Finding the vibrations of a Membrane, given some sound

[Sleeme]

Hey everybody,

I'm trying to figure out what the vibration modes of a circular membrane would look like if we put some sound through it.
As you may know does a circular membrane with clamped edges at radius a behave according to the wave equation.
The general solution thereof has the form:
[itex]
u(r,\theta,t) = J_m(\lambda_{mn} r)\left(C\cos m \theta + D\sin m \theta\right)\left(A\cos c\lambda_{mn} t + B\sin c\lambda_{mn} t \right)
[/itex]

Therefore given a vibration mode n,m we can calculate its frequency f:
[itex]
f_{mn}= \frac{\omega}{2\pi}=\frac{c\lambda_{mn}}{2\pi} = \frac{c\alpha_{mn}}{2\pi a}
[/itex]
where alpha are the zeros of the bessel funcion

So every mode has such a eigenfrequency.

Now working the other way around: when given a soundfile, how can I determine, how the membrane would vibrate? Mostly I'm interested in solving this discretly not analyticaly.
I tried to come up with a formulation like in the discrete Fourier transform:
A*x = b where b is the sound and in A (of size t,mn I have sin(\lambda_{mn}t) where t is different in each row. But when solving this for x somehow the solution has very high frequencies. For example for a simple sinus, so I'm obviously doing something wrong here

Any Idea? Am I completely wrong how I could solve this?


Selim

 

[Sleeme]

Does nobody have an idea? Would be really great, as I am working on a Project, that could reaaally use this. Just ask in case the question is not clear.Selim

 

[Vadar2012]

So isn't eigenfrequency just another term for natural frequency? I'm guessing you don't want to model it in an FEA program and just want to do it by hand?
So you could folllow something lsimple ike this:
http://hyperphysics.phy-astr.gsu.edu/hbase/music/cirmem.html

 

[Sleeme]

Thank you for the answer. To be a bit more specific: I use it together with a fluid simulation in order to model an experiment, where they placed some paint on top of the membrane, which was wrapped over a speaker. When they let sound through the speaker, the membrane starts to vibrate interacting with the fluid.
I've already modeled the membrane with the equations similar to your link, and the full setup is complete, but the problem is, that the vibration modes have to be selcted manually. It would be great if I could generate the modes by a sound file or microphone input and if possible physically correct.

Here's a sample video of the outcome, to give you an Idea what I do:
http://youtu.be/3XjPa_qplws?hd=1

 

[Gordianus]

Why don't you just take the FFT of the sound file and search for the different peaks in the spectrum?

 

[Sleeme]

Thanks for the answer Gordianus.
And what am I going to do with the peaks afterwards? How can I map them to the vibrations?
I mean If i get a frequency of 10, how should I map it to the other base? In the case of a circular membrane I get floating point values for each mode, for example 2.4/radius*c for the fundamental vibration mode. Thats why I tried to do something similar to a Fourier transformation, but I couldn't manage to find it out this way.

Selim

 

[Gordianus]

My mistake. I read the OP again and realized my proposal was wrong.
Know I think you want to know the amplitude of the mn mode. Is that right?
If so, the answer depends on where and how you excite the membrane. In loudspeaker jargon this phenomenum is known as "cone breakup".

 

[Sleeme]

Yea, your right, for all modes mn I'd like to know their amplitude if it is excited by a certein sound.
Let's just think for a moment that this is done by pure resonance and not by any direct physical impact of the air, so i don't excite the membrane directly. Just like in the video:

How could I find out the parameters in that case?

Thanks in advance

 

[AlephZero]

If the membrane is wrapped tightly over the speaker cone, you might be able to ignore the dyamics of the membrane and assume it moves the same way as the speaker. A "perfect" speaker should move in proportion to the applied voltage, i.e. exactly the same way as the graph of your sound file. You would need to find the scale factor between the sound file and the spealer motion, which obviously depends how loud you play the music. You could try playing a sine wave, putting a light object on the speaker cone (with the cone horizontal), and measuring when it starts to "bounce" out of contact with the speaker, which means the acceleration of the cone is > 1g. From the acceleration and the frequency, you can work out the amplitude of the motion.

If that isn't a good enough model for what you want to do, it would probably be easier to measure the frequency response curve of the membrane rather than try to calculate it. For example you could use the membrane as one plate of a capacitor (maybe painting it with conductive paint) and measure the capacitance changes agaisnt another "fixed" membrane. (You will have to make the fixed membrane small enoough so it does't block the air flow too much and interfere with the what you are trying to measure).

Trying to calculate the motion is a tough problem, because you would need to model the impedance of the air in contact with the membrane, not just the membrane itself. I don't know any closed-form solutions that would be useful for that. There are some closed form solutions for the impedance of a rigid circular object vibrating in air, but that's no use if your membrane is flexible.

Even making a finite element model of the fluid-structure interaction would be hard, because you have to model the infinite volume of fluid (air) outside of the speaker, without imposing unrealistic boundary conditions that reflect the sound energy back ontl the speaker again.

All this has been done, but it's not simple (and way beyond what could be explained in a post on PF). It would be a non-trivial project to figure out how to use a commercial software package (e.g. Abaqus) that has the capabilities to do this, let alone creating your own model from scratch.

And looking at your video, you don't actually want a model of the dynamics of the membrane on its own. You want a model of the dynamics of the membrane with a time-dependent amount of non-Newtonian viscous liquid (paint) attached to it, which may be quite different!

 

[Sleeme]

Hi alephzero,

thanks for your great insight and knowledge sharing,
As the soundfile can be user defined I cannot measure any values in advance sadly, so I have to come up with a computational solution.

In my model I did'nt include any fluid-membrane interaction besides the passive particle collision, so I can simplify the problem by only modeling the membrane.

If the speaker is no on the side of the membrane and not underneath it, then I assume that the membrane solely vibrates due to resonance as in the second video I posed. When this speaker now plays a sound file only from a frequency f, which is by chance exactly the same as the one from the fundamental vibration mode (2.4/radius/2pi*c) then I assume that the membrane would also vibrate in this exact same mode, correct?
This is why I wanted to come up with an equation like in the Fourier tranformation, where the soundfile from the spatial domain is tranformed into the basis of the frequency domain.
Now I wanted to transform in in the frequency domain of the vibration modes, which hopefully also provide a basis for this. Like I described in the OP, did I use a Matrix for this, which should be multiplied by the amplitudes of the vibration mode, to give me the soundfile.
One problem thereby is to define the values in the matrix, as multiple functions have a frequency f_mn from a vibration mode. For example A*sin(2pi/f*t) or B*cos(2pi/f*t) or any arbitrary superposition thereof.
This is were I'm stuck and can't come up with a solution.

Hope you guys can help me further,

Selim

 

[AlephZero]

If the speaker is no on the side of the membrane and not underneath it, then I assume that the membrane solely vibrates due to resonance as in the second video I posed. When this speaker now plays a sound file only from a frequency f, which is by chance exactly the same as the one from the fundamental vibration mode (2.4/radius/2pi*c) then I assume that the membrane would also vibrate in this exact same mode, correct?

If the sound source is off center, The sound pressure won't reach every point on the membrane at the same time because of the finite speed of sound. If the wavelength of the sound is the same order as the size of the membrane etc, this could be a significant effect. For high pitched musical notes, the wavelengths might only be of the order of 100mm or less.

This is why I wanted to come up with an equation like in the Fourier tranformation, where the soundfile from the spatial domain is tranformed into the basis of the frequency domain.
Now I wanted to transform in in the frequency domain of the vibration modes, which hopefully also provide a basis for this. Like I described in the OP, did I use a Matrix for this, which should be multiplied by the amplitudes of the vibration mode, to give me the soundfile.
One problem thereby is to define the values in the matrix, as multiple functions have a frequency f_mn from a vibration mode. For example A*sin(2pi/f*t) or B*cos(2pi/f*t) or any arbitrary superposition thereof.

That is what the "impulse response" means. If you apply an vey short pressure pulse to the system, if will excite all the modes in different amounts, and the response will be the combination of decaying vibrations at the different frequencies. You can consider your continuous sound input to be a series of impulses. The efficient way to do the calculations to take the FFT of the input, do a convolution with the response of the membrane in the frequnecy domain, then and inverse FFT to get the motion in the time domain. Second-level courses on dynamics (specifically, the dynamics of multi-degree-of-freedom or MDOF systems) or control theory will have the details. The mode shapes and frequencies that you found are the first step in that process. You might need to study a little bit of Lagrangian mechanics as well (i.e. what is useful for engineers to know about it, but not all the theoretical implcations and specialized notation that modern physics uses).