How *does* a Torsen measure slip angles?

[Scott Fisher <sefisher@cisco.com>]

Scott Justusson hints at something here that's been troubling me for
days:

> until you agree that slip angle differences across a center axle cause 
> the torsen to shift torque, [...]

First, my admitted limitations -- I have no experience with AWD vehicles
(well, not quite -- I *really* disliked the handling of the Mitsubishi I
test-drove two years ago, but we can discard that data point as
irrelevant to the discussion of Audis).  I have only a modest
understanding of how a Torsen works -- I'm aware that it exploits the
unidirectional transfer of torque between a worm gear and a helical
gear, and that's about the limit of my knowledge of how it works.  I
have a thorough understanding, both as a mechanic and a driver, of open
diffs, locked diffs (spools), and traditional limited-slip (clutch-pack)
diffs, BUT only in 2WD (and generally RWD) applications.

I also fully get slip angles, both theoretically (they're just vector
math on the friction circle: draw the forward force vector and the
lateral force vector and make a parallelogram out of them, and the
diagonal is your slip angle) and experientially (whee!).

So... where I'm having trouble following this is in understanding how
any device other than a 3D accelerometer, comparing against the steering
angles of the front wheels (which of course will be different given
Ackerman geometry, but let's not even GO there), can determine a tire's
slip angle, *wherever* that tire may be located.

Slip ratio -- yes, of course, that's what a clutch-pack LSD does, that's
what a Torsen does too, it reacts to, or limits, the difference in
rotational speed between two (or more...) wheels.  But slip *angle*
measurement per se eludes me.  The farthest I've come so far is
remembering what my codriver Miq and I used to call "setting the parking
brake" on our locked-rear RWD autocross car several years ago: if you
turned the steering wheel more than about 30 degrees in any direction
(engine off, or in neutral), the car would not roll forward or backward,
because the welded rear end would permit no differential rolling of the
rear wheels.  And with the sticky slicks on that car, it was a *bear* to
push it onto the trailer at the end of the day, because the rear wheels
only wanted to roll in a straight line.  Try to turn it, and the rear
wheel would eventually hop and skitter over the tarmac.

(I also remember driving that car, and other cars with locked
diffs/slicks: they have a tremendous tendency for power-on *understeer*
during corner entry because the torque is evenly split between the two
rear tires in a corner.  Essentially, the inside wheel pushes the nose
of the car towards the outside of the turn, until you get enough weight
transfer -- or you apply enough torque by pressing on the gas pedal and
approach wheelspin, hmmm -- that the outside rear tire's slip angle
becomes high enough and gives you power oversteer.  That happens when
the force vectors have swung the slip angle toward the outside of the
car and you get that delightful feeling of steering with the throttle,
which transfers weight rearward decreasing front grip and going back to
understeer... which is a smooth, controllable form of U-O-U, not the
"spider bite," which I've never experienced because I've only
raced/driven cars with FWD or RWD, not AWD.)

I'm hoping that someone here can provide an explanation other than "read
SAE885140," and take me the next step... because I sense that I'm really
close, and I can't quite make the mental transition to AWD.  It seems to
be buried in the center diff's response to the radius asked for by the
front wheels, compared to the rear wheels' tendency to retard slip when
they're asked to describe circles of different radii, and it all gets
very muddy from that point.  

But while I can see that the center diff would *respond* to cornering
inputs and *react* differently when the slip angles change, I can't
quite see yet how that constitutes measuring slip angle differences
across the center diff.  Is that it -- is it that when the slip angles
are low at the rear (e.g., on corner entry), the car wants to push (like
my "parking brake" scenario), and then as the outside tire's circle
becomes sufficiently larger than the inside tire's (which would be
accelerated by the outside rear slip angle increasing and swinging the
rear of the car wider), it breaks traction at the inside rear wheel for
a second (like our locked-rear autocross car wanting to hop its inside
rear wheel across the tarmac), causing a discontinuous loss of inside
rear grip (read sudden oversteer)?  Or am I completely off base, due to
my ignorance of something in the AWD dynamic that I simply have never
experienced?

So... my thanks to those of you who have stuck with the engineering
descriptions in the face of high emotions on all sides, and I hope that
addressing my own puzzlement provides clarification for others.

--Scott Fisher