Vibrations of a circular membrane with free ends

[alexao1111]

Hello,

As of this moment I am trying to get in the process of writing an Extended Essay on Chladni Plates, more specifically on a circular vibrating membrane with free ends. To begin with I thought the concept could be simplified to such an extent where I could take a cross-section of the plate and decribe it as a standard standing wave, however that does not seem to be the case. I wanted explicitly to try and calculate the speed of sound thorugh the medium of the vibrating plate using the chosen frequency and using the nodal lines in order to find the wavelength. After actually doing a trial the calculations seemed inconsistent and something had to be wrong. After further discussion with my supervisor, who does not have much knowledge regarding this topic, it appears the idea of investigating a vibrating ciurcular plate would deem to be rather complex. A big issue for me is also in terms of research because so to speak all sources I find regarding the topic only considers a drum head, which means that the ends are fixed.

I really want to make this topic work but I do lack any sense of direction. Is it anynone who would be able to point me in the direction of any papers or literature which talks about this? And also if possible clarify any misconceptions I seem to have about this phenomenon.

 

[nasu]

The pattern of standing waves is 2D. Looking at a cross section only is not enough to get the solutions.
The problem is indeed quite complex (mathematically) and make use of differential equations and special functions. You can find the solutions in different acoustic books or papers.
This may help maybe:
http://www3.ic.ac.uk/pls/portallive/docs/1/8649699.DOC

For rectangular plate you may do some simpler "deduction" of the modes, as in the above paper.
For circular, the solutions are in terms of Bessel functions and the frequencies are not in harmonic series. They depend on the zeroes of the Bessel functions, I believe.

 

[sophiecentaur]

Hi and welcome.
You certainly seem to be jumping in off the deep end with this one. The drum is probably what's analysed most because the the boundary conditions are 'easy'. If you have a non-fixed rim, there are a whole lot more possible modes if the excitation is not in the centre. I Googled Cymbal Vibration Modes and found a number of hits. Would any of those be useful for you? (The rim of a cymbal is not constrained so it could have a similar performance to your disc shaped membrane).
As nasu wrutes, the drum membrane can be modeled to give Bessel Functions as solutions - but that is a simpler case than yours.

 

[nasu]

The link I gave is about plates, not membranes. And a plate with free rim, if I understood correctly. I only skimmed through it.
Indeed the drum membrane is, well, a membrane. But the physics (and the math) is different for membranes and plates.
The restoring forces have different natures. They are treated in different chapters in acoustic textbooks.
I understand that the OP was about plates.

 

[sophiecentaur]

Problem with that reference is that it only addresses the case of a plate, excited in the middle. This is very much simpler than the general case (cymbals etc.) and gives almost the same solution (I think) as for a membrane with a fixed periphery. It would help to know what level the OP is coming from, though and to have some idea if the Maths of 2D modes is approachable for him/her. For me, the Bessel stuff has always be peripheral and I have just plugged in the numbers to the given Maths. My problem was always spotting the form of equation that will give me Bessel solutions. You are probably right about the different natures of the two problems.

 

[nasu]

I was just trying to give an example of how the problem is treated, with that reference. You can find more complicated cases in papers or acoustic books. But I don't think is relevant for the OP.The simplest case is still more complex than the approach proposed in the OP.
Definitely, the cymbals are more complicated than a flat plate. I don't think there is an analytical solution, just some formulas for the ferquencies of the modes, determined by experiments. But this is not so relevant to the OP, is it?

Some plate modes may look similar to the membrane modes, for sure. But this is is not to say they are interchangeable. The fundamental mode for a membrane with fixed edge is (0,1), which means one nodal circle and no nodal diameters. For a plate with free edge the fundamental is (2,0) which means two nodal diameters and no nodal circle (it's free edge).
And the expressions for frequencies will be different. For membrane they depend on the tension applied to membrane. For plates they depend on the structural elasticity of the plate.