Inertia calculation alternative

[Abasz]

Apologies for being unclear!

This video explains a lot, I think/hope. It depicts a "standard" way of measuring inertia. It is fundamentally more accurate than the "pendulum method". It is a "standard" undergrad physics experiment. Note that in the video, the rotation axis points vertical, wheras the rower flywheel axis points horizontal. This makes it even easier, since no idler pulley is required.

This thread shows the 12 encoder lines on the flywheel of my concept2 model C, causing 12 pulses per revolution. Normally used for determining the angular speed during rowing, but now I use it for determining $\frac{d\omega}{dt}$ to estimate the moment of inertia.

From de the delta-times I calculate $\omega = \frac{2\pi}{12\Delta t}$. Plotting $\omega$ versus time gives slope $\frac{d\omega}{dt}$ as a constant, which can be plugged in in $J=mr^2(\frac{g}{r\dot\omega}-1)$, were $m$ represents the mass attached to the string, $r$ is the radius of the "fan-drum" and g is acceleration due to gravity.

In my case it was particularly/surprisingly easy to carry out. The only thing I had to do is take off the screen from the flywheel housing (instead of just the front cover of my model C) and wind a string with weight around the "fan-drum". And I lifted the flywheel assembly 1 meter up such that I had more "drop-height". I did the experiment with two different weights: 172 [g] and 35 [g]. Results for $J$ were very similar, but I preferred the data obtained with the higher weight. I also had to make one or two tweaks in my analysis software, since during rowing, very low speed (<200 [rpm]) data is discarded.

@kamp169

[Abasz]

I have moved this conversation out of the other discussion.

I did some measurements, I repeated the same for 8 times.

• Flywheel has 3 magnets
• I used 105gramm weight
• My string was a bit stretchy actually
• Drop distance was around 1meter
• Used the ESP32 interrupt to log delta times
• I always rolled the flywheel as close to the first magnet without triggering as possible so I get an early impulse
• When the weight reached the flow I did not stop immediately the flywheel but as expected the next delta time was increasing compared to the last.
• I always got 4 impulses (hope that is enough) but I used 3 (one full rotation) so we can avoid magnet misplacement errors.
• My logged delta times across the repeated measurements were quite inconsistent they showed big deviation especially as speed increased.
• The CAD based inertia is around 0.07848

Questions:

1. Did you use the full range of the delta times or you picked the first?
2. Do I assume correctly that the distance element in the formula is calculated basically based on the number impulses vs. the drum diameter on which the string is rolled on
3. How big of an issue if the string is a little stretchy?
4. Do you adjust the velocity with any friction or air drag or you just take the time values as is?

My experience is that I am nowhere near the 3% (its more like 10% if I take out the obvious outliers from the 8 repeats, because I get 0.067 with this method). So I wonder what could improve this.

I will do the kayak erg as well as there I have a rather accurate value from the engineer that made it and see how that works.

EDIT:

I re-did the measurements with a strudier sawing string (previously I used knitting string that had some springiness). And there is a massive improvement. With the original weight I get 0.075 but with the increased weight (of 167gramm) it produced 0.0775 inertia which is very close to the CAD calculation. Especially if I round it up which should be plausible considering friction and air drag (while I suppose normally the latter should be nominal at such a low speed)

So one conclusion of mine currently is that a bit heavier weight produce better results especially on machines that have more friction (mine is rather old) which is logical as friction is constant and its constant effect on higher speed is relatively reduced.

[Abasz]

To cut it short: what is the weight of the steel disk? And what is the diameter of all the holes in it and at what radius are they positioned? And what inertia does Fusion produce (at density...) just for the steel disk? I take it that you also remove the central axle to get just the disk weight(?)

Part	Gramm
Disk	2,498
Clutch assembly	285
Clutch bolts	16
Screws	12
Plastic fan	845
Total	3,656

Clutch assembly is measured together with the bearings (its not possible to take it out), but I used different material and modelled the bearing as solid "pipes". Their weight is set based on subtracting the standard aluminium density and the remainder is assigned to the 3 bearing (1 clutch/one way bearing, two normal 6904ZZ rolling).

Disk used to be 2.5kg almost precisely but I drilled two small holes originally for the magnet so we lost that weight (holes are modelled) :D
Disk density: 7.825 g/cm³ to achieve this weight with the created solid body

Sketch:

note that the outer screw holes (that locks the plastic to the disk) only one has been sketch because its more efficient to do circular cut on the solid body, there 8 of these holes.

Plate is 2.5 mm thick I used 2.49 as it has a black coating.

[klamp169]

Abasz_data.xlsx
See my libreoffice calc based estimation (all in meters!) attached, based on the data that you supplied (thanks!). My result comes down to $J=0.074[kgm^2]$ (cell B88).
My estimation of the inertia (cell B85) of just the disk corresponds very well with your Izz. So the difference must be in the blade assembly.
I used my own fan-geometry, obviously. It will be worse than your approach, also obviously, but not so much that it can account for the difference between 0.074 and 0.078, I think. What is your opinion? See also my very rough estimate (cel E88) in the bottom right corner. It is already quite close, I think.

[Abasz]

Sorry for my late response.

Essentially this is a the method that Dave Vernooy did (https://dvernooy.github.io/projects/ergware/) but of course this is more detailed :) I am pretty sure this is what the CAD does just more fine grained.

Its clear that the difference comes from the fan blades. Those are tricky because their base is 3.4mm but their top is 5

Also they are curved (the purple dotted is the "bottom" and the black is the top.

But I think the biggest difference comes from the the ABS ring size and a little bit of clutch:

ABS ring mounted onto on steel disk with 8 screws:

• this is 392.3mm (diameter) while you used 365mm
• inner diameter is 360.4mm while you used 361

ABS outside ring

• outer diameter is 351.75mm you used 365
• inner diameter is 287.75 (its 32mm wide) you used 296.8mm

Clutch:

• you calculate outside radius with the 40mm but actually that is 57 (though there is a narrowing around half way through to 40.
• the inner diameter of the clutch is 37mm but the bearings have 20mm
• if you take diameter as 50 or something around that you will be around 0.0001 as contribution

These corrections (after correction to the ABS density to 1440.1 of course) yields total J as "0.0767" - "0.0768" (with clutch correction) which means that this yields less than 1% deviation so I would conclude this is accurate compared to the effort it takes to do a calculation vs model in CAD

PS: on the blades I forgot the thickness of the rings. Bigger ring is 3.9mm and smaller ring is 3.4 (where you used 3.2) but this does not change the inertia (in a noticeable way)

[klamp169]

Thanks! Your ABS parts are radially bigger than the concept2 fan, it seems. This brings the density down a little bit, but the inertia goes up.
I plugged in your numbers (mainly for the rings), did the goal seek again on the ABS weight and came very close to your inertia value.

I do agree that probably the CAD-Inertia is the one closer to reality. Critical there, I think, is that you have the weights of the various parts.

Can you reassemble the flywheel with good balance? I would not dare take the ABS part from the steel disk for this reason (not screwed on, btw, but riveted).

Still annoying that we cannot explain the difference with the measurements. Perhaps the friction model (Coulomb friction) is off?

[Abasz]

Can you reassemble the flywheel with good balance?

It is easy to take it apart because there are screw holes are threaded. So after disassembly I can assemble it the same way. Is it good balance? no. but its not the because reassembly. But because its a cheap clone :D

Perhaps the friction model (Coulomb friction) is off?

can be. Also balance would play a big role as that makes speed inconsistent at certain angles. But the data does not confirm this (as you correctly pointed out there are not big deviations in delta times). I mean while its possible that the different effects essentially balances things out so we dont see their effect, it is very unlikely. Yet I still think that the friction and the balance combination must be wrong.

While I was working on the kayak erg with the engineer I was told that the bearings can be bent when the aluminium assembly hole is correctly sized or damaged (bent) while pushing it in. But when I look at the spindown it moves evenly but its clear that there is a lot of friction.