Calculating fundametal oscillation of a springy plate?

[opticsman]

Hello All,
As a non-mechanical engineer i'd appreciate help on how to calculate fundamental oscillations(frequencies) from a thin,springy (weak) plate. like a leaf spring.I'm interested in a horizontal plate attached at its ends. From the Faq's i've seen the simple SHM formula seems to apply, approximately at least but how does the shape of the spring affect the result? I saw reference to a drooping cable which limits the fundamental oscillation so that the main mode has a node in the middle. This made me think that if I shape the spring ( make it wavy) the fundamental mode can be affected? any help or pointing to literature will be welcome

regards

Opticsman

 

[MikeyP]

It depends on the attachment conditions. ie if the plate is "simply supported" at both ends (displacement is fixed but free to rotate) or "clamped" (displacement and rotation is fixed). Ususally "somewhere in between" is a more realistic situation. In that case, the "clamped" and "simply supported" frequencies will give you an upper and lower bound. The book you need is "Formulas for Natural Frequency and Mode Shape" by Blevins. Its quite hard to get hold of but any university should have a reference copy.

Michael

 

[GregLocock]

I you make the plate sufficiently wavy then it gets very soft. Look at the diaphragm of the can on a barometer - it is steel, probably 0.4 mm thick, yet moves quite considerably in response to air pressure.

The bad news is that Blevins probably won't give you a direct answer. (ie I don't think there's a table of results in there for a wavy diaphragm)

The good news is that if you know the effective spring rate of the system, and the mass distribution, then you will get a very good approximation to the correct answer using the SHM result for a uniform beam of the same stiffness and mass distribution.

Your other option would be to make an FE model of it. Good luck. Mike's comments on end conditions are right on, they have a crucial effect on the performance of simple beams, but clever design can render them unimportant (ie you could decouple the diaphragm from the foundation by putting a big ridge just inboard from the foundation).

if you want to discuss this in more detail, please do. I have a copy of Blevins, even. Cheers

Greg Locock