Resonant frequency of an annular ring? Hookes Law?

[antsknee]

Hi,

I am trying to work out the resonant frequency of an annular ring, does anyone know a general equation for it?

For example the ring has an outside diameter = OD and inside diameter = ID. The ring is gently clamped at the outside diameter and a force F applied evenly at the inside diameter. At a particular frequency the amplitude of displacement will be maximum.

I believe it would be something like hookes law. The thickness of the ring would be a factor as well as the elastic modulus of the ring.

Thanks,

Anthony.

 

[antsknee]

I have been looking at tuning forks for inspiration.

The resonant frequency of a tuning fork is:

f=(1/l²)\sqrt{}(AE/p)

l=length of the prongs
A=area of prongs
E=youngs modulus of material
p=density of material

I will return with more thoughts as I have them.

 

[antsknee]

I have been searching these forums and it seems I need a Bessel function for cylindrical harmonics. They are used to understand musical instruments such as cymbals. The wikipedia page looks very complicated, I bet I won't get any replies to this thread :)

http://en.wikipedia.org/wiki/Bessel_function