# Simulating the physics behind a music box

Here's what the simulated sound of a music box sounds like:

## Some background

The comb teeth [3] in the picture below are the ones responsible for the notes you can hear.

The ratchet lever [1] rotates the cylinder [2], the pins pluck the comb teeth [3] which produces the music. The whole thing rests on the bedplate [4]. Source: wikipedia

## Version 1

The first model I made consisted of a uniform thin metallic plate (see the really cool gif below!). Problematically, there's some really strong coupling between the pins. Here's what happens when you apply a static deflection to one of the pins and let it go:

FEM model of a music box with a uniform plate thickness. Note the coupling of the pins.

As you can see, the adjacent pins start to vibrate after some time. The resulting tone is decent, but it's a bit "wobbly" due to the coupling between the pins:

## Version 2

The problem with the coupled pins must be something that would happen with real music boxes as well. Well then, how has this problem been solved before? What I noticed was that the thickness of the plate seems to vary. OK, so let's double the thickness of the plate everywhere except at the pins. Additionally, let's move the stiff supports a bit closer to the pins. Combined, these actions decrease the amount of sound that travels from one pin to another significantly. So, in practice, we get a tone that is a lot clearer. Great! After some minor additional changes, this results in the sound you can hear in the example at the very beginning of the post. There's still some coupling between some of the pins, but it's small enough not to be bothersome.

Refined music box FEM model. Note the absence of the strong coupling present in the previous model.

## Some theory

I used the Mindlin–Reissner plate theory, which should quite accurately describe the vibrating thin plate that causes the sound. I solved the problem numerically using finite element analysis using my own solver in python.

I started by calculating the static deflection of each pin when a force is applied at the very tip. I then "released" the pin, and calculated the resulting deformation in small time increments for each pin (using the GN22 algorithm). Then, I calculated the average position, velocity and acceleration of the plate at each time frame; if the whole plate is simplified and considered as a single mass with one degree of freedom the forces the plate directs at surrounding structures (such as a table, which I assume radiates most of the sound caused by the box to simplify things) should be somewhat proportional to the average acceleration of the plate.

The frequencies the pins vibrate at can be calculated according to the formula for the lowest natural frequency of a cantilever beam:

$\frac{3.52}{2\pi L^2}\sqrt{\frac{EI}{m}}$

where L is the length of the pin, E is the elastic modulus of the material, I is the moment of inertia and m is the mass per unit length. If $L_0$ is the length of the longest pin, we can now get the lengths of each subsequent pin using $\frac{L_0}{\sqrt{2^{i/12}}}$, where $i$ is how many semitones higher we wish to tune the pin to.

## Some thoughts

I'm really happy with the result, it sounds almost as good as the real thing. I added a bit of reverb, which is quite important for a natural sound. I think this is as far as I'll go with my music box simulations, but we'll see. 🙂