High Frequency Vibrations in a Stiff Problem

Abstract

The stiff problem here considered models the vibrations of a body consisting of two materials, one of them very stiff with respect to the other. We study the asymptotic behavior of the eigenvalues and eigenfunctions of the corresponding spectral problem, when the stiffness constant of only one of the materials tends to 0. We show that the associated operator has a discrete spectrum "converging", in a certain sense, towards a continuous spectrum in [0,∞) corresponding to an operator. We also provide information on the structure of the eigenfunctions associated with the high frequencies.

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.…”

On the vibrations of a plate with a concentrated mass and very small thickness

“…In fact, Theorem 5.1 proves that the high frequencies accumulate on the whole positive real axis. This is a common fact to a large class of spectral problems, which depend on a small parameter, with a discrete spectrum (see References [28,2,22,8]). We use the technique of the Fourier transform for time-dependent problems to derive spectral convergence and techniques of Spectral Perturbation Theory to prove the convergence of the corresponding eigenfunctions (see Section V.13 of Reference [8], and References [2,22], respectively).…”

“…The main results in this section are stated in Theorems 4.1 and 4.2 which give the convergence of the eigenelements of (19) towards those of (22). We prove the spectral convergence in Theorem 4.1 using the technique in Section VII.11 of Reference [8] which applies a result on spectral convergence for implicit eigenvalue problems depending on a parameter We state here this result and refer to in Section V.10 of Reference [8] As in Section VII.11 in Reference [8] (see also Sections IV.6 and IV.8, of Reference [8] and, References [3,10]) we introduce some boundary di erential operators associated with (19) and (22), their properties and the variational formulations of problems (19) and (22); we only outline here the technique.…”

“…We note that the solution U of (27), extended by U U | to R 2 − B, satisÿes (22). Let b(0; ·; ·) be the form deÿned by the left-hand side of (27) …”

“…For ÿxed n, the eigenvalues Á n = n = m−2 of (19) depend continuously on and converge, as → 0 to the eigenvalues Á n of (22).…”