Local problems for vibrating systems with concentrated masses: a review

Лобо, М.; Pérez, Eugenia

doi:10.1016/s1631-0721(03)00058-5

Cited by 48 publications

(58 citation statements)

References 19 publications

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“…Since the pioneering work [37] of E. Sanchez-Palencia, a lot of attention has been paid to mathematical analysis of vibrations of elastic bodies, with small parts wich are very heavy (e.g., pellets in an aspic or in a meat-jelly); see papers [4,9,12,25,30,33,39], as well as the monographs [34,38] in an incomplete list. Such problems are the best examples of the topping role of the boundary layer effect.…”

Section: Asymptotic Analysis Of Eigenvalues In Singularly Perturbed Dmentioning

confidence: 99%

“…If the stiffness matrix A and the density γ are measurable functions of the spatial variables x, and in addition uniformly positive definite and bounded, then the variational problem (12) admits normal positive egenvalues λ p , which form the sequence…”

Section: Vibrations Of Elastic Bodiesmentioning

confidence: 99%

“…The exchange of γ and A by γ h and A h from (17), respectively, transforms (12) in the integral identity for the body weakened by the defects ω h 1 , . .…”

Section: Vibrations Of Elastic Bodiesmentioning

confidence: 99%

“…the main terms of discrepancies, left by the field u in the problem (12) h for the composite body Ω h , appear in the system of equations in ω h j and in the transmition conditions on ∂ω h j . For the compensation of the discrepancies are used the special solutions of the elasticity problem in a homogenuous space with the inclusion ω j of unit size…”

Section: Formal Construction Of Asymptoticsmentioning

confidence: 99%

“…are vector eigenfunctions in the problem (12) corresponding to the eigenvalue λ p and orthonormalized by condition (14), finally the quantities γ j and |ω j | are defined in Lemma 3.2.…”

Section: Theorem 32 Let λ P Be An Eigenvalue In Problemmentioning

confidence: 99%

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On asymptotic analysis of spectral problems in elasticity

Sokołowski

2011

Lat. Am. j. solids struct.

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The three-dimensional spectral elasticity problem is studied in an anisotropic and inhomogeneous solid with small defects, i.e., inclusions, voids, and microcracks. Asymptotics of eigenfrequencies and the corresponding elastic eigenmodes are constructed and justified. New technicalities of the asymptotic analysis are related to variable coefficients of differential operators, vectorial setting of the problem, and usage of intrinsic integral characteristics of defects. The asymptotic formulae are developed in a form convenient for application in shape optimization and inverse problems. Keywords singular perturbations; spectral problem; asymptotics of eigenfunctions and eignevalues; elasticity boundary value problem.

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Section: Asymptotic Analysis Of Eigenvalues In Singularly Perturbed Dmentioning

confidence: 99%

Section: Vibrations Of Elastic Bodiesmentioning

confidence: 99%

“…The exchange of γ and A by γ h and A h from (17), respectively, transforms (12) in the integral identity for the body weakened by the defects ω h 1 , . .…”

Section: Vibrations Of Elastic Bodiesmentioning

confidence: 99%

Section: Formal Construction Of Asymptoticsmentioning

confidence: 99%

“…are vector eigenfunctions in the problem (12) corresponding to the eigenvalue λ p and orthonormalized by condition (14), finally the quantities γ j and |ω j | are defined in Lemma 3.2.…”

Section: Theorem 32 Let λ P Be An Eigenvalue In Problemmentioning

confidence: 99%

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On asymptotic analysis of spectral problems in elasticity

Sokołowski

2011

Lat. Am. j. solids struct.

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

Bakharev

Pérez

2017

Mathematische Nachrichten

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We consider a spectral problem for the Laplace operator in a periodic waveguide Π⊂R3 perturbed by a family of “heavy concentrated masses”; namely, Π contains small regions false{ωjεfalse}j∈double-struckZ of high density, which are periodically distributed along the z axis. Each domain ωjε⊂Π has a diameter O(ε) and the density takes the value ε−m in ωjε and 1 outside; m and ε are positive parameters, m>2, ε≪1. Considering a Dirichlet boundary condition, we study the band‐gap structure of the essential spectrum of the corresponding operator as ε→0. We provide information on the width of the first bands and find asymptotic formulas for the localization of the possible gaps.

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Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems

Лобо

Pérez

2009

Math. Meth. Appl. Sci.

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The interest in the use of quasimodes, or almost frequencies and almost eigenfunctions, to describe asymptotics for low-frequency and high-frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter e, has been recently highlighted in many papers. In this paper we deal with the low frequencies for a Steklov-type eigenvalue homogenization problem: we consider harmonic functions in a bounded domain of R 2 , and strongly alternating boundary conditions of the Dirichlet and Steklov type on a part of the boundary. The spectral parameter appears in the boundary condition on small segments T e of size O(e) periodically distributed along the boundary; e also measures the periodicity of the structure. We consider associated second-order evolution problems on spaces of traces that depend on e, and we provide estimates for the time t in which standing waves, constructed from quasimodes, approach their solutions u e (t) as e → 0.

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Local problems for vibrating systems with concentrated masses: a review

Cited by 48 publications

References 19 publications

On asymptotic analysis of spectral problems in elasticity

On asymptotic analysis of spectral problems in elasticity

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

Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems

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