Heh heh heh. Never fear, Treynor's here
<BLOCKQUOTE><font size="1" face="Verdana, Arial">quote:</font><HR> Te=Td*(wd/we)
So if the torque on the drum goes up with a relatively small decrease in the drum RPM via a gear swap from 3.07 to 3.55 you get a larger engine torque reading for a given RPM.
Going back to the equation HP=Torque*RPM/5252 this means that at a given RPM, your HP reading will be higher at the wheels for a 3.55 versus a 3.07.
We also cannot constantly increase the HP because we are limited by the physical size of the differential. To gain enormous HP from gearing we would have to install enormous gears. Sizes beyond the size of the differential design
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Your analysis is correct up to this point. The torque as measured at the dyno drum is equal to the torque as measured at the engine, multiplied by the effective gearing between the engine and drum (which includes wheels) minus frictional losses in the system, or as you put it Te = Td * (Wd/We).
However, the conclusion you then reach is incorrect. First, consider how the dyno calculates engine torque based on observed torque at the dyno. The dyno "knows" Td (from the observed acceleration of the dyno drum and the known rotational intertia of the drum), it knows Wd, and it knows We from the inductive monitor of the spark plug wire. So it calculates engine torque exactly by calculating Te from Td, Wd and We as you describe above.
Now, consider the change from a 3.07 to a 3.55 gearing, an increase in ratio of 15.6%. In this case, at a given engine RPM the torque observed at the dyno will have gone up by 15.6%, and the dyno drum speed will have gone down by 13.5%, a decrease of the same ratio, since (100% + 15.6%) * (100% - 13.5%) = 100%.
So what will the dyno calculate with the new gearing? The new Te' = Td' (which is 15.6% larger) * Wd' (13.6% smaller) / We (constant). But that is of course the same as the original Te, so the dyno isn't "fooled" and will report the same torque (and, by extension, horsepower) regardless of gearing.
Still confused? An example will make everything clear
Consider the ideal car (for this example, anyway) -- the
Doug Levin "Dominator" Viper. Measured at the engine, this beast has 1000 ft/lbs of torque @ 5000RPM, and is so well assembled that there is ZERO friction in the driveline. To put all this power to the ground, the owner installs the special Parts Rack "
TractionMaster(tm)" 2.00 rear gear set, then takes the car to the dyno for some brag sheets. As it turns out, the dyno drum on his local dyno is exactly 50 times as large as his rear wheels, so 50 revolutions of the rear wheel will result in 1 revolution of the drum. Thus, in the 1:1 4th gear the "Dominator" has a 2:1 drive ratio and a 50:1 wheel/drum ratio, so the dyno measures 50*2 = 100 times the torque being output by the engine and similarly notes that the dyno is rotating at 1/(50*2) or .01 times the speed of the engine. It thus dutifully reports that, at 5000 RPM,
Te = 100,000 ft/lbs (Td) * 50 RPM (Wd) / 5000 RPM (We), or 1000 ft/lbs of torque at the engine, exactly what we expect to see. Similarly, the dyno reports that HP at that RPM is (1000 * 5000 / 5252) or 952 HP.
Our fearless owner, hoping to get even more power, orders the Parts Rack "
Ulysses Horsepowerinator" rear gear set with a 20:1 ratio made possible through the use of special titanium cryo-treated ultra-heavy-duty gearsets. These installed, he heads back to the dyno (mostly in 5th, since 1st gear is now only good for 8 MPH) and fires up the beast. Again in 1:1 4th gear, the dyno now finds that the torque at the dyno is 50*20 or 1000 times the torque at the engine, and similarly the dyno is spinning at only 1 / (50*20) or .001 times the speed of the engine. Thus, it reports that:
Te = 1,000,000 ft/lbs (Td) * 5 RPM (Wd) / 5000 RPM (We), or 1000 ft/lbs of torque at the engine, and of course again 952 HP.
Moral:
Gears don't make horsepower. Gears multiply torque and divide velocity.