Why is my calculated power output different from the expected value?

[Mark Sullivan]

> Average velocity = 60rpm
> Average torque = (10 + 50 + 10 + 50) / 4 = 30TU=
> Power = 60rpm x 30TU = 1,800 power units

> However, if you calculate the power for each time sample (hence account for the different velocity recorded during each sample), then:
>
> Power = (62x10 + 58x50 + 62x10 + 58x50) / 4 = 1,760 power units

Help me get my head around this. I know the first is correct for power around an axis. The second is incorrect and is 40 power units under. If I do this Power = (58x10 + 62x50 + 58x10 + 62x50)/4 = 1,840 power units or 40 over. So why? It feels like the reason is just at the edge of my brain. Somewhere in school I must have learned the answer.

 

[jbriggs444]

The product of the averages is not, in general, equal to the average of the products.

 

[Mark44]

> Average velocity = 60rpm
> Average torque = (10 + 50 + 10 + 50) / 4 = 30TU=
> Power = 60rpm x 30TU = 1,800 power units

> However, if you calculate the power for each time sample (hence account for the different velocity recorded during each sample), then:
>
> Power = (62x10 + 58x50 + 62x10 + 58x50) / 4 = 1,760 power units

The calculation just above is not a weighted average. It doesn't take into account the different times for each sample, but instead just divides by the number of time intervals.
Taking the average of a bunch of averages doesn't produce correct results. To see why this is true, there's an old math problem that says it's 1 mile to the top of a hill. If you average 30 mph on the trip up, how fast must you go back down the hill to average 60mph for the entire trip?

The intuitive answer (which is wrong) is that the downhill trip should be 90 mph.

Help me get my head around this. I know the first is correct for power around an axis. The second is incorrect and is 40 power units under. If I do this Power = (58x10 + 62x50 + 58x10 + 62x50)/4 = 1,840 power units or 40 over. So why? It feels like the reason is just at the edge of my brain. Somewhere in school I must have learned the answer.

 

[Mark Sullivan]

Thanks, I did know that.

I am confused about a larger question that this is part of and I can't figure out which is right. It involves measuring power on a bicycle at the crank and whether elliptical or non round chain rings actually overweight/underweight a power reading when a crank speed is measured only once per revolution or non round chain rings are just an increase/decrease in the lever and well work is work and it is coming from your foot on the pedal which is on a circular radius. I lean to the latter but the counter argument is very good of which the above is part of it.

I guess I should start another thread but which forum physics or mechanical engineering?
Thanks

 

[Mark44]

I guess I should start another thread but which forum physics or mechanical engineering?

I lean toward mechanical engineering.

 

[Mark Sullivan]

Thanks, I started the thread to mechanical engineering as "Bicycle Crank Power Meters and Round and Non-Round Chainrings" in case anyone reading this thread is interested.