Saturday, April 2, 2016

Why spin-stabilized darts whirlybird so easily

Spin stabilization is one of the ways that the accuracy of Nerf darts can be improved. A spinning dart that would otherwise curve in flight - which happens due to damage or imperfect manufacturing tolerances - will instead travel in a corkscrew. The curvature of the dart's flight path is averaged out, and the overall path of the dart is a straight line.

Dr Snikkas' flywheel cages are a well-known way to put spin on a dart while firing it, and they do seem to improve the accuracy of people's blasters tremendously. The draugr team has been working on developing a mass-manufacturable flywheel cage that does the same, and, we've discovered something odd: spin stabilized darts whirlybird more. (A whirlybird is a complete destabilization of a dart characterized by the dart spinning rapidly like a pinwheel. A whirlybird can turn a shot that should have gone 100 feet into one that lands at your feet.) Sometimes a lot more: one of the test cages causes koosh darts to whirlybird almost all of the time.

Here are my (largely speculative) thoughts on why.

Edit: There was a mistake in my math in the first version of this post - this is what I get for working in a hurry and not double-checking.  It's fixed now.

As a primer on the relevant physics, I'm going to link to some scanned pages from a physics textbook - partially because they explain everything that I wanted to explain, and partially because I don't have time to re-write any of that into this post at the moment.

None of what I write here should make any sense if you don't read those scanned pages first. Since Blogger doesn't do math formatting (Boo! Hiss!), I'll use an underscore to represent subscripts and a "w" to represent a lowercase omega when referring to equations 17.1 through 17.6.

A Nerf dart has one principle axis of rotation which runs along its centerline - let's call this axis 1, with the moment of inertia I_1. The other two are degenerate; a dart can rotate in the absence of external force about any axis that runs through its centre of mass and is perpendicular to the aforementioned principle axis. Therefore, in order to apply Euler's equations to a Nerf dart, we need to pick any two axes that are perpendicular to the centerline and each other - let's call their moments of inertia I_2 and I_3.

Note that I_2 = I_3 (unless the dart is misshapen), and I_1 is much smaller than both of the other moments of inertia.

Applying Euler's equations to a dart with no spin (i.e. w_1 = 0) gives a boring result - in the absence of external forces, the angular velocity about all three axes does not change over time.

However, if there is some spin, Euler's equations present some interesting results - the change over time in w_2 is proportional to the product of w_3 and w_1, and likewise, the change over time of w_3 is proportional to the product of w_2 and w_1. One of these relationships is positive and the other is negative. This creates a situation where w_2 and w_3 cycle into each other - and, the higher w_1 is, the faster this will happen. This seems to create a funny interaction with the air that leads to a whirlybird.

Given this, why don't all darts whirlybird eventually? After all, it would only take a very small initial angular velocity on each axis for things to eventually spiral out of control (pun intended). The answer is simple: drag stabilization tends to damp out small w_2 and w_3. A dart that is traveling through the air experiences an external torque which pushes it into alignment with its direction of travel. The magnitude of this torque is difficult to estimate, but, given that darts usually don't whirlybird, it has to be enough to stop whatever initial angular perturbation darts experience from growing.

A Dr Snikkas cage puts a fair amount of spin on darts that go through it - i.e. w_1 is high - but it also launches the darts with an extremely low w_2 and w_3 due to its high-tolerance pseudobarrel and concave flywheel surfaces. A cage which puts a similar amount of spin on darts, but with flat flywheels and no pseudobarrel, spits out whirlybirds.

This explanation predicts that a cage with more canting will, other things being equal, be more prone to producing whirlybirds. Aside from this, I can't think of any predictions that we could use to test this explanation - different darts have both different weight distributions and different drag stabilization coefficients, the latter of which are unknown and therefore confound our ability to make predictions.

If I am right about what is causing these whirlybirds, there are only two ways that we could solve this problem: put less spin on the darts, or launch them with less irregularity. The former would be the easiest solution, and the latter would be the best for accuracy.

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