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Given 2 metal shapes which have the same mass and identical material, a circular torus and an elliptical torus:

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which of these two shapes would likely have the higher natural frequency, if the mass and material is the same in both?

James
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    Is the radius of the material cross section the same? i.e. we can have a really elongated elliptical torus by making the material cross section smaller. IMO, the question leaves free parameters which might have impact on the answer. – AJN Sep 11 '24 at 17:37
  • @AJN To make a clear contrast, let's have a really elongated elliptical torus (like a hairpin shape) versus a circular torus of the same mass and same material... which of these should have a nigher natural frequency? – James Sep 12 '24 at 00:45
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    If the material is the same and the mass is the same, and the diameter of the wire is the same then intuition says the bending mode of the elongated ellipse will be at a much lower frequency than the likely first mode of the circular ring, which might be a bending mode or a twisting mode. https://www.abebooks.co.uk/book-search/title/formulas-natural-frequency-mode-shapes/author/blevins/ is your friend – Greg Locock Sep 13 '24 at 06:55
  • @GregLocock thank you. I have also heard it mentioned that the material cross section will affect natural frequency, smaller material cross section results in lower frequency while larger cross section results in higher frequency. So the ellipse being the same mass but stretched should have the lower natural frequency compared to the circular torus, which matches your intuition. – James Sep 13 '24 at 07:33
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    You can keep the wire dia the same if you reduce the minor axis of the ellipse. Ultimately you'd end up with two parallel rods just touching each other, whose first mode is easy to calculate. – Greg Locock Sep 13 '24 at 08:32

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