Boring post alert!
I've been asking for opinions on flight arrows and one suggestion was to select the spine by getting 'em to fly straight without flights. That's fair enough, so why do we need flights at all?... well the back end may well waggle in flight or move round in a circle, I've seen it with an arrow with wet flights the point went straight and the nock was describing a 3 inch circle!
The flights will also damp down the vibration/oscillation of the arrow caused by it flexing round the bow. This begs the question, how long does the vibration last?
So, I took one of the shafts I'm using and clamped one end in the vice, I pulled the free end out by one inch (a reasonable deflection) and let go... blimey the damn thing was vibrating significantly for at least 4 seconds! Well even at at 150fps it would still be vibrating after travelling 200 yards!
Ok there are loads of errors and approximations there:-
1. The mode of vibration is more likely to be with either end stationary (nodes) and the middle moving (anti-node), rather than the one end fixed (node) and the end an anti-node.
2.The deflection would probably be less.
So lets assume the oscillation dies down twice as quick and was half as big, so we'll divide the time by 4. That still gives 1 second which is 150 feet or 30 yards as we sometimes call it.
That pretty much ties in with the sort of waggle you can sometimes see.
Ha! Just found a better test. I tried to hold the ends and twang the middle but it wasn't really feasible... but clamping the middle and twanging both ends simultaneously produces the same result from a topological view point! Yes the vibration dies down quicker, but does go on for about a second... there, I'm not as daft as I look! :-)
Ah, but the fletchings aren't at the anti node, so they won't be damping out the movement, Hmmm I'm more confused than when I started. maybe I should try having fletchings in the middle of the arrow too? just kidding... or am I?
I'm not suggesting that this is by any means a good mathematical model, but maybe it's indicative of the usefulness of the fletchings. It's a trade off between stabilising the arrow rapidly and increased drag for the rest of the flight.
I'm pretty sure that actual testing is the only way to optimise the arrow to the bow, no way you can buy an "off the shelf" flight arrow.