The skin effect in vibrating systems with many concentrated masses

Abstract

We address the asymptotic behaviour of the vibrations of a body occupying a domain \documentclass{article}\usepackage{amsfonts}\begin{document}\pagestyle{empty}$\Omega\subset\mathbb{R}^n, n=2,3$\end{document}. The density, which depends on a small parameter $\varepsilon$\nopagenumbers\end , is of the order $O(1)$\nopagenumbers\end out of certain regions where it is $O(\varepsilon^{‐m})$\nopagenumbers\end with $m>2$\nopagenumbers\end. These regions, the concentrated masses with diameter $O(\varepsilon)$\nopagenumbers\end, are located near the boundary, at mutual distances $O(\eta)$\nopagenumbers\end, with $\eta=\eta(\varepsilon)\rightarrow 0$\nopagenumbers\end. We impose Dirichlet (resp. Neumann) conditions at the points of $\partial\Omega$\nopagenumbers\end in contact with (resp. out of) the masses. We look at the asymptotic behaviour, as $\varepsilon\rightarrow 0$\nopagenumbers\end, of the eigenvalues of order $O(1)$\nopagenumbers\end, the high frequencies, of the corresponding eigenvalue problem. We show that they accumulate on the whole positive real axis and characterize those giving rise to global vibrations of the whole system. We use the fact that the corresponding eigenfunctions, microscopically, present a skin effect in the concentrated masses. Copyright © 2001 John Wiley & Sons, Ltd.

Asymptotic expansion of eigenelements of the Laplace operator in a domain with a large number of ‘light’ concentrated masses sparsely situated on the boundary. Two-dimensional case

“…By contrast with the papers [47,48,49,50,51,52], it is assumed that the masses are situated on the boundary rather sparsely, as was assumed in [25,26,27,28,29,30,31,32], when the distance between the masses is substantially greater than their diameter. The distance between the masses is assumed to be equal to ε, the diameter of the masses equal to a ε, where a = a(ε) → 0 as ε → 0, and the density is assumed to be equal to ε −m , m < 2.…”

Spectral convergence for vibrating systems containing a part with negligible mass

“…Let * be an eigenvalue of (14) and U * an eigenfunction associated with * , U * ∈V, U * of norm 1 in L 2 (B). Let C * be deÿned by (36) with U * i ≡ U * , namely:…”

“…In Section 6 we provide a sample which illustrates the interest of the results in Sections 5 in order to describe the asymptotic behaviour of the spectrum of an eigenvalue problem associated with a vibrating system with many concentrated masses. Let us mention References [6,[10][11][12][13][14] in connection with these vibrating systems (see also Reference [6] for more references).…”

Spectral gaps for the Dirichlet‐Laplacian in a 3‐D waveguide periodically perturbed by a family of concentrated masses

“…[39,40] for the case where 0 < ⩽ 2, [45] for = 2, and [8,32] for Neumann and periodic boundary conditions with > 0. Also, introducing very many concentrated masses, at a distance between them which depends on and converge towards zero, changes qualitatively the asymptotic behavior of the low frequencies and that of the associated eigenfunctions: see [10,[27][28][29][30]36]. See Section 1.1 for more details.…”