Calculate Bell Ringing Frequency | 8.5cm Diam, Cast Brass

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How do I calculate the frequency of an 8.5 cm diameter bell ?
The bell is cast brass for resonance, but I intend to activate it with a buzzer so sound will last for more than a few seconds. Does the weight affect anything other than volume ? I intend to use it inverted, supported near the "top".
 
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You don't.
For a real bell you would need to use numericial methods to solve the equations (unless you have a VERY simple bell, i.e. something very symmetric) but even then you are not very likely to get a useful answer, partly because real bells are note made of a "perfect" material; various imperfections etc (which in turn depends on how the bell was cast) can have quite a large influence on the resonance frequency.

Is there any reason why you simply can't measure the frequency? It should be easy, all you need is a microphone connected to the sound card of a computer+some suitable software.
 
"a traditional English-bell profile "
What I need is the "dominant strike chord"

Any suggestions on "some suitable software" ?
 
Well, all you need is some software that can show you the frequency spectrum of the signal from the microphone. Sigview comes to mind and you can download a free trial version.

However, you could also try something VERY simple such as AP Tuner which is a program used to tune guitars, by default it will only show you what note you are playing (i.e. A, C# etc) but if you look n the "Recording Device and Calibration" it will also show you the last detected frequency.
Or, alternatively, you could just look up what frequency e.g. C# corresponds to (chances are that a bell will be reasonably well "tuned" to a note).
 
I need piezo(?) alarms to ring my bell. I need a resonant frequencies of 392 hz and 1700 hz, 12 volts would be best. Also need a 30 minute timer.