Multiple Rotating Masses in Revolution

[JaredJHuffman]

I've simplified my question to make it easier to explain; I'm trying to find a way to determine the motion of a freely turning body as resultant of 2 counter-weights on rotating arms.

Here's my write-up:

Description: Electric motors turn both orange shafts clockwise at constant angular velocity. Both weights start out closest to the center blue shaft as shown in Position 0 and rotate with their respective orange shafts. The orange shafts turn freely with respect to the green arms, and the green arms turn freely with respect to the blue shaft. All objects’ masses are negligible except for the red counterweight blocks, and the blue shaft is grounded. See pictures for positions 0, 1, and 2 for the relative motion of the red arms at 90 degree intervals.

Question: What is the rotational position of the green arms over time? What factors will affect this motion?

Thanks for any help you can offer! Let me know if there's anything I can clear up about the situation.

 

[Filip Larsen]

Welcome to PF!

I assume you mean, that the motors turn the red arms relative to the green arms with constant angular velocity (and not freely as you said). If so, conservation of the initial zero (angular) momentum will make the line from the CM of one block to the CM of the other stay fixed in inertial space, meaning the green arm will swing out to a maximum when the red arm is at 90 deg to the green arm (picture 2) and then back in when the angle is 180 deg (picture 3). Seen from above the red blocks would appear to slide directly away from the blue center and then directly back in. To the extend the green arms do have some mass, they will carry some of the momentum making the red blocks (as seen from above) describe ellipses instead of straight in and out. The more mass the green arms have relative to the red blocks the less the green arms swing out and in and the more the red blocks describe a circle as seen from above.

PS: I assume this is not homework. If it is, you should ignore my answer and post in the homework section of the forum.

 

[JaredJHuffman]

Thanks for the explanation! I'm actually trying to impart a vibration by the oscillation of the green bars, so this is the motion I was hoping to get. If I understand correctly, I could get less sweep out of the green bars by shortening the red bars or by lengthening the green bars (or both)?

 

[Filip Larsen]

If I understand correctly, I could get less sweep out of the green bars by shortening the red bars or by lengthening the green bars (or both)?

Yes.

If this is going to be a practical device for vibrating the green part you most likely need to ensure that the angular position of the two red axis relative to the green arms are synchronized. If you don't and one of the axis gets 180 deg out of phase with the other, the green part will no longer rotate back and forth around the blue axis but just deliver a translational force that rotates.

If you are only interested in delivering a rotational vibration I would think it should be possible to make a simpler device that requires only a single drive engine and a single moving part.

 

[JaredJHuffman]

You're absolutely right about the timing; even a few degrees out of time would cause it to drift. Thanks for all the help!