eigenvalue - What is the meaning of pre-tension for a stiff membrane?

On one hand, I know that the tighter a drum head is stretched, the higher its natural frequencies. This relation is given by:

$$f_{ij}=\frac{k_{ij}}{2\pi R}\sqrt{\frac{T_0}{h\rho}}$$

where $k_{ij}$ are roots of the Bessel function of the first kind, $R$ is the membrane radius and $T_0$ is the pre-tension.

On the other hand, if my membrane is a stiff solid that is clamped to edges of a cavity, no pre-tension is required to produce frequency-response when a disturbance is introduced. Even if we neglect the induced pre-tension by gravity, we would still expect our stiff membrane to respond to an impulse, because tension will be generated "internally" upon deflection due to the impulse (see also: "Chladni plate", which has free boundary conditions).

However, I cannot use $T_0=0$ in the equation above, since it provides only the trivial solution. So what is the meaning of pre-tension for a stiff membrane?

Answer

You cannot use that formula for a stiff plate - you have a fundamentally different situation.

In the stiff plate, the restoring elastic force comes from the stress arising from the strain induced elastic deformation of the plate's material. This is related to the materials' modulusses (Young's, Shear and Bulk Modulusses) and Poisson ratio. See how these ratios do NOT appear in your formula.

In the stretched membrane, on the other hand, the material is assumed to be infinitely flexible and flimsy. At rest, all the tension is in the membrane's plane. As the membrane is deformed sideways, there is a component of this roughly constant tension opposing the deformation: a two dimensional analogy of the stretched, infinitely flimsy string. Other than its being infintely flimsy, no other material properties matter in this analysis.

I can't give you a formula for what you need, but you would have to work out orfind the analogy for Euler-Bernoulli beam theory or Timoshenko beam theory. Timoshenko Beam theory applies to the stiff beam, and this is quite a different theory to that of the Stretched flimsy string.