Vibrating plates with concentrated masses and very small thickness: Low frequencies
by
Delfina Gómez
Dpto. Matemáticas, Estadística y Computación. Universidad de Cantabria (SPAIN)
Coauthors: Miguel Lobo (Dpto. Matemáticas, Estadística y Computación. Universidad de Cantabria, SPAIN), Eugenia Pérez (Dpto. Matemática Aplicada y Ciencias de la Computación. Universidad de Cantabria, SPAIN)

We consider the vibrations of an elastic, homogeneous and isotropic plate occupying the domain [`(\Omega)]×[-h<sub>0</sub>/2, h<sub>0</sub>/2] of R<sup>3</sup> that contains a small region of high density, the so-called concentrated mass. The size of this region, \epsilon[`B]×[-h₀/2, h₀/2], depends on a small parameter \epsilon; the density is of order O(\epsilon^-m) in this part and O(1) outside; m is a parameter, m > 0, and h₀ is the plate thickness. We consider the associated spectral problem in the framework of the Reissner-Mindlin plate model. We look at the asymptotic behaviour of the eigenvalues and eigenfunctions (\zeta^\epsilon, u^\epsilon) of this spectral problem when the parameters \epsilon and h tend to zero, h being h²=h₀²/12. We assume a relation between \epsilon and h; in particular, we perform the study in the case where h=\epsilon^r with r >= 1.

For each fixed n, we study the asymptotic behaviour, as \epsilon --> 0, of the eigenvalues \zeta_n^\epsilon, the so-called low frequencies, and their corresponding eigenfunctions. We show that there is a different behaviour depending on whether m is m < 2, 2 <= m <= 4 or m > 4.

Date received: June 12, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calt-14.