Asymptotic behavior of eigenfrequencies of a thin elastic rod with non-uniform cross-section

Abstract

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 $\varepsilon$ 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 $\varepsilon \rightarrow 0$.