A2-spoke dumbbells unstable?

On 8/28/14 7:37 AM, 'Combs, Mike' mikecombs@... [spacesettlers] wrote:

> And speaking of which, Im getting lots of entertainment from
> spectacular fails due to the new factoring-in of spin instability in the
> simulations. One thing I seem to have found is that 2-spoke dumbbells
> are spectacularly unstable. The first and simplest one I did began
> quickly spinning on the wrong axis. Ive also gotten much more chaotic
> results.

OK, I've been looking into this a bit. It sure is a weird result, but
I'm not yet sure whether it's wrong.

A 2-spoke dumbbell, with a central hub, is like a deck of cards: the
three moments of inertia is all different. If not for the central hub,
it would be symmetrical around the long axis. But the central hub adds
a little bit more mass along on perpendicular axis than the other.

You can't see this in High Frontier, because the inertia display gets
rounded to a few significant digits. But peeking at a barbell with bar
length 50, bell radius 15, and hub radius 10, I find the moments of
inertia are:

I_x = 248133300000.0
I_y = 248049100000.0
I_z = 1504614000.0

Y is the (desired) spin axis, but since that's the middle of the three
moments, this is unstable due to the Dzhanibekov effect
(http://highfrontierblog.com/2014/07/30/rotational-dynamics). Just
*barely* so, but true nonetheless. This is unlike, say, a cylinder,
where two moments of inertia are exactly the same (and we're spinning
around the third). So, this would tend to cause the axis to oscillate
periodically.

There is another effect I've seen mentioned in dynamics texts, though I
don't really have an intuitive understanding of it yet: in a case where
two moments of inertia are the same, the spin is only stable around the
third. In this case, I_x and I_y are *almost* the same, and I_z is
quite different, so this effect might drive towards a rapid, stable spin
around Z (which is what you observed).

Finally, of course, there's the energy-loss effect, which for a barbell
drives it towards any flat spin (though for this, orientation around Z
doesn't matter -- it doesn't matter which way the hub is pointing).
This effect is opposite the other two. I had a two-bell bar stuck in
its rapid spin around the Z axis mode, and then hacked the code to
substantially increase the rate of energy loss. The colony promptly
flopped back down into a slow, flat spin. I conclude that with the very
small rate of energy loss the code normally uses, it wasn't enough to
counteract the other two effects -- though it is probably what knocked
it out of balance initially.

So what does it all mean? I think the shapes you were experimenting
with really are unstable -- or at the very least, their dynamic behavior
is quite hard to predict, because there are several effects at work
which lead to different final states. But when you have three or more
bells, or widen the hub so much that it's a flat (rather than tall)
cylinder, then the moments of inertia sort themselves out so that I_y is
the largest, and the stability problem goes away.

Best,
- Joe