Natural frequency equals one over two pi times the square root of stiffness divided by mass. Stiffness can be calculated by subjected the cylindrical shell to a unit load and calculating the deflection. Depending upon the geometry of the shell and the manner in which it is supported this may be nothing more than a simple span beam or cantilever. Local effects may be neglected depending upon diameter and thickness of shell. The mass is simply the mass of the cylinder and any other fixed weights.
Steve1's formula looks pretty good to me for the fundamental natural frequency of a beam.
I have my doubts whether it can be easily applied to a shell or arch, where a uniform radial load would give pure axial stress, and no transverse deflections. (Presumably your worst mode would be with one half span deflecting downwards and the other half upwards).
If I could not find a reference text which gave me the solution, I would model unit length of shell as a simple pinned arch, and analyse the vibration modes using any of the commercial programs available to you.
If you have no access to such a program, try downloading a copy of CADRElite. That does simple vibration problems very well. (Its demo of a truss bridge vibrating was very impressive). I seem to recall that the downloadable copy will not permit you to save your model, but for a one off requirement that would not have to be too much of a problem.
The best reference for shell vibration is "Vibration of Shells" by Arthur Leissa, published in 1973 as NASA SP-288. This had been out of print for years, but the Acoustical Society of America is now reissuing this and the companion "Vibration of Plates". See
Even for a simple cantilever under self load only, Steve1's formula is inadequate since the modal mass is not the full mass of the shell., or if you prefer, the mass does not see the full tip deflection.
Blevins says that for a plain uniform cantilever the first mode is at 3.52/2/pi/L^2*(sqrt(E*I/m)) Hz
where m is the mass per unit length.
I must confess I have run into trouble with this formula in the past, it does not agree with another published source, but most people seem to agree with this one.
There seems to be some misunderstanding concerning the definition of a "cylindrical shell simply supported at its edges". I have in front of me a drawing of a fuel gas heater. It consists of a horizontal 20 inch diameter schedule 40 pipe, simply supported at it's quarter points, along with some attached piping. This can be described as a cylindrical shell, but I propose that it behaves as a beam, and I would calculate its natural frequency as I previously indicated. There is a somewhat fuzzy line that distinguishes the response of a structure from being based on global characteristics (beam action) as opposed to local effects (plate action). There was not enough information provided in the original question to know what the particular case was refering to. I thought I clearly stated that the response depends upons geometry (diameter to thickness ratio). Perhaps I misunderstood this question.
