# Torque calculation, to achieve clean spin+tumble

Here's a pencil-like robotic spaceship carrying an experiment, it is a solid mass 100m long, 100 inches thick and weighs 1000kg. We're in deep solar space 100au above the sun.

Assume we can apply any platonic torque to the object.

Notice the global unchanging XYZ axis shown.

If the body is rotating around it's own long axis, we'll call that "spinning".

If the body is rotating only around the global Z axis, we'll call that "tumbling".

To be clear (i): if it is "tumbling", then the "spin" would not always be around the global X axis. "spin" is around the long axis of the object: that axis would change globally as the object moves in other ways.

To be clear (ii): what we describe as "tumbling" would take place only in the global YX plane, with no movement to-or-fro in the Z direction.

Now:

what we want the spacecraft to do is spin and tumble cleanly, at the same time.

To be clear: we want the want the angular momentum to be the sum of two items. A constant component along the global z axis. And a rotating component which is perpendicular to the z axis. (Indeed, along the body length for clarity.)

Of course ... that's absolutely impossible because of gyroscopic effects.

If you give it two naive torques to spin it, and, make it tumble, in fact there will always be a "wobble" in the third global axis (ie: seen from overhead it will rotate slightly back and fore in the y axis).

So - what torque would you have apply

# over time

to get it "spinning" and "tumbling" cleanly with no wobble in the Y axis?

In a word:

what is the torque you must apply over time to force a body, to have two angular momentums, one causing the body to rotate at h HZ along the Z axis, and the other causing the body to rotate at i HZ along an axis which is rotating at h HZ (ie, in sync with the first-mentioned rotation) in the XY plane, with no other motion or wobble?

I imagine the answer is something quite simple like "oh you must apply sin(time) torque here" or some other cohesive solution.

This would seem to be an everyday issue to engineers and the like, so I imagine there is some well-explored solution. But I couldn't find it.

Joriki has kindly pointed out that the solution is a torque where: the axis of the torque initially points along the global y axis, and then, the axis of the torque rotates in the global XY plane matching perpendicularly the z-rotation of the object.

However I ran this in a simulator and it doesn't work. If you set the z-rotation of the torque axis to some fixed value, it just twists the object somewhat randomly. If you set the z-rotation of the torque axis to zero and slowly increase the z-rotation of the torque axis, again it largely just twists it around randomly.

It occurs to me that the formula to get the object to do the behaviour described, would have to, in fact, take in to account the current angular momentum of the object at any moment. Perhaps? It is very confusing to see how you would start the motion or describe maintaining said movement. Both problems - starting or maintaining - seem astoundingly difficult.

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I suggest to avoid words like "tricky" in the title; this is a very subjective assessment. – joriki Jul 8 '12 at 8:31

The torque is the time derivative of the angular momentum. If I understand correctly what you mean by "spinning and tumbling cleanly with no wobble", you want the angular momentum to be the sum of a constant component along the $z$ axis and a rotating component perpendicular to the $z$ axis, along the body axis. The time derivative of this sum is a vector perpendicular to both the $z$ axis and the body axis, which rotates in sync with the body. In your snapshot of the motion, it would point along the $y$ axis.
@Joe: I doubt that I fully understand all of your questions; you might want to try framing them in more formal language; but one thing I would reply is that what you describe as a wobble isn't actually happening, so it's not relevant whether the wobble would go back and forth if it happened; what you want to know is what torque to apply for the wobble to not even begin, and that torque does indeed rotate continuously and uniformly with the body around the $z$ axis. – joriki Jul 8 '12 at 9:41
@Joe: Yes, you're right, the torque is always in the $x$-$y$ plane; it rotates about the $z$ axis at the same angular speed as the body axis, remaining perpendicular to it. – joriki Jul 8 '12 at 11:14