Weird Power Measurements with a Northern Lights Airforce Airdyne Cycle

[rdbm]

I attached Assioma power meter pedals to a Northern Lights Airforce Airdyne Cycle that looks identical to the classic Assault Bike, but it's much harder to pedal for a given RPM. Here's the result for pedaling with only the legs {{RPM, Console Readout, Assioma Readout, Difference}, ..., }: {(30, 50, 59, 9), (40, 99, 129, 30), (50, 176, 251, 75), (60, 288, 437, 149), (75, 536, 902, 368)}. The equations work out to P (in watts, console) = A x (RPM / S) + B x (RPM / S)^3 and P (in watts, Assioma) = A x (RPM / S) + B x (RPM / S)^3 + C x (RPM / S)^4. For my bike A = 37.33, B = 250.66, C = 151.88, and S = 60. I would imagine others have tested their Assault Bikes in this fashion and got the usual, just purely cubic results and not the quartic one, probably due to the fact that you'd have to pedal the Assault Bike really fast to even spot this. The quartic fits the difference value to the RPM almost like a glove! Ideas?

 

[Mark44]

A few points:

• The data you show is very confusing. That data would be easier to parse if laid out in table form.
• Your equation would greatly benefit from being written using LaTeX (or rather, MathJax, the variant we use here at PF. In your equations, what is S?
• What do you intend to show with your cubic equation?

 

[Mark44]

Regarding LaTeX, at the lower left corner of the input pane there's a link to our guide for LaTeX.

 

[rdbm]

The Airdyne has a console that tells you the power and pedaling speed you are outputting. I replaced the pedals (that came with the Airdyne) with special pedals (Assioma Power Meter Pedals) that measure power with strain gauges. The results are given in the following table:

Cadence (RPM)	Airdyne Reading (Watts)	Assioma Reading (Watts)	Difference (Watts)
30	50	59	9
40	99	129	30
50	176	251	75
60	288	437	149
75	536	902	368

"S" is the reference cadence I used for the equations that I fitted to the data. I set S = 60 in the equations. The Airdyne registered 288 watts on its console at 60 RPM. Assuming a constant angular velocity of the pedal cranks, I determined the values of A and B for the Airdyne in the first equation by experimenting with different numbers for A and B (using simultaneous equations) and got 250.66 watts (B) due to the air resistance at the fan blades and 37.33 watts (A) due to the linear friction due to the Airdyne's moving parts at 60 RPM. The first fitted equation, P (in watts, console) = A x (RPM / S) + B x (RPM / S)^3, yields the watts under the "Airdyne's Reading" column that match closely. I did the same for the second equation, P (in watts, Assioma) = A x (RPM / S) + B x (RPM / S)^3 + C x (RPM / S)^4, which describes the Assioma readouts. Notice how the Difference increases quartic-like with respect to the cadence (e.g., 30 x (75/40)^4 = 371 (close to 368)). The human pedaling stroke is not constant; there's slowing down and speeding up all the time. This makes it harder! It's kind of like running 100 meters is much easier if done at a constant speed as opposed to accelerating over one 10 meter stretch from a stationary start and then decelerating over the next 10 meter stretch to a stop and repeating four more cycles until the 100 meters has been covered at the same average speed. Something like this seems to be adding to what the console thinks the power output is. Or could this be just some uncanny coincidence.

The fitted two equations are given below:

$$P(in watts, console) = 37.33(RPM/60) + 250.66(RPM/60)^3$$
$$P(in watts, Assioma) = 37.33(RPM/60) + 250.66(RPM/60)^3 + 151.88(RPM/60)^4$$

 

[erobz]

The Airdyne has a console that tells you the power and pedaling speed you are outputting. I replaced the pedals (that came with the Airdyne) with special pedals (Assioma Power Meter Pedals) that measure power with strain gauges. The results are given in the following table:

Cadence (RPM)	Airdyne Reading (Watts)	Assioma Reading (Watts)	Difference (Watts)
30	50	59	9
40	99	129	30
50	176	251	75
60	288	437	149
75	536	902	368

"S" is the reference cadence I used for the equations that I fitted to the data. I set S = 60 in the equations. The Airdyne registered 288 watts on its console at 60 RPM. Assuming a constant angular velocity of the pedal cranks, I determined the values of A and B for the Airdyne in the first equation by experimenting with different numbers for A and B (using simultaneous equations) and got 250.66 watts (B) due to the air resistance at the fan blades and 37.33 watts (A) due to the linear friction due to the Airdyne's moving parts at 60 RPM. The first fitted equation, P (in watts, console) = A x (RPM / S) + B x (RPM / S)^3, yields the watts under the "Airdyne's Reading" column that match closely. I did the same for the second equation, P (in watts, Assioma) = A x (RPM / S) + B x (RPM / S)^3 + C x (RPM / S)^4, which describes the Assioma readouts. Notice how the Difference increases quartic-like with respect to the cadence (e.g., 30 x (75/40)^4 = 371 (close to 368)). The human pedaling stroke is not constant; there's slowing down and speeding up all the time. This makes it harder! It's kind of like running 100 meters is much easier if done at a constant speed as opposed to accelerating over one 10 meter stretch from a stationary start and then decelerating over the next 10 meter stretch to a stop and repeating four more cycles until the 100 meters has been covered at the same average speed. Something like this seems to be adding to what the console thinks the power output is. Or could this be just some uncanny coincidence.

The fitted two equations are given below:

$$P(in watts, console) = 37.33(RPM/60) + 250.66(RPM/60)^3$$
$$P(in watts, Assioma) = 37.33(RPM/60) + 250.66(RPM/60)^3 + 151.88(RPM/60)^4$$

We are accelerating the fan and various gear train components with our non constant cadence too.

 

[jrmichler]

I attached Assioma power meter pedals to a Northern Lights Airforce Airdyne Cycle

When analyzing an airdyne exercise bike, it's best to start with the physics by listing possible sources of resistance to pedaling.

1) Fan power is proportional to RPM^3. This is a basic fan law.
2) Possible friction that is constant at all speeds. This would show up as power proportional to RPM.
3) Possible viscous friction. Viscous friction torque would be proportional to RPM, so the power would be proportional to RPM^2.

These bikes turn easily with light force on the pedals, so start by assuming that 100% of the pedal power is used to spin the fan against air resistance with the other two sources negligibly small. Start by fitting a cubic curve to the console power readings. The best fit curve has the equation $Watts = 0.0013 * RPM^3$. The resulting curve, along with the original data, is shown plotted below.

The equation closely follows the console readings, with a small linear difference. Next, do the same with the Assioma pedal outputs. That best fit cubic curve is $Watts = 0.00211 * RPM^3$. Plotting the Assioma data with the Assioma best fit cubic curve results in the following:

While both equations follow their data well, the Assioma equation follows the Assioma data with less error. The Assioma readings are based on strain gaged pedals, so originate with direct measurements of torque and RPM. The Airdyne readings are apparently based on a factory calibration curve because they closely follow the predicted cubic trend. The difference between the two sets of measurement can be explained by a large calibration error in the airdyne readings. You can test this by fastening cardboard disks to both sides of the airdyne fan, then repeating the test. The cardboard disks should reduce the power at all speeds. The Assioma readings should show less power at each RPM, while the Airdyne readings would be expected to be the same as the readings without the cardboard.

 

[DaveC426913]

_{Gotta be honest, when I saw 'weird power' I thought it gonna be the ability to grow your fingernails instantly, or to telepathically repel soap bubbles. A little disappointed here.}

 

[rdbm]

Nice one!

 

[rdbm]

When analyzing an airdyne exercise bike, it's best to start with the physics by listing possible sources of resistance to pedaling.

1) Fan power is proportional to RPM^3. This is a basic fan law.
2) Possible friction that is constant at all speeds. This would show up as power proportional to RPM.
3) Possible viscous friction. Viscous friction torque would be proportional to RPM, so the power would be proportional to RPM^2.

These bikes turn easily with light force on the pedals, so start by assuming that 100% of the pedal power is used to spin the fan against air resistance with the other two sources negligibly small. Start by fitting a cubic curve to the console power readings. The best fit curve has the equation $Watts = 0.0013 * RPM^3$. The resulting curve, along with the original data, is shown plotted below.
View attachment 358923
The equation closely follows the console readings, with a small linear difference. Next, do the same with the Assioma pedal outputs. That best fit cubic curve is $Watts = 0.00211 * RPM^3$. Plotting the Assioma data with the Assioma best fit cubic curve results in the following:
View attachment 358924

While both equations follow their data well, the Assioma equation follows the Assioma data with less error. The Assioma readings are based on strain gaged pedals, so originate with direct measurements of torque and RPM. The Airdyne readings are apparently based on a factory calibration curve because they closely follow the predicted cubic trend. The difference between the two sets of measurement can be explained by a large calibration error in the airdyne readings. You can test this by fastening cardboard disks to both sides of the airdyne fan, then repeating the test. The cardboard disks should reduce the power at all speeds. The Assioma readings should show less power at each RPM, while the Airdyne readings would be expected to be the same as the readings without the cardboard.

Good graphs. If this extra pushing contributes independently to the fan’s cubic power, it would explain this. If you double the fan speed, force quadruples, and the other viscous? force doubles: the result would be eight times the force and sixteen times the power: in line with the results I was getting on the Assioma pedals. (Magnetically braked cycles also exhibit quadratic power, which would result in the third term being cubic instead.) Seems that the pure math that I brought to the table reflected the interplay between two separate forces that, together, were acting like one quartic force. This extra added force must be generated by the jerky pedaling motion of the human pedaling power curve.

My Assioma power pedals were calibrated and tested against a good Zwift trainer and its numbers are accurate to within 2% or so. The Airdyne was calibrated at the factory using a 20 hp motor attached to the drive train, and it spun the whole engine (chain, cogs, cranks, and fan). I don’t have access to this kind of testing equipment, so I verified things another way: using only my arms, I pulled as hard as I could on the arms of a Schwinn Evolution Comp (no longer available) that was certified accurate enough (within 2%) for medical use: I sustained 63 RPM (200 watts) for 90 seconds. Forty eight hours later, I tried to see how many strokes I could do in 90 seconds on my Northern Lights Airforce Bike: 54 RPM (216 watts). Close enough. By using my arms, I could keep constant tension throughout the range of motion of the arms of the cycle. There was only a very small fraction of a second of deadtime during which the arms switched directions, which greatly reduced the size of the ‘quartic’ terms. I also tried fan dampening and observed a reduction in the power output on the Assioma pedals.

Looks like my pedals are working properly and they’re not pulling my legs (pun intended).