I was recently cutting down a tree and, even though I’m ashamed to admit it, it fell in the wrong direction. It fell in the opposite direction as to what was intended. Well, lesson learned. Next time I will be more careful and use a rope if I want to be completely sure it falls in some specific direction.

Cutting down a tree (from google, that’s not me)

So, let’s investigate things. Not by cutting down more trees, but instead by simulating doing it!

So, what’s happening?

The tree in static equilibrium. The center of mass is marked to the left of the trunk.

To simulate this, i drew a very simple tree-like structure in 2D (to defend myself I’ll have to say that the tree I cut down looked a lot more centered than the one above, where it looks quite obvious that the tree would fall to the left). The tree has a lot more branches on one side as compared to the other. The center of mass is shown to to the left of the trunk. The resulting average force will force the tree to tip over to the left as it’s located to the left of the trunk, no matter how we cut the main trunk (if there is some magic trick I’m not aware of, please let me know).

The deformation of the tree after being cut

After cutting a dent in the trunk with the purpose of allowing the tree to fall to the right, we can see that it starts to lean to the left. Whoops. This moves the centre of mass even further to the left. From the figure above, you can also see that the tiny narrow partition to the the left of the dent is carrying the whole tree; all the major forces (in red) are now centered around the dent.

Conclusion

As long as the centre of mass is located to the right of the tiny partition left after the dent, we should be fine. This is not always easy to see, especially if the tree has a lot of branches.

The theory

I used my solver in python with plane elements to calculate the figures above. The colors represent the von Mises stress.

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. 🙂

Here are some nice images of vibrating window frames (ooh, wow)! The finite element model I made with Python will hopefully enable me to do two things: calculate the radiation coefficient of the window frame, and calculate the effect of coupling the wooden frame with the windowpane (although shear locking is definitely a problem for the model I’m using at the moment). These things should be very central when one considers the computational airborne sound insulation of the structure as a whole.

I’m also thinking of making a more detailed model of the music box I wrote about earlier using this model at some point. That should make a fun topic for the blog.

In this post, I’m going to examine a hypothetical small open plan office, and the optimal way to treat the space acoustically. Check out the publication related to some of the theory I’m going to base this on here. I’m going to make the example geometrically simple, so the result will be clear and somewhat intuitive.

The setup

Small open office setup

A hypothetical simplified small open plan office is shown in the picture to the left. The spheres and cubes represent the possible positions for the office workers.

I’ll assume that sturdy office screens are placed air-tightly against the wall and floor, so that sound doesn’t leak through the edges of the screens. I’ll also ignore any sound diffracted over the screens.

Reflections

First degree reflections

The sturdy office screens isolate sounds very well; this means that sound doesn’t travel directly from one position to another, but instead through diffraction (which I assume to be negligible) and reflections.

First and second degree reflections

First degree reflections are relatively easy to predict. Second degree reflections are already significantly harder to predict. Third degree reflections are very hard to predict without computer simulations. Third order reflections (and above) are often already far from intuitive.

The goal

I wish to hear as little as possible of my coworkers. The office screens already attenuate direct sound. But this is not enough. If I don’t consider the other routes the sounds travel from one position to another, the screens will function as little more than visual barriers.

So what do I want to do? It turns out that early reflections are almost always the most important reflections to consider when one wishes to affect speech intelligibility. Another important factor is the background noise level, but I’ll assume that the ventilation provides a decent amount of masking noise. Keep in mind that by early reflections I mean reflections arriving early on in time, without taking any notice of how complicated the path the reflection has traveled is.

I’ll make the following goal: I want to get rid of the early reflections as effectively as possible, using a relatively small amount of absorbing material, such as acoustic panels. Let’s assume that I can’t place anything on the floor, as it would make cleaning (and walking around the room) too difficult. What is the optimal way to place the absorption?

The result

The result

The figure to the left shows the places where absorbing material should be placed, with dark blue representing the most important positions. There are two places where the placement of absorbing material is very central in this example; the ceiling above the office workers and the wall on the opposite side of them. In this simple case, the answer is fairly intuitive. For more complex situations, this is not always the case.