthogonal
matrix (once again, see
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 Ri, we compute the coordinates, PRi, of the point P in ROADi 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 ROADi to coordinates in EARTH:
The last set of coordinate frames is WHEELi. As with ROADi, there is one instance per wheel. WHEELi is centred at the wheel hub.
Under normal rolling, the coordinates of its origin in ROADi are WRi = (0, 0, Ri), where Ri is the loaded radius of the tyrewheel
combination. Pedantically, Ri 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 WHEELi. Consider the plane occupied by the wheel itself. This plane intersects ROADi in a line that defines
the X direction of WHEELi, 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: WHEELi
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, Fx, on the contact patch, which we need for the magic formula, must be computed in WHEEL. It will be
smaller than the total weight, Wi, 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.