Rigid body outline

I’m about to teach the dynamics of rigid bodies in my theoretical mechanics class, and I wanted to get down my plan. I developed and used this first a couple years ago, and I think I’m not going to change too much. Mostly it’s driven by the fact that most of my students in this class don’t take Linear Algebra.

1. Study physical pendula This is a mix of reminding them what they’ve done with rigid bodies before, but also showing the Lagrangian approach for finding the period of oscillation. Some big ideas:
   1. The axis is fixed
   2. Size affects period for realistic pendula
   3. While the speed of every particle might be different, they’re all related to the angular frequency
   4. For the heck of it, I might ask them to analyze how frisbees can bounce, or why discs fade left for right-handed backhand throws.
2. Consider L and omega There are lots of situations where the angular momentum and angular rotation vector are not parallel. Sometimes L is fixed and omega moves around it, and sometimes it’s the other way around. Omega is fixed if the axis is fixed (like step 1 above). L is fixed if there’s no torque (like free fall with tumbling). Big ideas:
   1. L isn’t always parallel to omega
   2. Wheel balancing saves the bearings that have to provide the torque if it’s unbalanced.
   3. Tumbling objects don’t always rotate around the same body axis
3. Orientation Next we work out how to describe the orientation of an object (either about a fixed point or the center of mass if there is no fixed point). There’s the typical Euler rotations, but last time my students just worked out how to do it with the chairs in the room. It was funny because they picked a different “line of nodes” than our book did. It wasn’t their fault, it’s truly arbitrary. It meant I had to whip up some notes for them to show how the equations in the book changed, but it wasn’t that big of a deal for me to do that. Big ideas
   1. It takes 3 angles (actually, I’m not sure if that’s strictly true, but I feel like it must be, since 2 put the ultimate axis in place, and the third rotates about that).
   2. If something tumbles, the dynamical variables are those three angles
4. Tackle a simple system We consider a system composed of a small number of masses that are rigidly connected. We figure out their position at any time by doing the appropriate rotations using the dynamical variables. This leads to a Lagrangian approach for those angles.
   1. The kinetic energy is a heck of an ugly expression involving the angles and the orientations of all the points.
   2. It doesn’t scale very well (8 points -> 10 seconds in Mathematica. 20 particles, more like a minute)
5. Show a better way finally we get around to the inertia tensor, pointing out that the body frame is a much better way (computationally) to consider the problem.
   1. You have to do 6 sums with your particles. That’s it. It’s pretty fast, even for 100’s of particles. Their orientations are set and these 6 sums make up the (fixed) inertia tensor
   2. You have to re-describe the omega vector in the body frame. This is, by far, the hardest part.
   3. You solve it in the weird rotating body frame, but make pictures/animations in the lab frame. Luckily we’ve already done that (using the same variables) in step 4
6. Revisit L vs omega The animations/simulations from the previous 2 steps show the complicated tumbling that can happen for non-symmetric systems. Why does that happen? Because the tensor isn’t diagonalized. So are there any axes you can rotate around that aren’t crazy? I have the students rotate the system, constantly recalculating the inertia tensor until they get zeros where they want them (they typically focus on the third column, since that’s how we’re set to spin the system). If you can get zeros (you can!), then you discover axes that won’t tumble strangely.
   1. You can find those zero’s systematically (it’s not just random hunting)
   2. Sometimes you get zeros and the system still tumbles crazily. That’s because you have small instabilities when rotating about the axis with the middle principle moment (middle as in the not the smallest nor largest value). This is the famous tennis racket problem (hold it so the face is pointing up and flip it (like a juggler). The racket will always twist in midair.
   3. For motion about those axes, L and omega are parallel
7. Add torque Now look at systems with a fixed point and some initial rotation. You’ll see precession and nutation. Have fun!
   1. It’s a one line change in the code to add in gravitational torque. You just set the potential equal to m g times the height of the center of mass.
   2. You can look at Feynman’s idea of falling to precess.

 

Well, that’s the plan, anyways. Any thoughts? As is often my custom, some sample comments you can choose from:

1. What the heck are you doing teaching this without Linear Algebra as a pre-requisite?
2. Cool, how do you have them manipulate the structures to minimize the tensor terms?
3. Whoa, wait, are you not deriving all the precession and nutation equations in class?
4. How do your students figure out the orientation part?