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January, 2020 Asymptotic behavior of eigenfrequencies of a thin elastic rod with non-uniform cross-section
Shuichi JIMBO, Albert RODRÍGUEZ MULET
J. Math. Soc. Japan 72(1): 119-154 (January, 2020). DOI: 10.2969/jmsj/81198119

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 $\varepsilon \rightarrow 0$.

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Shuichi JIMBO. Albert RODRÍGUEZ MULET. "Asymptotic behavior of eigenfrequencies of a thin elastic rod with non-uniform cross-section." J. Math. Soc. Japan 72 (1) 119 - 154, January, 2020. https://doi.org/10.2969/jmsj/81198119

Information

Received: 30 August 2018; Published: January, 2020
First available in Project Euclid: 16 October 2019

zbMATH: 07196500
MathSciNet: MR4055092
Digital Object Identifier: 10.2969/jmsj/81198119

Subjects:
Primary: 35P15
Secondary: 35J15, 35P20

Rights: Copyright © 2020 Mathematical Society of Japan

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Vol.72 • No. 1 • January, 2020
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