2020 Volume 72 Issue 1 Pages 119-154
We study the eigenvalue problem of the elliptic operator which arises in the linearized model of the periodic oscillations of a homogeneous and isotropic elastic body. The square of the frequency agrees to the eigenvalue. Particularly, we deal with a thin rod with non-uniform connected cross-section in several cases of boundary conditions. We see that there appear many small eigenvalues which accumulate to 0 as the thinness parameter 𝜀 tends to 0. These eigenvalues correspond to the bending mode of vibrations of the thin body. We investigate the asymptotic behavior of these eigenvalues and obtain a characterization formula of the limit equation for 𝜀 → 0.
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