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I wonder if there is a connection between fundamental frequency and Young's modulus of a material. For example, how to calculate the Young's modulus of a glass bar by knowing its frequency spectrum?

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The frequency is a function of the dimensions of the bar and its Young's modulus.

You need to know what mode of oscillation you are exciting in your bar - there is a hug difference between the flexural and longitudinal modes.

If the rod is bending, you can find the formulas here. The derivation goes on and on... but you should be able to use the formula on the first page (for free-free):

$$f = \frac{1}{2\pi}\left(\frac{22.373}{L^2}\right)\sqrt{\frac{EI}{\rho}}$$

In this formula, $I$ is the second moment of area of the rod - see the wiki article for an explanation and to find the appropriate value for the shape of your rod.

If you have a higher mode, you can find the position of two fixed nodes and use the fixed-fixed equation instead.

And if you have longitudinal vibration, the answer is much simpler - you just have to look at the transit time of the sound wave from end to end. One round trip corresponds to the fundamental frequency, so

$$f = \frac{v}{2L} = \sqrt{\frac{E}{4L^2\rho}}\\ E = \rho\; \left(\;2\;L\;f\;\right)^2$$

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    $\begingroup$ Closely related, the classic way to measure crystal elastic constants is ultrasound resonances along different crystal orientations. $\endgroup$ – Jon Custer Jul 15 '15 at 14:09
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    $\begingroup$ This is a good answer, but I think it would be good to also point out that, depending on the geometry and the mode of vibration, moduli other than Young's modulus (e.g. the shear and uniaxial strain moduli, which for isotropic materials can be expressed in terms of E and the Poisson ratio) will come into play. There's a lot more to material stiffness than Young's modulus. Specifically, if the aspect ratio is not much more than 1, both these expressions have problems. $\endgroup$ – elifino Jan 31 '16 at 6:56

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