Re: Torsen tech

["Lawson, Dave" <dlawson@ball.com>]

(warning will robinson...   mathematical analysis follows)

>Answering my own question on the speed of the car for the 38/62% case:
>
>Radius = 40m
>lateral acceleration = 4 m/s^2
>v^2/40 = 4 (lateral acceleration is v^2/R)
>v = 12.6 m/s or about 45 km/h or just over 28 mph.

That's the same speed I had posted a few weeks back. Wow,
reproducible results.

>This unfortunately represents just one moment in time
>since the paper claims the car is accelerating at 4 m/s^2.
>If they maintain constant radius, then lateral acceleration
>must be increasing.  If they maintain constant lateral acceleration,
>then their radius must be increasing.

In interpreting this section, I was following the ground rules that
the vehicle was "following a circular path with a radius R = 40 m
at a lateral acceleration of aq = 4 m/s2". So in order to follow the
circular path, ie constant radius and constant lateral accel, the
vehicle was moving at a constant velocity. I do agree that if the
vehicle had a linear acceleration term, that one of the ground 
rules would change, like you said, either the corner radius or 
lateral acceleration increases.

What I interpret from this section is that when driving around a 
dry skid pad in a circle with a 40 m radius at a constant speed of
28 mph, the torque distribution is biased to the rear at 38/62,
all because the front wheels are following a larger circle than
the rear wheels. When you do the trig of this scenario, you 
can calculate that the difference in the circular paths which
the front and rear wheels follow is indeeed 0.2%.

>From practical experience, I think we can all attest to the 
fact that in a torsen, the front and rear wheels are not *locked* 
together which forces them to rotate at the same speed. Anyone
of us can drive at 28 mph in a 262 ft diameter circle in a dry
parking lot and not get the forced wheel slip which would come
from forcing the front and rear wheels to rotate at the same 
speed. From this practical example we can determine that the
torsen does indeed allow output shaft speed differences. I 
would conclude that the plot that Dave has posted is a viable 
model for torsen behavior, it allows relative output shaft speed 
differences all while distributing torque up to the bias ratio.

Then I got to thinking, given all the information presented in
the infamous paper, constant radius circles with constant lateral 
acceleration, and identifiying the car used, an 80 quattro, 
there is enough information to calculate 3 points on the plot
of bias ratio vs. relative wheel speed difference in radians per
second for this specific car.

1st point - 50/50 torque split and shafts rotating at the same 
speed, comes up with x=0 rad/sec, y=1 BR

2nd point - 40 m circle, lateral accel = 4 m/s2, vehicle speed
of 12.65 m/sec, results in a BR of 68/32 = 2.125.
Given that the front and rear wheels have a forced slip of
0.2%, knowing the size of the tires and the speed of the
vehicle, we can compute the delta wheel speed between
the output shafts. Running through the math I get a result
of x=0.085 rad/sec, y = 2.125 BR

3rd point - 15 m circle, lateral accel = 4 m/s2, vehicle speed
of 7.75 m/sec, results in a BR of 75/25 = 3.0. We can compute 
the "forced slip" of this case from the radius of the circle and
the wheelbase of the 80 quattro. Following this tighter circle,
I come up with a forced slip of 1.47% between the front and 
rear wheels. Then knowing the tire circumference and vehicle 
speed, I calculate the delta wheel speed between the output 
shafts to be .385 rad/sec. So x = 0.385 rad/sec, y = 3 BR.

Plotting these 3 points up we can determine that this relationship
in the torsen is not linear, meaning you can't draw a straight line 
between these 3 points using a linear scale. You could fit a 
polynomial to the points, but I have not done this yet.

What does all this data mean? I can make the following 
conclusions:

1) the torsen in the 80 quattro has different operating characteristics
than the torsen in a humvee.

2) the torsen does indeed allow wheel speed differences when
operating within it's bias ratio. And the wheel speed differences 
follow the data points I presented above. It does not *lock* the
output shafts together where they must rotate at the same speed.

3) ??? Anyone else?

-
Dave Lawson