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Attempting to Predict Tension on a Circular Membrane

  1. Oct 22, 2015 #1
    I have been wanting to write an equation which predicts the tension on a circular membrane (AKA drumhead). However, I'm not sure that my answer is on the right kind of track, if it's even correct.

    As for the procedures I took, first I started with the equation for tension on a string, which I obtained from the website of guitar string company D'Addario:

    T = UW ( 2 L f )^2 ,

    where

    'UW' is the unit weight of string per unit of linear measure,
    'L' is the speaking length of string, and
    'f' is the fundamental frequency produced by that length of string when set in motion.

    Next, I defined 'UW' as

    UW = V ρ ,

    where

    'V' is the volume of the string and
    'ρ' is the string's material density.

    Then, I defined 'V' as

    V = A l ,

    where

    'A' is the cross-sectional area of the string and
    'l' is the unit length of measure (a variable which apparently doesn't contribute to the unit of measure of the equation at all).

    Now *that* was a string, but if you were to take that string and stretch it over a drum so that it intersects the drum's center, could you not abstractly say then that the string now represents a diametric force line that the drumhead naturally experiences when under tension?

    For a drumhead we go back the part where we define the geometry of the body under tension, 'V', but instead of the very long, thin cylinder of a string before we now have a short, very large cylinder.

    V = A h π ( D / 2 )^2

    where

    'A' is the square unit of measure (which, like with the string, does not contribute to the equation's overall unit of measure), and
    'h' is the height (or perhaps thickness) of the drumhead.

    Substituting this back into the equation, and instead of 'L', being the speaking length of string, writing 'D', the speaking diameter of the drumhead, we arrive at

    T = h π ( D / 2 )^2 ρ ( 2 D f )^2

    Simplifying...

    T = π h ρ f^2 D^4

    but this doesn't have the right measure of units yet- there's one extra power of the diameter than we need to have our measure in Newtons, so there's something else we have to do: we need to divide everything by half of the drum's circumference, π ( D / 2 ) . This ends up with

    T = 2 h ρ f^2 D^3

    For ease of use by the layman musician, we can furthermore divide by the force of gravity to get a value in either pounds or kilograms.
     
  2. jcsd
  3. Oct 23, 2015 #2

    Simon Bridge

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    The string equation relies on the string being much longer than it is wide... the fundamental frequency f in the equation is dependent on the length of the string and the string tension in the equation is along that length.

    Have you searched online for other peoples attempts at the same problem? It is a common exercize in tension fields. BTW. Quite a hard one.
     
  4. Oct 23, 2015 #3
    Doesn't the same hold true for the drumhead's diameter being many times greater than its thickness?

    'Googling' tension field theory, I've found one such article so far that seems to stand out:

    http://maeresearch.ucsd.edu/~vlubarda/research/pdfpapers/ActaMech2011.pdf

    While I am doing this, though, I have also drawn up a very simple apparatus for getting real-world values for comparison, using spring scales attached to turnbuckles, which are then attached to the tuning bolts running through the metal ring which tensions the drumhead.
     
  5. Oct 23, 2015 #4

    Simon Bridge

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    No. The approx for the string makes it a 1D problem, for the drumhead, its a 2D problem.

    ... you cannot do physics by analogy.
     
  6. Oct 24, 2015 #5
    The state of stress of a string is different from the stress analysis for a sheet (as Simon points out) because, in the case of a string, stress only exists along the length of the string and it points along the length, while, in the case of a drum membrane, there are stress components in two principal directions. There is stress in the radial direction in the drum membrane, but there is also stress in the circumferential direction. If the membrane has been stretched radially before securing at the edges, these stresses are equal (a so-called transversely isotropic state of stress).You can find analyses of drum membrane oscillations in the online literature that take all this into account.

    Chet
     
  7. Oct 24, 2015 #6
    @simon - Okay. Based upon the apparatus described above, would it not still be possible to derive an approximate equation for tension on a flat circular plane based on frequency (for a single case of a set height, diameter, and density)?

    @chet -

    Were you referring to the linked article I posted in my last comment?
     
  8. Oct 24, 2015 #7
  9. Oct 24, 2015 #8
    Ok. I'm sorry. Unfortunately for me, the last time I took a calculus class was 2008, my senior year of high school, and we never got into Bessel functions or the Laplacian or anything like that. Just derivatives, integrals, chain rule, disk and shell method, and I *think* the start of graphing differential equations, so I wasn't really sure how to interpret what was written. I'll keep looking, though.

    Earlier today I was looking through the recommended list of related threads on PhysicsForums, which is where I found this:

    https://www.physicsforums.com/threads/vibration-of-a-circular-membrane-equation.501848/
     
  10. Oct 24, 2015 #9
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