Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses |
| |
Authors: | G A Chechkin T A Mel'nyk |
| |
Institution: | 1. Department of Differential Equations, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, , 119991 Moscow, Russia;2. Department of Mathematical Physics, Faculty of Mechanics and Mathematics, Taras Shevchenko Kyiv National University, , 64, 01033 Kyiv, Ukraine |
| |
Abstract: | 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 of ?‐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 . The density of the junction is of order on the rods from the second class and outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ? → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ? → 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. |
| |
Keywords: | homogenization asymptotic approximation spectral problem thick cascade junction rapidly oscillating boundary concentrated masses |
|
|