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Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses
Author(s) -
Chechkin G. A.,
Mel'nyk T. A.
Publication year - 2013
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2785
Subject(s) - eigenfunction , cascade , mathematics , rod , eigenvalues and eigenvectors , mathematical analysis , operator (biology) , laplace transform , order (exchange) , asymptotic expansion , geometry , physics , chemistry , quantum mechanics , medicine , biochemistry , alternative medicine , chromatography , pathology , repressor , transcription factor , gene , finance , economics
A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number 5 N = 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 density of the junction is of order O ( ϵ− α) on the rods from the second class and O ( 1 ) outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ϵ → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ϵ → 0, namely the case of ‘light’ concentrated ( α ∈ (0,1)), ‘middle’ concentrated ( α = 1), and ‘heavy’ concentrated masses ( α ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated ( α ∈ (1,2)), ‘intermediate heavy’ concentrated ( α = 2), and ‘very heavy’ concentrated masses ( α > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.