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The picture below shows an isolated system with a fairly massive wheel at one end, attached via its axle to a long shaft, like a bike tire on a bike frame, but the bike frame is merely a low mass 'truss.' At the other end of this long shaft is a mass of roughly the same magnitude as the wheel, so the center of gravity (CG) is roughly in the middle of the shaft. This mass is fixed to the shaft with no rotation possible. The assembly is floating motionless in 'space', with no contact to anything else.

Diagram of apparatus

If a motor on the 'truss' causes the wheel to spin in one direction, the truss would begin to spin in the other direction about the common center of mass. The net angular momentum would thus be zero both before and after the spinning commences. The wheel and the truss would rotate in the same plane. The actual rotation rate of each part is dependent upon the individual masses, wheel radius, and the length of the shaft. It is getting complex, but so far easy enough to envision in general.

Now, place into the 'truss' a mechanism that causes this long shaft to rotate on its long axis (which would be perpendicular to the spin axis) while it is still undergoing the above rotation relative to the wheel. Upon rotation at this new joint, a force is applied to the axle of the spinning wheel (and of course to the mass at the other end of the truss, too) causing each part to try to turn in the opposite direction. Given that the net angular momentum is still zero, the vector sum of the individual parts (the spinning wheel and the body as a whole rotating around its CG) must be equal and opposite. I cannot envision the result, however, nor do I have the background to derive the equations of motion of this 'whirly gig.' The best I can imagine is that the plane of overall rotation should change, with the plane of the wheel's rotation changing in the opposite direction. This would seem to result in a state where the wheel is no longer spinning in the same plane as the overall assembly, but that would seem unsustainable and perhaps unstable. How can the behavior of this system be understood?

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It would help a lot if you could draw a picture to visualize the situation. As we all know, a picture is worth a thousand words. – Vijay Murthy Apr 16 '12 at 20:22
I tried to add an image, but wasn't allowed to since I am a new user and they don't allow anyone with fewer than 10 posts to include an image. Spam prevention. I did try to improve the text, however. I would be happy to email a powerpoint or pdf image to whomever asks for it. – Tom Apr 16 '12 at 21:14
After thinking more about this, I am beginning to convince myself that, once the joint in the shaft is turned, the plane in which the larger assembly rotates would begin to wobble, essentially keeping the axis of this spinning plane 180 opposite from the axis of the wheel. Both axes would thus precess around their original spin axis. It would be a pretty wild system. I am by no means certain of this thought experiment, however. Can anyone confirm? – Tom Apr 16 '12 at 21:29
Tom, if you can upload the image anywhere on the web and put the URL in the post or in a comment, then someone else can easily include the image for you. If you don't have access to any other website where you can post it, you can email it to me at the address listed in my profile and I'll upload it for you. – David Z Apr 16 '12 at 22:39
Aha! I think I got it now. Starting from the support (near the CG) there is a 2DOF joint allowing rotations in X-Y axes (Z is coming out of screen). The support structure is placed on this joint with one end on the right having a fixed counterweight and on the other end another joint along then local Y axis (up) where a disc is spinning. This last joint is powered by a motor. The question is to describe the motion of this machine. – ja72 Apr 17 '12 at 19:00

The drawing you generated is much better than my poor attempt, and accurately shows the important aspects of the problem. Thank you. There is one difference, however. My system is literally floating in outer space, with no contact holding it up. This could be approximated in the apparatus you drew by having the support be, say, a ball joint which allows for more freedom of motion of the entire assembly so that it can wobble. Or it could be hung with a u-joint at the c.g.

To set up the problem, it starts with everything motionless. When the motor is started to spin up the disk, the rest of the assembly starts to rotate in the opposite direction to maintain the net angular momentum at zero. The complicated part comes next.

My suspicion is that, once the joint in the horizontal shaft at the blue support is twisted (not continuous spinning, just tilted), thus tipping the spinning disk, the entire length of the shaft will start to wobble as it rotates. This would be in response to the fact that the moment of the spinning disk has tipped, and an equal but opposite tilt in the moment of the spinning assembly must occur for the net angular momentum to remain at 0. As it spins, both these moments will precess, but will continue to cancel each other out. And when the joint at the blue support is returned to the original orientation, this wobbling will stop and the assembly will return to spinning in the original plane. It might make an interesting toy, but that is not what I am aiming for. My goal is to find a way to manipulate the spinning apparatus so that the plane of rotation of the entire assembly can be changed to a different (but stable!) plane. Replacing the rotating joint with a simple 'knee' joint would seem to result in the same behavior. Despite the fact that there is no net angular momentum, the system seems to resist my attempts to 'capture' it in a new orientation.

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If it is floating in space, why have the support collar at all? I would just remove all mention of supports and joints near the cg. There is floating bar with a disk spinning on one end. Simple! – ja72 Apr 18 '12 at 13:59
I agree. The design is simple. Building and testing such a thing on Earth requires some support however. I'd love to find a '0-g room', but they don't exist. The behavior of this simple design seems fairly complex, however, and my goal is to find a design which allows one to adjust the plane of rotation without expending propellant to overcome angular momentum. The ultimate goal is to use such a design to provide artificial gravity in space. There are many other constraints in such a design. Preference is to not have to stop it to adjust the plane, but that can be done with this design. – Tom Apr 18 '12 at 16:04
LOL! When I did my undergraduate research I was looking for a frictionless plane without gravity also to test impacts on articulated mechanisms. Those kind of things are really hard to construct. I suspended my mechanism from a 3rd story balcony and video taped it from above as it moved. It worked nicely in the end. – ja72 Apr 18 '12 at 17:30

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