Visualizing Rotation of a Rigid Body

My fans (yes, both of you! Or am I down to one? zero?) may be wondering where I’ve gotten to. I’ve been working up a visualization program for a conference in Nanjing, China.

Back in grad school, I was an aerospace engineer, working in spacecraft dynamics and control. Torque-free rotation of a rigid body is one of those basic physics topics that we studied, on the way to figuring out how to control rotational motion of satellites and such.

My PhD advisor has gotten a big award, to be presented at a conference in Nanjing, and they’d like to have his students and colleagues come to present. Not having worked in the aerospace business for a while, I don’t have any research results to present, but I thought it would be fun to use OpenGL to visualize some of the analytical solutions to the rigid body rotation problem.

Recall what a spinning football does. If you get it spinning exactly around the long axis (the “perfect spiral”), it just spins around that axis. More often, it gets that funny wobble. How do we figure that wobble out?

One way to visualize the motion is to think about the angular momentum and angular velocity vectors (not the same, unfortunately, unless you have a spherically-symmetric object, which footballs aren’t). The angular momentum vector is constant in inertial space (what you’d see outside the vehicle, if you could see angular momentum vectors), but isn’t in a body-fixed frame (what you’d see if you were strapped into the vehicle).

What’s the angular velocity vector? If you think of the object as spinning on an imaginary axle, the angular velocity vector points along the axle, and its length is proportional to the speed of rotation. This axle usually has to be imaginary, since its direction can change relative to the body.

So I’ve worked up a simulation of a rotating, rigid body, with angular momentum and angular velocity vectors added.

A rotating body, with its angular momentum shown as a yellow arrow, and its angular velocity shown as a purple arrow.

This is a frame of the animated image. The yellow arrow is the angular momentum vector, and the purple arrow is the angular velocity vector. The red and yellow ice cream cone thing is the “spotlight” that illuminates the scene.

More images and explanations to come as I make progress with the visualizations!

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