TY - JOUR

T1 - High frequency oscillations of first eigenmodes in axisymmetric shells as the thickness tends to zero

AU - Chaussade-Beaudouin, Marie

AU - Dauge, Monique

AU - Faou, Erwan

AU - Yosibash, Zohar

N1 - Publisher Copyright:
© 2017 Springer International Publishing.

PY - 2017

Y1 - 2017

N2 - The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2ε) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue λ(ε) and the eigenvector angular frequency k(ε) 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(ε) 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(ε) tends to infinity as ε tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide λ(ε) and k(ε) and demonstrate by numerical examples the different behavior as e tends to zero.

AB - The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2ε) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue λ(ε) and the eigenvector angular frequency k(ε) 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(ε) 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(ε) tends to infinity as ε tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide λ(ε) and k(ε) and demonstrate by numerical examples the different behavior as e tends to zero.

KW - Axisymmetric shell

KW - Developable shell

KW - Koiter

KW - Lamé

KW - Sensitive shell

UR - http://www.scopus.com/inward/record.url?scp=85013941528&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-47079-5_5

DO - 10.1007/978-3-319-47079-5_5

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AN - SCOPUS:85013941528

SN - 0255-0156

VL - 258

SP - 89

EP - 110

JO - Operator Theory: Advances and Applications

JF - Operator Theory: Advances and Applications

ER -