Sunday, January 29, 2006

Power and Force Revisited

Ok, so I've been doing some number crunching, and here's what I've come up with:

horsepower is related to torque and rpm.

Allow me to illistrate:

One horsepower is 33,000 ft-lbs per minute (330 lbs lifted 100 ft in 1 minute)
therefore:
33 lbs lifted 1000 ft in 1 minute equals 1 horsepower
66 lbs lifted 1000 ft in 30 seconds equals 1 horsepower
3300 lbs lifted 10 ft in 1 minute equals 1 horsepower

something a little more down to earth:
hp = horsepower
ft-lbs = foot pounds
rpm = revolutions per minute
5252 = magic number (constant)

Engine A puts out 150 hp at 3500 rpm.
Engine B puts out 150 hp at 6500 rpm.
Engine C puts out 150 hp at 10,000 rpm.

At rated speed, Engine A puts out 225.1 ft-lbs of torque.
At rated speed, Engine B puts out 121.2 ft-lbs of torque.
At rated speed, Engine C puts out 78.8 ft-lbs of torque.

Notice that as rpm increases, torque output decreases, even though horsepower remains constant. Now, check this out:

Engine A, Engine B, and Engine C are each geared down to produce an output speed of 1000 rpm. What is the new torque output for each engine?

Engine A:
150 hp @ 3500 rpm = 225.1 ft-lbs
3500 rpm / 1000 rpm = 3.5:1 gear ratio
225.1 ft-lbs * 3.5 = 787.8 ft-lbs

Engine B:
150 hp @ 6500 rpm = 121.2 ft-lbs
6500 rpm / 1000 rpm = 6.5:1 gear ratio
121.2 ft-lbs * 6.5 = 787.8 ft-lbs

Engine C:
150 hp @ 10,000 rpm = 78.8 ft-lbs
10,000 rpm / 1000 rpm = 10:1 gear ratio
78.8 ft-lbs * 10 = 787.8 ft-lbs

*NOTE: All calculations were done without rounding, then the final answer was rounded to one decimal place.


Now, lets look a little deeper:
Engine A:
150 hp @ 3500 rpm = 225.1 ft-lbs (peak power)
282 ft-lbs @ 2200 rpm = 118.1 hp (peak torque)
225.1 ft-lbs * 3.5 = 787.8 ft-lbs (peak power rpm geared down to 1000 rpm)
282 ft-lbs * 2.2 = 620.4 ft-lbs (peak torque rpm geared down to 1000 rpm)

Engine B:
150 hp @ 6500 rpm = 121.2 ft-lbs (peak power)
155 ft-lbs @ 4000 rpm = 118.1 hp (peak torque)
121.2 ft-lbs * 6.5 = 787.8 ft-lbs (peak power rpm geared down to 1000 rpm)
155 ft-lbs * 4.0 = 620.0 ft-lbs (peak torque rpm geared down to 1000 rpm)

Engine C:
150 hp @ 10,000 rpm = 78.8 ft-lbs (peak power)
103.4 ft-lbs @ 6000 rpm = 118.1 hp (peak torque)
78.8 ft-lbs * 10 = 787.8 ft-lbs (peak power rpm geared down to 1000 rpm)
103.4 ft-lbs * 6.0 = 620.4 ft-lbs (peak torque rpm geared down to 1000 rpm)

Summery:
When engines with the same hp rating, each running at rated rpm, are geared to the same output rpm, the torque outputs will be identical (for all intents and purposes).
When engines with the same hp rating, each running at PEAK TORQUE rpm, are geared to the same output rpm, the torque outputs will be identical (for all intents and purposes).

When two (or more) cars, with identical hp ratings at different rated rpms, are geared to achieve the same OUTPUT rpm in the same gear, assuming the weights are similar, the performance of the vehicles will be similar.

However, this assumes custom gearing for each gear, taking into account such things as tire size, engine performance data, etc. However, as the rated rpm for rated hp increases, peak torque decreases. This in turn, when torque is applied through a FIXED gear ratio, produces LESS FORCE at the tire, effectively REDUCING accelleration. This problem could be overcome by the use of a variable ratio system, which would allow for the application of PEAK OUTPUT torque at all or most speeds.