"Bell" modes? for a circular tube [Son of Chime] 2

[Dinosaur]

Guys,

I'm back to discuss the design of a chime, a circular tube to make a sort of bell like sound during a church service. For you old timers, we talked about this about four years ago. Back then, my time ran out and I decided to order a single chime tube from an orchestral chime producer. Makes me feel like an engineering wimp that I couldn't do the calculations, but a wise man once said (e.g. Clint Eastwood) "A man's got to know his limitations." But seriously folks ...

It is my belief that to make a proper chime, you cut a tube to the length corresponding to the acoustic resonance length for an open-open tube resonating at the requisite frequency. For a concert A of 220 Hz for example,

344 m/s / (2 x 220 Hz) = 0.782 m -or- 30.78 inches

The exact frequency is not important, but let us assume we will be shooting for the concert "A" for the purpose of this discussion. (220 Hz = 1382.3 rad / s)

So the tube will be 30.78 inches long. However, my experiments led me to conclude that the mode of vibration I am seeking in the design is not the vibration of a free-free rod in its second mode, which I originally believed, but rather a "bell" mode as suggested in this forum years ago. I interpret that to mean a shell mode of vibration of the tube wall through the thickness.

If this is the case, then I guess that if the tube were considered narrow relative to its length, the only variables we are working with are the material properties of the tube (the mass density and Young’s modulus) as well as the diameter and the wall thickness.

I still have some aluminum tube from the original experiment so I would like to stick with the aluminum for now. (E=10,000,000 psi & p=165 pcf) The diameter of the tube is around one inch. Using these values, what would the wall thickness have to be to get a good "Bell" mode? Is it possible to describe the arithmetic in a post? Thanks for everyone's help. - Dinosaur

 

[electricpete]

Whoops. I didn't mean to imply that excel was free. You know what I meant.

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[electricpete]

I guess for completeness a 4th factor which could affect which mode is dominant is the location of a strike (is it near a node or an anti-node of a particular mode).

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[electricpete]

I programmed the transfer matrix method in scilab.

As I increased the number of segments, the solution converged to the beam solution. I guess I should have known that - the underlying math of the method is the beam assumption. The strength of that method would be in analysing beams/shafts that have changes in cross section and one or more springy supports (bearings) along their length. Not a smart tool to apply to the chime problem which can be solved exactly (to within the same assumptions) using the beam formla. But at least now I have a good test case to verify my program.

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