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Resonant frequency of millimetric objects

  1. Feb 28, 2007 #1
    Hello everyone!

    First post here!

    I have quite a specific question. I am investigating resonant frequency of biological matter with dimensions of about 1mm cubed.

    What I would like to know is...

    1) does anyone know of any good journals which look into resonant frequencies of small objects and perhaps methods of measuring the resonance. [given that the resonance will be hard to detect as the mechanical vibrations will be of small amplitudes] [well i imagine this to be true]

    2) I believe that I will be working in the ultra sound frequencies and would like to know if anyone knows how dimension will effect the resonant frequency.

    ie. How is resonant frequency related to radius [letting the object to be a sphere of uniform density]

    Hope someone can help me. I have tried to find some papers on this, but I may just be rubbish at looking.

    Thanks for reading. Rob
  2. jcsd
  3. Feb 28, 2007 #2


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    It's not clear to me what you mean by "resonant frequencies of small objects". I was under the impression that the resonant frequency, and whether something HAS a resonant frequency, was a function of the material rather than the size.
  4. Feb 28, 2007 #3
    I was under the impression that dimentions of the object will determine the resonant frequency. Such that, if a peice of glass is in resonance, and you then shorten the piece of glass it will no longer be in resonance.

    I think this is demonstarted by SAW waves on piezo electric substrates [ie frequency filters]

    I may be wrong.

  5. Feb 28, 2007 #4

    Claude Bile

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    They do in the case of a mechanical resonance, but material matters as well. For example, if you were to use a different material with the same dimensions, it would most likely possess a different mechanical resonance.

    Predicting the resonance of a biological sample would be extremely difficult simply because the composition of biological matter is complex. In addition, I'm not sure if principles that apply to homogenous, rigid materials would apply so well to biological matter.

  6. Feb 28, 2007 #5
    The roots of my question are thus;

    1) I am doing a 3rd year project on mechanical resonance in millimetric objects with possible application to insect resonance as a method of pest control.

    I dont think that finding resonance in bio matter will be easy, however, the insect in question has a very rigit outer shell which should exhibit some mechanical resonance.

    I guess i am asking if anyone knows of an equation which will relate;
    f: frequency
    p: density
    a: 3d dimentions
    y: youngs modulus
    s: bulk moduls

    or anything else related to an object of uniform density

    I am going to start with plactic objects and work onto others, glass beads, etc

  7. Feb 28, 2007 #6

    Claude Bile

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    A good first step would be to find the formula for velocity of a mechanical wave inside a medium, as any resonance properties ultimately depend on this figure.

    Figuring out the modes from there is not as simple as the wave-on-a-string case due to the spherical symmetry of the problem. In optics, such spherical resonances are called "Whispering Gallery Modes", I haven't been able to find any reference to a mechanical analogue, but this link;


    Contains some mathematical analysis that is quite analogous to the mechanical case, and could at the very least give you some idea of where you need to head in terms of finding the mechanical resonances of these spheres. (You'll also notice some similarities to atomic physics, purely due to the spherical symmetry).

    Good Luck.

    Last edited: Feb 28, 2007
  8. Feb 28, 2007 #7
    I'm a little confused by this. Since when are animals of uniform density?
  9. Mar 1, 2007 #8
    "possible application"

    I am starting with isotropic structures and then refining my model and mathematical modeling [with FEMLAB finite element analysis software] to biological matter with near isotropic shell structure.


    thanks for the feelback so far guys, esp the pdf!
  10. Mar 1, 2007 #9
    It is possible to use lasers to detect micro-deflections but you would probably need to silver/gold plate one face of your specimen and that may affect the response. If you have an electron microscopy department they often have the ability to gold plate very small objects.
    Basically you split a laser beam into two paths. One reflects off the specimen, the other path is direct. You then combine the two beams together in phase. If your specimen moves the two beams come out of phase proportional to the size of the deflection and you can detect the phase change.
    The set-up needs to be absolutely rigid as the laser path has to be unaffected by any outside influences, so find a proper laser table to set it up on.
  11. Mar 1, 2007 #10


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    You don't usually need special surface preparation or rigid mounting for laser doppler vibrometers (LDVs). The work fine to measure things like vibrations of running machinery. I've seen LDV measurements done on small objects (approx 1mm cubes) with vibration amplutudes less than 1e-3 mm at frequences of 10 KHz, without any special surface preparation etc.

    In fact the idea of "specal fixing" makes no sense for measuring vibrations, because restraining the object would change the frequency and vibrating shape that you wanted to measure.

    http://www.dipmec.unian.it/misure/strumenti/LDV/ldv_en.html [Broken]
    Last edited by a moderator: May 2, 2017
  12. Mar 1, 2007 #11
    When I did it about 10 years ago I was looking at serious small deflections so we built one on a laser table as there was nothing commercially available to meet our requirements. Nice looking package on the link, shows how quickly specialist lab equipment arrives in the commercial sector. You'll be able to rent one from the tool hire shop next.
  13. Mar 1, 2007 #12
    Thanks for the help guys.

    The laser detection looks like a really good way of doing it. A big part of the problem is determining if resonance has been reached in such a small object. I dont suppose you can remember what meterial you were woking with which had a resonance of 10kHz?

  14. Mar 1, 2007 #13


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    We were not trying to measure resonance - FWIW we were measuring friction effects at high frequency, where the elementary "static and dynamic coefficients of friction" model doesn't fit what actually happens.
  15. Mar 1, 2007 #14

    Claude Bile

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    I would explore other, simpler avenues before resorting to laser detection - as a laser brings in plenty of outside complications such as vibration isolation.

    My initial thought is that an electro-mechanical setup would be simpler and cheaper. I would use one piezoelectric crystal to vibrate the sample, and another to detect the response of the material. Piezoelectric transducers capable of doing this can be as cheap as a few dollars.

    Of course there may be engineering issues that prohibit this type of setup, but I think it is worth looking into in any case.

  16. Mar 2, 2007 #15


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    Using a piezoelectric transducer to vibrate the sample seems like a good idea but trying to attach two transducers to such a small specimen could be difficult. The inertia of the transducers is going to have a big effect on the response.

    You might be able to use just one transducer, if you measure both the voltage and current input and look at the phase differences between them for the transducer on its own, and with the specimen attached to it.

    It's probably a good plan to start testing something more inertia (e.g. a 10mm ball bearing) and reduce the specimen size when you find out what works.
  17. Mar 2, 2007 #16


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    I thought of a very simple way to excite resonance and measure the frequency - at least for moderate size metal spheres.

    Just take two identical spheres, make them collide with each other, and record the sound they make with a microphone.

    You could use a "Newtons cradle" type of device for the collision, or do something like roll the spheres down two identical ramps facing each other, so they leave the ramps with equal and opposite velocities and collide in mid air. No problems with attaching exciters, measurement transducers, etc.

    This might not work well for very small spheres but it would be a quick way to get some measured data for larger objects.
  18. Mar 4, 2007 #17

    Claude Bile

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    On the same line of thought - Perhaps it would be simpler to ping the spheres and measure the response using a piezoelectric transducer? A fork type setup should be fairly passive, and a FFT of the output would yield the natural frequencies present.

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