Resonant frequency of round cantilever

Hi.

I would like to know how to calculate the fundamental resonant frequency (flexural) of a round cantilever (cylinder clamped at one end, free on other).

Please provide formula and units of variables as well as an example using the following data:

Cylinder: Ø=2mm, L=20mm
Material: Spring steel (see below)

Young's modulus is 210000 MPa
Density is 7850 kg/m³

(More properties available at http://www.matbase.com/material-categories/metals/ferrous-metals/spring-steel/material-properties-of-67sicr5-din-17221-spring-steel-grade.html#properties)
JasonMewesAsked:
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d-glitchCommented:
This looks like an academic assignment.  So you can get help and hints, but not complete solutions.

There is a relevant textbook on line:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&ved=0CD4QFjAFahUKEwjUuYjTudjHAhWGph4KHd45BNg&url=https%3A%2F%2Fperso.univ-rennes1.fr%2Flalaonirina.rakotomanana-ravelonarivo%2FStokey_chapter7.pdf&usg=AFQjCNFUI8GOB4IhQpcttl5bMurQHihvUg&sig2=FeHynSgSAhZrEPt0R4pX7A&cad=rja

With a section on
  LONGITUDINAL AND TORSIONAL VIBRATIONS OF UNIFORM CIRCULAR RODS
JasonMewesAuthor Commented:
This is not an academic assignment.

This a company account and I am a senior software engineer well beyond my academic years.

However, I am not a mechanical engineer and not educated in this area. As such I wish to check my calculations for a device utilizing tiny motions of a resonant needle based on a piezoelectric XY scanner. I am using audio frequency DACs and ADCs we need to keep the resonance of the needle in the range 1kHz+/- 500Hz to provide adequate temporal resolution so that we can extract useful data from the error-stream (difference of output vs. feedback amplified).

A cylindrical cantilever is the closest (simple) approximation of the shape of the needle and the range of acceptable resonance is wide enough to use such an approximation. The shape of the needle can be tuned over prototyping iterations, but it would be nice to at least hit the ballpark on the first run.

This is my current setup based on formulae from the web:

Y=210000000000 Pa (Young's modulus)
d=7850 kg/m3 (Density)
a=0.002 m (Diameter)
L=0.05 m (Length)

f1=0.162*(a/(L²))*sqrt(Y/d)

f1=670.316 Hz (Fundamental resonant frequency)
d-glitchCommented:
I get a resonant frequency of 579 Hz using the formula for a tuning fork from Wikipedia.
I have not tried to figure out where the differences come in.

You could also measure the frequency directly, by clamping one of your needles in a vise and plucking it.  A trained ear or a microphone and a scope should provide the data you need.
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JasonMewesAuthor Commented:
I can't measure it at this point, as I have yet to produce the mechanical parts and the turn-around is about one month.
This is why I wish to find the correct dimensions, so that we can hit fairly close on the prototype production run.

Cantilever resonance algorithms I've found appear to be designed for square/rectangular bar cantilevers, not round cylinder ones.

The electronics system we are building for the device will be able to tune in to the exact resonance using a PLL-like algorithm - so we do not need to measure it by any other means although we certainly could if necessary.

Said system will however only work for frequencies in the range of a few hundred hertz to a few kilohertz which is why I am looking for a formula that will give us a good approximation of what to expect. This setup will provide auto-calibration of variations in the machined parts due to precision limits during manufacturing as well as variations in material properties.

Your calculation is certainly close enough to strengthen my confidence, but a mathematically accurate answer for a cylinder cantilever would be the goal of this question.
d-glitchCommented:
I forgot to post the link:    https://en.wikipedia.org/wiki/Tuning_fork

When I simplify the Wikipedia formula for the frequency of cylindrical prongs I get   freq=0.140*(a/(L²))*sqrt(Y/d)

The variable dependence is the same, but the numerical factor is slightly different.  I don't know why.  Where did your formula come from?  

Ref [7] might be helpful.   But I suspect either of these formula would be close enough.

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JasonMewesAuthor Commented:
Yeah, I derived the same formula and got identical results to yours using that.
I now have three different calculations yielding three similar, but different, values.

Regardless, three different calculations yielding similar results and your external feedback has me convinced enough that we are "in the zone", so I'll consider this answered to within the acceptable margin of error :)
d-glitchCommented:
I expect that your electronics will have more than adequate range and resolution to drive a needle at resonance.  Once you have a prototype, you will be able to develop expertise rapidly.

What Q are you expecting?  That will determine the resolution requirements.
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