Calculating notes of twanged strips

Consider a twanged metal ruler, or the "keys" of a thumb piano (kalimba)...

I'm in the middle of designing A Thing, and I am sure that there ought to be a way of calculating the note produced when you twang the end of a strip of metal.

Assuming I know the Young's modulus, length, width & thickness of a strip of metal (rectangular cross-section), is there a formula into which I can plug these numbers and predict the note produced?

Or are there other properties I need to know?

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Well, I tried asking Google(r) about this, about the "natural frequency of a vibrating beam", or maybe "a vibrating cantilever". I think it suggested that word "cantilever", or maybe it seemed like that was the word I should be using since it came up in a lot of the results.

Anyway, one of the results near the top of the pile was this lab handout from a mechanical engineering course at MIT,


and if you look at equation 7.10 or 7.11, I think this is equation you want, although I am not sure if that equation is a realistic model for the physics of your thing.

omega = (zetastar)^2*L^-2*((E*I)/(rho*A))^0.5

where E is Young's modulus

I is moment of inertia (calculated from beam's cross-section shape)

A is cross-section area

rho is mass density

zetastar is 1.875104 (dimensionless)

I dunno. I do not really grok this equation completely. However, the most striking feature of this equation, to me, is that it looks like this fundamental frequency is proportional to L^-2, the reciprocal of the length of the beam squared.

So if I had a set of cantilevers of different lengths, but same material, same cross-section shape, I might try looking for that dependence on L^-2.

Since you have the "Thing" in your possession, you might try recording the sound each cantilever makes, then doing a FFT (fast fourier transform) on those sound samples for to look at them in frequency domain. This is the sort of thing you can do with a number-crunching tool like MATLAB or Octave. Octave is essentially the free, open source, equivalent of MATLAB, which is not free.

I mean doing that you would, be able to "see" the sounds your twangy cantilevers are making. I am guessing that when you look at the sound of something like this, there will be some spikes, and the tallest of those spikes will correspond to the fundamental frequency.

The FFT for a pure tone, a sine wave in time domain, looks like a delta function, just one big spike, in frequency domain.

Actually, I think the Wikipedia page for FFT,


has some pictures of FFTs, and some explanation of this.

Or you might have other tools for measuring audio frequency, like an oscilloscope, or frequency counter, or one of those electronic guitar tuning gizmos.

Here are some links to other people that give that same formula for the frequency of a vibrating cantilever:




Also there is a Wikipedia page that mentions this too, in the section titled, "Examples: cantilever beams"


Sort of the thing I keep seeing are this list of the first few smallest x that solve:

cos(x)*cosh(x) = -1

The first six of these xn are:
x1 = 1.8751
x2 = 4.69409
x3 = 7.8539
x4 = 10.99557
x5 = 14.1372
x6 = 17.279

and it turns out for n larger than about 3, they're quickly converging to
(2*n -1)*(pi/2)

And the frequency of the nth mode has a factor of xn squared in it.

The lab handout I linked to previously, just mentioned the smallest of these, namely for mode 1, x1=1.8751

It is funny stuff. I mean, it is a lot more complicated than the ways (modes) a stretched string can move.

For that it is just integer multiples (harmonics) of the lowest frequency.
f1 = 1*f1 (of course)
f2 = 2*f1
f3 = 3*f1
f4 = 4*f1
f5 = 5*f1



Kiteman (author)  Jack A Lopez1 month ago

I should have expected headaches...

Thanks, though, I think I can get it from that.


There are also some GREAT apps for your phone that will do the spectral analysis

There are several apps available that claim to tune musical instruments through the microphone.
Might not be as accurate as calculating but I think the rate of error would be about the same.
Since your stips are hand made and with uneven ends I guess checking with an app might be the best starting point.
Once you got some reference lengths you can use them to tune the next model before putting it together.
The fine tuning should be done by ear anyway.

Kiteman (author)  Downunder35m1 month ago

I was wanting to be able to predict what note would come out of the thing before I made it - the manufacturing process I want to try is too expensive to be throwing away loads of items.

iceng Kiteman1 month ago

So they cannot be manufactured with shape or mass adjustment areas ?

Kiteman (author)  iceng1 month ago

I might end up filing bits off, but I will be out-sourcing the actual manufacture.

iceng Kiteman1 month ago

Looking forward to the ible :-)

Measure the set you have, adjust what you need - use that as the baseline ;)
I did not say to measure every unit you build ;)
I assume you want 8 individual notes and very little to no adjustment.
Using what you have to get a basiline would accomplish this IMHO.

Kiteman (author)  Downunder35m1 month ago

Ah, I've accidentally mislead you - this will only have one twangey bit - I just used the thumb piano to illustrate the twang.


It is an art form; with electronics its math, with organic and other materials its art, so Pluck, Listen, Trim, and repeat until tuned.