walterpump wrote: ↑October 16th, 2022, 6:45 pm
Hi Nomath: Do you know of any relatively simple math formula for calculating Drag Factor based on the deceleration of the flywheel? I would like to try to calculate the drag factor for my Model A, just so I know what it is. Thanks for any info you can provide, and happy rowing. Jim
Here is a method to measure the Drag Factor experimentally. I have never come across a Model A, so I don’t know if this method is feasible. Let's see.
Because the deceleration also depends on the moment of inertia of the wheel, you need to determine the moment of inertia as well.
What you need is a stopwatch, a 0-1 kg scale, a ruler, two bicycle chains and a chain tool. The stopwatch should be capable of recording and storing split times. The stopwatch in an Apple iPhone is fine.
The first practical requirement is that you can give the wheel a good spin by hand. The second requirement is that the wheel keeps spinning for at least 10 turns before it halts
The theory behind the measurement is that the spinning wheel is slowed down by air drag. When there is no driving force on the wheel, i.e. while spinning freely, the rotation period increases linearly with time. In a mathematical expression :
dTr/dt = 2π * DF/J
Tr is the duration of a full turn [sec].
t is time [sec]
DF is damping factor [kg m²]
J is the moment of inertia of the wheel [kg m²]
Attach a bright sticker to the rim of the wheel. Give the wheel a good spin by hand. Record the rotation periods by pressing the stopwatch when the sticker passes the reference point, e.g. the fork that holds the wheel. Record six consecutive rotations.
You can now make a graph of the rotation period against the time elapsed.
The slope of the best-fit line through the points, S1, is the value of the ratio 2π * DF/J
To determine DF, we also need to determine the moment of inertia J. There are experimental methods to measure the moment of inertia of a wheel directly, such as a tri-filar pendulum, but they are difficult to implement. The trick that I will explain below is much easier.
The circumference of the Model A wheel is probably roughly 2 meter, which is longer than one bicycle chain. So we connect the two chains with a common chain link (e.g. SRAM, Shimano, KMC) and then reduce the length of the combined chains to the circumference of the rim bed. The two chain ends should not overlap! Measure the weight of the chain on the scale, M. Measure the distance from the rim bed to the wheel axis, R, or calculate R from the length of the chain L (R=L/2π). Tighten the chain ends with a tiewrap, so that the chain cannot move.
Now again give the wheel a good spin and record 5-6 consecutive rotations. Because of the added mass, the wheel will slow down less. Again make a plot of the rotation period against time en determine the best-fit slope, S2. Clearly, S2 is smaller than S1.
Now back to the physics.
Adding a mass M at a distance R from the rotation axis, has increased the moment of inertia of the wheel in the second experiment to J+M*R².
So we have two equations :
(1) S1 = 2π * DF/J
(2) S2 = 2π * DF/(J+M*R²) (i.e. assuming that the chain does not affect the aerodynamic drag).
Resolving for J gives : J = M*R²/(S1/S2-1)
and : DF = J * S1/2π
The trick with the bicycle chain is thus to add a known amount to the moment of inertia of the wheel. From the effect of this on the slope, the original moment of inertia can be deduced and the damping factor.
More accurate than a stopwatch, which is difficult to operate accurately for rotation periods of less than 2 seconds, is a video camera of known frame speed. Count the number of frames before the bright sticker returns to the reference position.
Let me know if it works.
