Perturbation of the eigenvalues of a membrane with a concentrated mass. (English) Zbl 0685.73025

Summary: We study a vibrating membrane with a distribution of density depending on \(\epsilon\), which converges, as \(\epsilon\) \(\searrow 0\), to a uniform density, plus a point mass at the origin. We establish local vibrations at the vicinity of the origin and global vibrations of the membrane. The asymptotic study for \(\epsilon\) \(\searrow 0\) is performed using the method of matched asymptotic expansions.


74H45 Vibrations in dynamical problems in solid mechanics
49R50 Variational methods for eigenvalues of operators (MSC2000)
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
74G70 Stress concentrations, singularities in solid mechanics
35P20 Asymptotic distributions of eigenvalues in context of PDEs
74K15 Membranes
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