VIBRATIONS OF A THICK PERIODIC JUNCTION WITH CONCENTRATED MASSES

Abstract

The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a mixed boundary-value problem for the Laplace operator in a plane thick periodic junction with concentrated masses is investigated. This junction consists of the junction's body and a large number N=O(ε-1) of thin rods. The density of the junction is order O(ε-α) on the rods (the concentrated masses if α>0), and O(1) outside. The results depend on the value of the parameter α(α<2, α=2, or α>2). There are three kinds of vibrations, which are present in each of these cases: vibrations, whose energy is concentrated in the junction's body; vibrations, whose energy is concentrated on the thin rods; and vibrations (pseudovibrations), in which each thin rod can have its own frequency. The frequency range, where pseudovibrations can be present, is indicated. The asymptotic estimates for the corresponding eigenfunctions and eigenvalues are proved.