Calculate the 'Feel' of a Drumhead?

[Chrono G. Xay]

As another of my personal music projects, I have wondered if it would not be possible to calculate the 'feel' of a drumhead (i.e. the amount of 'give' expressed as transverse displacement 'z' that an equally pre-tensioned circular membrane of radius 'r' experiences when struck on its plane at a point a radial distance 'd' from its center, where 0 < d < r ) .

I tried to imagine just how it would deform when struck, and the most simple mathematical model that I can come up with is when it is struck in the center, represented by the graph of 'sqrt( x )' revolved about the y-axis. I don't have my notes with me right now (I'm on my honeymoon, but wanted to finally put this question up here), but in a related--complete--project I've been working on, which involves calculating the 'relative' angular displacement and torque needed to execute any rhythm with any sticking at any tempo, I was able to tie it into this one using Hooke's Law (or at least the Law of Conservation of Energy, given the implement's moment of inertia, center of mass, terminal angular velocity, and, technique-wise, the use of only the fingers, a la 'Gladstone'- the use of the wrist, or rather any set of joints besides the fingers, would obviously require additional maths).

Anyway, with this project I'm more interested in obtaining a more generalized value for the 'give', where the angular momentum is a set value like 1 rad•kg•m/s, the idea being that each time a drumhead is virtually 'struck' in this simulation it is exactly the same.

I know I was able to construct an equation that would allow me to find the 'feel' of a drumhead at any point between the center of its plane and its edge, the goal being a graph with respect to 'd' that would explain the general change in 'feel' much more easily for the layman musician, but for the time being went with the most mathematically simple model, where d=0

(For what it's worth, I do understand, going in, that the dynamical level of a sound, whose scientific magnitude is measured in deciBells, is logarithmic in scale, base 10.)

I'll be back home in another few days, where I can post some pictures of my work so far.

 

[Chrono G. Xay]

I started with the equation for Elastic Modulus,

E = /frac{σ,ε} = /frac{F / A_i,ΔL / L_i

=> F = /frac{ΔL,L_i}*E*A_i

and from there further defined the terms as they seemed to be:

ΔL = L_f - L_i => /frac{ΔL,L_i} = /frac{L_f - L_i,L_i} = /frac{L_f,L_i} - 1

L_f = L_{1f} + L_{2f}
L_i = L_{1i} + L_{2i}

/frac{L_f,L_i} - 1 = /frac{L_{1f} + L_{2f},L_{1i} + L_{2i}} - 1

F = T

'T' - Tension on membrane

However, I know I need to tie this in to

F_s = -k*x

and in my case 'x' would be more like 'z'.

I arrived at an equation for theoretically predicting the tension a drumhead by manipulating the geometries I input when constructing the equation for tension on a string- a thin, large cylinder instead of a small, long cylinder. I have yet to construct a physical apparatus for testing it.