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Lets say I have a mechanical system whose mechanical resonances (mode shape and frequency) I can measure with perfect accuracy. Is this theoretically equivalent to knowing the materials parameters, including spatial variation, with perfect accuracy? My intuition says that the answer is yes, but I am not sure how to prove it. For concreteness, here is a specific example.

Given a right circular cylinder such as the one shown below; will measurements of its resonant modes allow me to reconstruct its spatially non-uniform Young modulus and Poisson ratio?

enter image description here

Chris Mueller
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  • Related: http://physics.stackexchange.com/q/22735/2451 Related quantum mechanical inverse scattering problem. – Qmechanic May 14 '14 at 16:21
  • An initial thought, that I'll try to turn into a full answer later: This sounds suspiciously like the tomography problem. That might give you at least a partial answer. – Colin McFaul May 14 '14 at 17:05
  • @Qmechanic Thanks for the links. The problem of hearing the shape of a drum is slightly different than what I am interested in. My question in the parlance of a drum would be: Given the shape of the drum can one infer the material parameters of the skin from the sound? – Chris Mueller May 14 '14 at 18:56
  • @Qmechanic This text from one of the answers to the inverse scattering problem which you linked has a more analogous problem. They study the ability to recover the non-uniform mass density of a vibrating string from its spectra. It turns out that "the spectra for two sets of boundary conditions are necessary and sufficient to determine [the mass density uniquely]." – Chris Mueller May 14 '14 at 19:50
  • If the material is linear elastic, the free-vibration modes are the eigenvalues of a system that has the form $K(x) u(x) - \omega^2 M(x) u(x) = 0$. You question appears to be the following: If you know the mode amplitudes $u(x)$ and the frequencies $\omega$ can you find $K(x)$ and $M(x)$? The problem appears to be ill-posed at first glance but regularization techniques could possibly be used to arrive at a rough solution. – Biswajit Banerjee May 15 '14 at 02:46

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