Drum head vibrations and bessel funtions.

topgun2021

Gold Member
I am doing a math project of the vibration of a drum head. I my research, I have found what I need for single head vibrations, but I can't find anything on an entire drum.

Does anyone have literature on the mathematics behind the vibrations of an entire drum (both heads, shell, etc), and how that related to the frequency (Hertz value!?) it gives off?
 

B-squared

Silver Member
I have some idea of what you're trying to do, but I think you need to look at something like finite element analysis to model a drumhead or a drum. Doing it using pure differential equations would be almost impossible given the odd geometry and boundary conditions.

Even if you use FEA, it would take some time to assemble your material constants. Getting things like Young;s modulus and thickness for the shell wouldn't be too tough but finding them for the head would be a chore. Are you including the shell in your mathematical model?

I've done some of this before in graduate school and it's really interesting. There probably won't be too many people following this thread, but I will. Please keep posting.
 

PQleyR

Platinum Member
There isn't one frequency but lots, obviously. I imagine it's like church bells where there's several tones at once without a single dominant fundamental tone.
 

topgun2021

Gold Member
I have some idea of what you're trying to do, but I think you need to look at something like finite element analysis to model a drumhead or a drum. Doing it using pure differential equations would be almost impossible given the odd geometry and boundary conditions.

Even if you use FEA, it would take some time to assemble your material constants. Getting things like Young;s modulus and thickness for the shell wouldn't be too tough but finding them for the head would be a chore. Are you including the shell in your mathematical model?

I've done some of this before in graduate school and it's really interesting. There probably won't be too many people following this thread, but I will. Please keep posting.
I am doing the presentation in mid May, starting research now.

I Really want to focus more on the math of the single batter head, then shortly explain what else you would have to do for the entire drum.

If you still have the work you did, I would like that. I'll put your name in my citations.
 

B-squared

Silver Member
A good place to start would be determining the resonant frequencies (primary and maybe a couple more). Try the batter head first. Using radial coordinates you could model the head with pinned edges. The shell can be connected at the pinned nodes in FEA. Let me think a little about the differential equation possibility. It may not be as bad as I first thought if you use radial coordinates and solve only for primary frequency. You would have a sin function across the shell with a circle outside for the first mode.

What I did, was model chimes using FEA and then tried to use my model to find the length of a tube necessary to produce given frequencies (like a concert A = 440 Hz). I found that using a very thin tube allowed for the best results. I actually had one of the chimes I modeled cut to size in a machine shop. I came pretty close in the end. It isn;t exactly what you're doinf, but it;s close.

Check the community link. I think my email's in there. If not, I will put it in there.
 

B-squared

Silver Member
My email should be there. I thought about your problem a little more and I am starting to look at it like a 3-D guitar string. The tension in the head can be applied radially and the spring constant for the head should be E (Young's Modulus). If the head deflection is the z-axis, and the radius is R, then one boundary condition is delta z is zero at r=R. Does this sound like the direction you are headed?
 
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