Update: here’s a small tune I made to demonstrate what the resulting sample would sound like as an instrument:
I’ve always been fascinated by how things, especially sound waves, look like in slow motion. By examining something in slow motion, somehow even complex phenomena start to seem intelligible.
In this post, I’m examining the behavior of a very simple instrument in slow motion. The physics are very much the same as when examining vibrations in structures, but instruments are (very much) nicer to listen to. The principles shown here are naturally applicable to larger scale phenomena that occur in structures of different kind (buildings, bridges, etc).
The instrument will consist of a cantilever beam made out of steel, with a length of 20 cm. The cross-sectional dimensions of the beam are 2 cm x 2 cm. The instrument will thus resemble something like a simple vibraphone / glockenspiel / large music box.
We’ll use the same principle to get the sound of the beam as an electric guitar uses to pick up the sound of a string, by placing a (virtual) pickup at a position 3/4:th to the left of the tip of the beam (or 1/4:th from the left of the beam, whichever suits your fancy).
What will happen if we hit the very tip of the beam, with a very sharp, impulse-like force (even sharper and quicker than what’s possible with the hammer in the picture)?
First, it should be noted that the same principle is used as when an electric guitar picks up the sound of a string. The pickup transforms the deflection at a certain point on the string (or in this case, beam) directly to sound. The resulting sound for the beam is as following:
To see what the beam looks like directly after being hit by the sharp impulse, examine the resulting deformation of the beam as a function of time here (click on the blue area to load the content). The sound is generated from the movement of the circle in the direction of the y-axis.
Some things to notice when watching the time lapse of the deformation:
- There’s a transient part at the very beginning (slightly visible in the 0.0015 s time span) which attenuates very quickly. This is the part of the response where standing waves (resonances) have not yet formed.
- The two lowest natural frequencies (resonances) can be seen clearly; one at 464 Hz and another at 2910 Hz. The third natural frequency, 8150 Hz, can be seen at the very beginning of the response.
- For this setup, the higher frequencies attenuate quickly. In the end, only the lowest natural frequency, 464 Hz, can be heard. This gives the distinct pitch you can hear in the sound sample above.
I’m using finite element analysis to examine the behavior of the beam. The theory is the same as in the previous post, but I’ve also calculated the mass matrix for the beam. I’ve used 25 elements for the beam, thus solving a 50-degree of freedom system. Note that I’m using the Euler-Bernoulli beam theory, so some simplifications are made.
The damping was done using Rayleigh damping. For those familiar with this type of damping, the value for $$\alpha$$ was 1e-05 and the value for $$\beta$$ was 1.5e-06. I chose these values simply on the basis of listening to which damping values sounded better than others, not much thought were put into them. They seem surprisingly small as compared to other values I encountered online, maybe someone more acquainted with Rayleigh damping could offer me their opinion on this?
I used the Newmark algorithm for time-stepping, with a value of 0.5 for both $$\beta_1$$ and $$\beta_2$$.
I used Python for the calculations. Python is ideal for such an endeavor, and free! SymPy provided me with the tools I needed to solve the necessary equations, while NumPy did the calculations for me.
I saved the resulting calculations as a binary file, which only contained the necessary information. For example: only one byte / element describes the deformation at each time step in the animation seen above, as 8 bits is more than enough in this case.
Thoughts for the future