# Finding normal modes of a 3D object

carllacan
Hi.

I'd like to find the normal modes / harmonics, displayed in a dB - Hz graph, of a given 3D object, namely a wind instrument shaped like a hyperbolic con with holes. I'm trying to perform simulations on Comsol Multyphysics, but I don't know how to do it. Is Comsol the best suited software for this task? If so, could you point me to a guide were they do something similar?

Thank you.

Homework Helper
Since the is a "wind instrument", most likely you really want to find the normal modes of the air vibration, not of the material of the instrument. That is a tough problem to solve exactly, because you need to include the effect of the infinite amount of air OUTSIDE the instrument in your model.

One way to do that is to use an approximate model for the acoustic impedance of the holes in the instrument and couple that to a model of the air inside.

Another way is to model the instrument inside a large but finite volume of air (i.e. model the instrument being played inside a closed room), and then ignore the modes that correspond to the room and not the instrument.

If you go that way, it may be better to do a frequency response analysis exciting the air in the same way that the instrument is played, which will "automatically" pick out the instrument modes in preference to the room modes. With a frequency response analysis, you can also include "no-reflection" boundary conditions at the walls of the "room", to suppress the response from the room itself (i.e. you are modeling the room as an anechoic chamber).

If you also want to include the flexibility of the instrument itself, you need to do a coupled fluid-structure analysis.

I've no idea how to do the above in COMSOL, but it may give you some clues about what to look for.

Finding the vibration modes of the instrument itself (vibrating in a vacuum) is much easier, but probably not what you are interested in - unless you just want to check the instrument doesn't have any vibration modes in its playing frequency range, which might produce "wolf" notes when the air and instrument vibrations are coupled together.

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carllacan

A friend of mine is trying to build a wind instruments (this one, if you are curious) and asked me how could he perform simulations on Comsol, so that he didn´t have to do everything by trial and error. I told him what you said and see if we can get something out of it :-)

I am really interested on this topic. Could you recommend me some books (or sites, or whatever) on acoustics or physics simulations so that the next time I have to do something like this I can do it by myself?

Again, thank you for your time.

Gold Member
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Homework Helper
carllacan
If your main interest is postioning the finger holes to accurate tuning, this is a "practical" level of calculation, rather than trying to do a complete simulation of the physics:
http://www.chrysalis-foundation.org/flute_tone_holes.htm

Actually I was trying to build the instrument, but that seems really interesting. Thank you :-)

carllacan
Sorry for reviving this post, but I've finally been able to get my hands on Comsol, and I have already built a geometry for the instrument. I also created a cube of air for finding the harmonics. Now I have 2 doubts:

1- How should I model the "playing"? I mean, I've been wandering around the options and I decided it must be either by creating a vibration at a mass of air at the beginning of the instrument or by simulating a flow of air. Which one would be aproppiate, if any?

2- The material for the instrument. There are not much types of wood on Comsol, so I'll have to create one similar to the one it is normally used. Which parameters (density, hardnes...) do you think will influence more the harmonics? Density for sure, and I am thinking elasticty, Young model and so on... any suggestions?

Gold Member
You need to solve the wave equation for pressure in the fluid volume of the instrument. The structural resonance and its interaction with the vibrating air is negligible, so the main material you need to consider is air.

AlephZero said:
That is a tough problem to solve exactly, because you need to include the effect of the infinite amount of air OUTSIDE the instrument in your model.

Actually, you can neglect the effect of the air outside of your instrument. There is a small interaction effect at the open ends of your instrument called the end effect, but to include it you need to introduce part of the outside air and some special boundary conditions. It's also mentioned briefly here, where they say the end effect effectively adds 0.6*radius to the total length of your tube:
http://www.phys.unsw.edu.au/jw/woodwind.html

Here is a link that might be useful, The acoustic muffler problem they discuss can easily be modified first to a simple tube with open ends, so you can try if you can reproduce the analytic results of a simple 1 m straight tube, then you can proceed to your real geometry: