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

Leal, C.; Sanchez-Hubert, J.

Q. Appl. Math. 47, No. 1, 93-103 (1989).

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.

MSC:

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

Keywords:

vibrating membrane; local vibrations; global vibrations; method of matched asymptotic expansions