An interesting data point on multi-stage flywheels

A few years ago, I wrote a post exploring the possibility of stacking flywheel cages to achieve higher velocities. This was written mostly from a theoretical perspective. At the time, there was very little actual data on how much velocity could be achieved with such systems - very few people had made multi-stage flywheel systems, and those had a pesudobarrel in the middle which dropped the velocities that could be attained well below the theoretical maximum.

As you might recall, the simple model that I used predicted that the glass ceiling velocity of such a system would be equal to the square root of the number of flywheel stages times the glass ceiling velocity of a single stage - and that the spacing between flywheels would have no effect on this velocity.

498 Nerf has obtained some very interesting results from a two-stage flywheel system with varying spacing between the stages. A video explaining these results, along with links to the chrono results, can be found here.

In summary, he obtained:

• 140 fps with the flywheel stages jammed as close as possible together
• 155 fps with the flywheel stages moved a bit further apart
• 165 fps with the flywheel stages spaced such that a dart enters one just as it leaves the other

All of these results were obtained with the same motors and voltage.

This last result is pretty close to what we should expect based on my model -  but the variation in velocity with flywheel spacing demands explanation. I can say that the explanation that 498 proposes is either incorrect or incomplete. While increasing the spacing between flywheels will result in the dart spending a greater total time in contact with at least one stage, it will also decrease the time that the dart spends in contact with both stages - and, mathematically, these effects should cancel.

Kysan 16180 motors running on 4S have a no-load speed of 22,200 RPM, and rhinos running on the same voltage have a no-load speed of 44,400 RPM, if we neglect energy lost due to air resistance on the flywheels assume a nominal voltage for the battery. These are greater by a comfortable margin than my calculated lower bounds on the critical RPM for each flywheel stage of 20,200 and 28,500 RPM for the first and second stage respectively. So, the numbers check out - it looks like 498's multi-stage flywheel system is supercritical, or at least pretty close.

However, the first flywheel stage has a free-running RPM that is less than the second-stage critical RPM - and I think that this is what explains the drop in velocity when the flywheels are spaced closely together. When the flywheels are spaced close together, the dart is accelerated by the second stage while it is still in contact with the first stage - and it is accelerated beyond the flywheel surface speed of the first stage. This results in the dart dragging the flywheels forwards instead of the flywheels dragging the dart forwards - hence the drop in velocity.

So: while these results may at first glance appear to contradict my model, they actually support it on closer examination. Given that the velocity attainable with a single flywheel stage is already pretty close to the maximum velocity that people outside of NIC wars usually want, this is largely just a matter of curiosity - but it's still nice to finally see some experimental verification.

Update: 498 has done some further experimentation, which provides more confirmation - his original setup, which had the motor cages right next to each other, yielded a higher velocity with first-stage motors with a higher RPM (Kysan 17114, which turn at about 30k RPM at 4S), indicating that his original first-stage motors were not supercritical.

Interestingly, he obtained an average of 152 or 160 fps (depending on whether a few shots, believed to be misread, are discarded), which is still less than the 165 fps that we would expect. Perhaps 30k RPM is still not quite supercritical - or perhaps the difference of ~5 fps in the average results is just an artifact of small sample sizes. Either way, these results strongly suggest that the 'dragging' effect is real.