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

TL;DR

This paper analyzes the behavior of the first eigenmodes in thin axisymmetric shells as their thickness approaches zero, revealing different asymptotic behaviors in acoustics and elasticity models through novel expansions.

Contribution

It introduces a new asymptotic expansion method to determine eigenvalue and eigenvector behavior in thin shells for both acoustics and elasticity models.

Findings

Eigenvalues and eigenvectors exhibit distinct asymptotic behaviors as thickness tends to zero.

In acoustics, the angular frequency remains zero, while in elasticity it tends to infinity for certain shell geometries.

Explicit asymptotic formulas are provided for cylinders and elliptical shells, supported by numerical examples.

Abstract

The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2$\epsilon$) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue $\lambda$($\epsilon$) and the eigenvector angular frequency k($\epsilon$) 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($\epsilon$) is always zero. In contrast to it, for the Lam{\'e} system of linear elasticity several different types of shells are defined, characterized by their geometry, for which k($\epsilon$) tends to infinity as $\epsilon$ tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide $\lambda$($\epsilon$) and k($\epsilon$) and demonstrate by numerical examples the different behavior as $\epsilon$ tends to zero.