High frequency oscillations of first eigenmodes in axisymmetric shells as the thickness tends to zero

Marie Chaussade-Beaudouin, Monique Dauge, Erwan Faou, Zohar Yosibash

The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness ($2\varepsilon$) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue $\lambda(\varepsilon)$ and the eigenvector angular frequency $k(\varepsilon)$ for shells with Dirichlet boundary conditions along the lateral boundary, and natural boundary conditions on the other parts.

First, the scalar Laplace operator for acoustics is addressed, for which $k(\varepsilon)$ is always zero. In contrast to it, for the Lamé system of linear elasticity several different types of shells are defined, characterized by their geometry, for which $k(\varepsilon)$ tends to infinity as $\varepsilon$ tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide $\lambda(\varepsilon)$ and $k(\varepsilon)$ and demonstrate by numerical examples the different behavior as $\varepsilon$ tends to zero.

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In “Recent Trends in Operator Theory and Partial Differential Equations - The Roland Duduchava Anniversary Volume”,
Operator Theory Advances and Application (Birkhäuser/Springer) Vol. 258, 89-110 (2017).
On line DOI: 10.1007/978-3-319-47079-5_5

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