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Hi Tom,

I did poke around with numbers last night. My conclusion is that to get
maximum acceleration from my '87 S4 5-speed, I should upshift at redline.
This won't be true of all vehicles, rather it's because the S4 has a flat
torque curve relative to gear ratio spacing. Here's my analysis, and I'd be
pleased for anyone to poke holes in it:

First principle: The acceleration of a wheel-driven vehicle is a function
of: 1) the force that the tire applies to the pavement, 2) the weight of
the vehicle, and 3) friction, drag, and other stuff we'll ignore. The force
that the tire applies to the pavement is a function of torque at the rear
axle and the radius of the tire, assuming for our purposes perfect traction.
For this analysis, I will consider vehicle weight and tire radius constant,
so I claim that acceleration is proportional to rear-axle torque, more
rear-axle torque = greater acceleration.

Second principle: Torque at the rear axle is a function of torque at the
crankshaft and the torque-multiplying effects of the transmission gearing,
the rear axle gearing, and the torque-converter of an automatic
transmission. Since I'm a 5-speed kind of guy, I'll ignore the
torque-converter. My S4 has a 2.2:1 differential, a 1st gear ratio of
4.067:1, 2nd gear ratio of 2.714 to 1, 3rd gear ratio of 1.934:1, 4th is
1.463:1, and 5th is 1:1. So if the engine is delivering its peak torque of
317 lb-ft, in 1st gear the transmission delivers 317 * 4.067 * 2.2 = 2836
lb-ft to the rear axle. In 2nd gear, it delivers 317 * 2.714 * 2.2 = 1893
lb-ft to the rear axle. So, at fixed engine torque, the transmission
delivers 1.5 (4.067/2.714) times more torque to the rear wheel in first gear
than in second, with proportionally higher acceleration.

So, at any given engine speed, we can accelerate faster in a lower gear
because we get better torque at the rear axle. When, then, should we
upshift and sacrifice the relatively better torque multiplication of the
lower gear? It seems to me that we should upshift when increased torque
available at the engine speed in the higher gear is sufficient to offset the
reduced torque-multiplication of the higher gear. In my car, I should
upshift to second when 1.5 times the engine torque is available at the lower
engine speed, to offset the reduced torque multiplication. It turns out
that on the S4, this never happens. At 6500 RPM, (according to the owner's
manual,) my engine is making about 250 lb-ft, so rear-axle torque is still
250 * 4.067 * 2.2 = 2237 lb-ft, or more than *peak* rear-axle output in
second gear. Put differently, my car pulls harder at redline in first than
it ever does in second. I upshift to avoid throwing a rod, not to get
better acceleration. For my car, it turns out that this is true in every
gear. This is a tribute to our engines, they continue to produce pretty
good torque up to redline. In fact, there's only about 25% variance in
engine torque in the range between 1500 and 6500 rpm!

So, you see the root of my quarrel with Ed Ruiz' analysis on the other list:
engine torque is important, but it's only half the story. Gearing is the
other half. One gets better acceleration running the engine well beyond the
torque peak, as long as the drop in torque is not proportionally worse than
the penalty in torque-multiplication from upshifting.

There's an intuition about horsepower in here too: An engine produces more
power producing a given torque at a high rpm than at a low rpm. (It makes
sense, as it's burning more fuel and liberating more energy with more
explosions per second.) A car with a continuously variable transmission will
produce maximum acceleration running the engine at the horsepower peak, and
not torque peak, because the torque multiplication of the transmission will
be better there.

I have a graph in mind to illustrate this point about when to shift. I'm
going to make one for my 928 and one for my motorcycle, just for contrast.
When it's done I'll stick it on the web and post a URL.

We'll see how this maps into the real world at the races!

Regards,

Phil