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I Modal frequencies of a vibrating circular membrane? (DRUM)

  1. May 22, 2017 #1
    I am in the process of trying to develop a modal drum synth. I have the following graphics as references for the frequencies of some of the first modes relative to the fundamental:

    drummodes.jpg

    mem.gif

    This is a good start. But I want to be able to model more modes than just that.

    What is the formula required to calculate any mode as a multiplier of "f1" like above?

    Eg. If I am wondering what the (7,1) mode is, how would I calculate that?

    This article gets into it from what I can tell, but it's way over my head mathematically:
    https://courses.physics.illinois.ed...p406pom_lecture_notes/p406pom_lect4_part2.pdf

    Ideally I'd like a formula I can punch (2,1) into and get 2.14 (for example).

    Any help? Very appreciated if so.

    Thanks,
    Mike
     
  2. jcsd
  3. May 22, 2017 #2
    Here is the excerpt from that Physics 400 level course that I think summarizes the equations that are needed:

    bessel.png

    I never went past first year physics (biology & medicine guy) so I don't understand how to use these.

    Any help?

    If anyone has any medical questions I'm happy to answer those in return. :biggrin:
     
  4. May 22, 2017 #3
    Your issue is how to compute ##x_{mn}##, which is the n-th zero of the m-th order Bessel function. There is no simple formula to do this, however there are many resources that will do the dirty work for you. A Bessel function of the first kind, ##J_m\left (x\right )##, looks kind of like a periodic function but whose y values cross the x axis at non-periodic x values. You don't need to know all about Bessel functions to solve your problem. There are online calculators that input m-n and spit back ##x_{mn}##. Google "zeros of Bessel functions".
    If you know how to program c++, the boost library has a function that does this as well.
     
  5. May 22, 2017 #4
    Wow. This is way harder to do than I thought it would be. Okay. Thanks.

    So I've looked into what you're describing a bit, and I can find three approaches to getting it done (none of which I actually understand o_O?:)). I can't code anything except HTML lol.

    (If you're wondering how I'm building a drum synth without knowing how to code, it's because I'm doing it in Reaktor which doesn't require coding so much as linking modules in a logical fashion to get the signal you want.)

    Anyway, here is what I've found:

    1) Calculators
    I found two Bessel calculators:
    http://keisan.casio.com/exec/system/1180573472
    http://www.mhtl.uwaterloo.ca/old/courses/me3532/js/bessel.html

    The Casio one says it's for calculating zeros, but I don't know what to put in there to get what I want (or if it can give me what I want).

    2) C++ with Boost
    As you suggested, Fred, I found this page which seems to summarize a lot about Bessel functions and C+:
    http://www.boost.org/doc/libs/1_62_0/libs/math/doc/html/math_toolkit/bessel/bessel_root.html

    Is there a simple bit of code of command I could just plug in to get what I want?

    3) Python with MPMath
    I found this seemingly simple technique using "MPMath" for Python:
    http://fredrik-j.blogspot.ca/2010/07/sage-days-23-and-bessel-function-zeros.html

    He gives the following example of its utility:

    >>> besseljzero(1,100)
    314.9434728377671624580656

    Is that what I want? Or would it get me what I want?

    Which of those would likely be the easiest way to get what I want? Thanks again. Any further help would be appreciated. I feel like a toddler.
     
  6. May 22, 2017 #5
    I think the easiest implementation for you is to go with the Casio calculator. In order to work that calculator enter m in the box that says "order ##\nu##" and n in the box that says "ordinal number s", then press the "execute" button. Divide your answer by 2.4. Give it a go and compare with the ##f_{mn}## you already have.
    Edit: The answer is by the red dot.
     
    Last edited: May 22, 2017
  7. May 22, 2017 #6
    Ha!

    Well that is super easy. You just made my day Fred. That was a million times simpler than I was preparing for.

    Thanks. Have a good one. :smile:
     
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