ryoung

I originally prepared a spreadsheet similar to the one below to assist with tuning the triple adjustable shocks of our Formula Continental. This car is about as far from an Audi as you can get -- a 975 lb mid-engine rear-drive full racecar with wings and on sticky slicks versus my 3840 lb AWD S8 on street tires. It dawned on me, though, that all I had to do was change the inputs, and it would model the S8's transition thru a corner just as well.

The inputs:

Vehicle Weight -- 4000 lb (wishful thinking here, that the driver weighs only 160 lb)
Wheelbase -- 113.5 in
Front Track -- 63.6 in
Rear Track -- 63.2 in
Front Weight -- 60%
CG Height -- 22 in (an educated guess)

The assumptions:

1. The grip available from the tires follows the principle of the friction circle -- whether braking, cornering, acceleration, or combinations. The grib available from each tire is calculated individually based on vertical load. [See chart at the end of the post.]

2. The shocks do not affect load distribution (and in steady state, they don't).

3. No sway bars installed.

4. Spring rate (actually, wheel rate) is proportional to static weight distribution.

5. The height of the roll centers is the same front and rear.

6. Corners are right turns.

7. Identical front and rear tires.

8. Front brake bias is 69.1% This was chosen to force the "Excess Grip" to zero both front and back (or how much brake bias you'd have if ABS corrected an imbalance).

Without having the exact location or coordinates of the ends of all the suspension elements, it's impossible to know the height of the front and rear roll centers, which would allow calculating the exact effect of spring or roll bar changes. The height of the roll center determines how much load is transferred to the contact patch thru the springs vs directly thru the suspension elements. In any case, though, the quantitative affect of input changes can be seen, along with qualitative affect of spring, shock, or sway bar changes.

Except for brake bias, they were absolutely no fudge factors used to generate this spreadsheet. The test of the '01 S8 in "Road & Track" reported 0.87g of cornering. The spreadsheet shows that the S8 is capable of 0.88g (0.875g if a 3rd decimal is displayed!) in steady state cornering. I was truly surpised at the agreement.

In the table, other than seeing what happens to the loads on each tire during cornering, the "Excess Grip" column is interesting. If the values are zero front and rear, the car is perfectly balanced; if the front has excess grip, the car will oversteer; and if the rear has excess grip, the car will understeer.

<img src="http://pictureposter.audiworld.com/17157/weightdistribution.jpg">

Observations:

1. With maximum straight line braking -- 78% of the load and 69% of the braking is being handled by the front tires ... no surprise here

2. Hard trail braking into the corner -- With 60% of the load on the front, the car's going to be loose. However, weight is moving back and out, from the inside front to the outside rear. Softening the rear shocks in bump or stiffening the front rebound will reduce oversteer in this phase of the corner. It improves the weight distribution across the rear tires, and makes it worse across the front.

3. Easing off the brakes into the corner -- Balance is perfect at 0.26 g of deceleration. Weight has moved to the rear primarily from the inside front.

4. Steady state corner -- Based on the identical front and rear vertical/horizontal loads, the car should be perfectly neutral. Remember, though, that the coefficient of friction of the tire does not increase linearly with load, especially at high slip angles. As a result, the front has less grip than the rear, and the car understeers. [Our Formula Continental has larger tires in the rear, and it's model takes the larger tire contact patch into consideration.]

5. Squeeze power on in corner exit -- Again no surprise, bad things are starting to happen. There's much less load on the front tires than the rear, and the car will push badly. Load is moving primarily from the left front to the right rear, so the front shocks should have minimum rebound to keep the problem from getting totally out of control. The benefits of a rear sway bar are obvious -- weight would move off the inside rear to the other three. The increased weight on the front, along with the poorer weight distribution in the rear, would both contribute to reducing understeer. Stiffer rear springs would also help for similar reasons.

6. If you don't trail brake but instead do all your braking in a straight line, you can look at what happens to corner weights with no lateral loads (parked) and then going to steady state cornering. Since weight is just moving out - softening front rebound, stiffening rear rebound, stiffening front bump, or softening rear bump are all shock adjustments that would improve turn-in.

The data in the chart below was taken from charts in "Going Faster" by Danny Sullivan and from an article in the July '03 edition of "Racecar Engiineering" -- they agreed surprisingly well. The black line is actual, the coefficient of friction between the tire and road drops as load increases. The blue line would be the theoretic constant cf of 1.0. The formula in the chart is just a polynomial curve fit of the data used to determine "Available Grip" in the spreadsheet.

And if you gotten this far and are still with me ...

Sensitivities -- changing some of the inputs and seeing what happens,

1. Increase vehicle weight to 4400 lb (i.e., '04 A8L): Steady state cornering drops from 0.88 to 0.84g.

2. Reduce vehicle weight to 3600 lb (I wish): Steady state cornering increases from 0.88 to 0.91g.

3. Improve weight distribution from 60/40 to 50/50: Steady state cornering increases from 0.88 to 0.96g! Straight line braking improves from 0.95 to 1.02g.

<img src="http://pictureposter.audiworld.com/17157/tiregrip.jpg">

Well ... enough for now, any questions?

smallTTs

I would expect it to be more like 21-22 inches.
It's tough to measure, but 21 inches is close for most road going sedans.


Good calculations.

.6 g aceleration maybe in first gear, but probably lower in other gears.

My $.02

ryoung

.
that was a best guess, and it could easily be higher. Maybe the S8's relatively lower ride height and all the aluminum isn't worth as much as I'd hoped.

You also pretty much nailed acceleration g's. Software I'm using predicts 1/4 times within 0.1 sec and final speed within 1 mph of actual. It also predicts max g's of

0.8 in 1st at 0 mph
0.6 in 2nd at 41 mph
0.4 in 3rd at 65 mph
0.2 in 4th at 95 mph
0.1 in 5th at 142 mph

smallTTs

measure that height from the ground. That's also about the top of the wheel rims (not top of tire).

There is a lot of vehicle above that line which is about the 21-22 in. I mentioned.

Just for fun, are those max g figures somewhat near max torque rpm in each gear?