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

Gómez, D.; Лобо, М.; Pérez, Eugenia

doi:10.1002/mma.342

Cited by 12 publications

(6 citation statements)

References 21 publications

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“…Remark A. 23 Using continuity of embeddings into spaces on the boundary, it follows immediately that…”

Section: Appendixmentioning

confidence: 98%

“…In order to simplify the system (23), one is led to first solve the equation (where we replace λ with −λ)…”

Section: Theorem 311mentioning

confidence: 99%

“…Remark 3.14 It is not difficult to obtain the differential expression for the first and the third equation in (23). (Note that for the third equation one uses (24).)…”

Section: Theorem 311mentioning

confidence: 99%

“…In [19] the limit spectrum of order h 2 is studied, starting from the problem of three-dimensional homogeneous isotropic elasticity, while in [21] a quantitative analysis is provided for the same problem for two different spectral scalings in the corresponding subspaces (membrane and bending). In [29] an asymptotic analysis of the spectrum is carried out starting from a threedimensional heterogeneous plate that is possibly heterogeneous in the transversal direction, while in [23] the authors study a Reissner-Mindlin plate with a mass concentrated at the origin and provide an asymptotic analysis of its spectrum in different regimes.…”

Section: Introductionmentioning

confidence: 99%

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Spectral and Evolution Analysis of Composite Elastic Plates with High Contrast

Marin

Cherednichenko

Velčić

et al. 2022

J Elast

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We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes.

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“…Remark A. 23 Using continuity of embeddings into spaces on the boundary, it follows immediately that…”

Section: Appendixmentioning

confidence: 98%

“…In order to simplify the system (23), one is led to first solve the equation (where we replace λ with −λ)…”

Section: Theorem 311mentioning

confidence: 99%

“…Remark 3.14 It is not difficult to obtain the differential expression for the first and the third equation in (23). (Note that for the third equation one uses (24).)…”

Section: Theorem 311mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spectral and Evolution Analysis of Composite Elastic Plates with High Contrast

Marin

Cherednichenko

Velčić

et al. 2022

J Elast

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“…IV. 6 and IV.8 of Ref. 25 for the variational formulation of problems (1.1)-(1.2) and (1.1)-(1.3) in H 1 (Ω 1 ).…”

Section: Introduction and Statement Of The Problemsmentioning

confidence: 99%

On the Whispering Gallery Modes on Interfaces of Membranes Composed of Two Materials With Very Different Densities

Pérez

2003

Math. Models Methods Appl. Sci.

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We consider a vibrating membrane occupying a domain Ω of R 2 , composed of two materials, with very different densities. These materials fill two domains Ω 1 and Ω 2 of R 2 , and Γ is the interface between them: Γ = ∂Ω 1 ∩ ∂Ω 2 . We look at the associated spectral problem. We prove that there are modes which concentrate in a small neighborhood of Γ, the whispering gallery modes. We address the cases where Ω 2 , the part with negligible mass, is either a bounded or unbounded domain (Ω 2 = R 2 −Ω 1 ), and the case where Ω 1 is a concentrated mass: Ω 1 = εB, with ε → 0, and the density in Ω 1 very much higher than elsewhere.

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