# Acoustic resonance frequency of a cylinder with elastic walls under tension

I have to find the parameterized resonant frequency of this cylinder whose walls are elastic and have a tension T. The question asks to make any assumptions as necessary. I am not sure how to proceed. I have looked up resonant frequency computation for strings with tension T but can't relate the methods to a 3D object such as this. Any suggestion on the way to go about it or a link explaining how to do such derivations would be helpful.

Presumably this is a hollow cylinder, otherwise it doesn't have "walls".

Also, presumably this is supposed to model solid-fluid interaction with a flexible solid body, otherwise the tension is irrelevant.

If those assumptions are correct, the basic point is that the natural frequencies of the cylinder (ignoring its interaction with the fluid) depend on the tension in the walls.

In the vibration equation $M \ddot x + \omega^2 K x = 0$, the stiffness $K$ of the cylinder has three components, $K = K_e + K_\sigma(T) + K_L(P)$ where

• $K_e$ is the elastic stiffness of the cylinder (presumably, using small displacements and a linear elastic material model).
• $K_\sigma(T)$ is the stress stiffness of the cylinder - analogous to the stiffness component of a vibrating string which is proportional to the tension in the string.
• $K_L(P)$ is the load stiffness of the cylinder, which is caused by the fact that the fluid pressure always acts normal to the surface, and therefore changes direction if the cylinder deforms and becomes non-circular as it vibrates.

The "tension in the cylinder" is ambiguous - it's not clear whether it is an axial or circumferential tension, or some combination of the two.

How you put all these pieces together and then couple the solid model to the fluid, depends very much on what software you are using, but all the theory is well-known and explained in advanced finite element textbooks, and also in the theory manuals of most commercial FE software packages that can handle such a model.

A real masochist might try to model the whole situation analytically and derive a parametric model directly. Alternatively, set up a computer model, run it with a selection of different parameter values, and fit a statistical model to the results. Both methods should get to the same end point!

• Note, for a simple model of a vibrating string, you can assume $K_e = 0$ because the elastic stiffness of the string is negligible compared with $K_\sigma$. In a first course in dynamics the fluid-solid interaction is not even mentioned, and so $K_L$ is also ignored. – alephzero Jun 5 '17 at 0:58
• Seems like a parametric model derivation will be complicated. I'll try modeling this in COMSOL. Thanks. – user1155386 Jun 5 '17 at 1:01