I’m closing in on the end of my semester of teaching Theoretical Mechanics (often called Classical Mechanics – essentially an upper-level course for physics majors that revisits mechanics with more math). We’ve got two more chapters to cover and I wanted to post some thoughts here about the last one, the chapter on normal modes of oscillation.
This chapter is a great lead in to the concept of resonance of a mechanical system. We start with just a few particles, typically connected by springs, and try to find solutions to the equations of motion that have every particle oscillating at the same frequency. We try to find the frequencies where that happens and also try to find the relative motion of the particles when it happens.
Here’s a few animated gifs that show the response of a simple 2-mass, 3-spring system when excited on either resonance and when excited by a non-resonant frequency:
It’s (somewhat) clear that the system responds with a much larger amplitude when you excite it at one of its resonances (top two) than when you don’t (bottom one).
Typically this chapter is a lead-in to continuous systems where you study a continuum of resonance issues but it’s not uncommon to end this course just after this chapter, as I’m doing.
Here’s my problem: to really teach this stuff for finite particle systems, you have to hit linear algebra really hard and you have to diagonalize two different matrices at the same time. Even if I didn’t have the misfortune to be teaching this class without linear algebra as a prerequisite it would still be quite a tough and long ride.
Ok, so here’s my cop-out hare-brained scheme: Instead of first linearizing and then diagonalizing the system, we’ll simply model it with Mathematica and study the following.
- Where is the equilibrium location? We’ll put in a ton of viscous friction and let the system settle down.
- What are the eigenfrequencies? These are the resonant frequencies using fun German prefixes. We’ll re-model the system without friction and start all of the particles at a position nearby to their equilibrium positions. Then we’ll Fourier Transform the motion of various particles to find the resonance peaks.
- What are the normal modes? This is the description of how each particle moves when executing one of the eigenfrequencies. We’ll either filter the previous motion allowing only one peak of the spectrum or excite the system somehow like in the above animations.
After working really hard on Lagrangian dynamics earlier this semester, point (1) is super easy, at least in principle, for my students. For point (2) I’ll have to show them how to do the transforms in Mathematica. Point (3) will take some work for the first suggestion and just some ingenuity for the second.
Here’s an example of how we might go about doing it for a system of a 2-mass, 2-spring system hanging from the ceiling:
I’m excited for this way of teaching it as it seems to be what happens in real life when you look for resonances of a system. Basically you hit it and see where the peaks in the Fourier spectra are. I’m nervous about the lack of mathematical rigor, though, so please feel free to push back in the comments. If you really think I’m making a mistake you have two weeks to change my mind for this year :)