Enhanced spatial skin–effect for free vibrations of a thick cascade junction with “super heavy” concentrated masses

Abstract

The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses is studied. This cascade junction consists of the junction’s body and a great number 5N=O(ε−1) of ε-alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order O(ε). The mass density is of order O(ε−α) on the rods from the second class and O(1) outside of them. There exist five qualitatively different cases in the asymptotic behavior of eigen-magnitudes as ε→0, namely the case of “light” concentrated masses (a∈(0,1)), “intermediate” concentrated masses (α=1) and “heavy” concentrated masses (α∈(1,+∞")) that we divide into “slightly heavy” concentrated masses (α∈(1,2)), “moderate heavy” concentrated masses (α=2), and “super heavy” concentrated masses (alpha>2). In the paper we study the influence of the concentrated masses on the asymptotic behavior of the eigen-magnitudes in the cases α=2 and α>2. The leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions are constructed and the corresponding asymptotic estimates are proved. In addition, a new kind of high-frequency vibrations is found.