Vibrations of thin plates with small clamped patches

A. E. Lindsay, Wenrui Hao, A. J. Sommese

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Eigenvalues of fourth-order elliptic operators feature prominently in stability analysis of elastic structures. This paper considers out-of-plane modes of vibration of a thin elastic plate perforated by a collection of small clamped patches. As the radius of each patch shrinks to zero, a point constraint eigenvalue problem is derived in which each patch is replaced by a homogeneous Dirichlet condition at its centre. The limiting problem is consequently not the eigenvalue problem with no patches, but a new type of spectral problem. The discrepancy between the eigenvalues of the patch-free and point constraint problems is calculated. The dependence of the point constraint eigenvalues on the location(s) of clamping is studied numerically using techniques from numerical algebraic geometry. The vibrational frequencies are found to depend very sensitively on the number and centre(s) of the clamped patches. For a range of number of punctures, we find spatial clamping patterns that correspond to local maxima of the base vibrational frequency of the plate.

Original languageEnglish (US)
Article number20150474
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume471
Issue number2184
DOIs
StatePublished - Dec 8 2015

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thin plates
Thin Plate
Vibrational spectra
Patch
eigenvalues
Vibration
Perforated plates
vibration
Eigenvalue
Eigenvalue Problem
Geometry
elastic plates
Dirichlet conditions
Elastic Plate
Algebraic Geometry
Spectral Problem
Spatial Pattern
vibration mode
Elliptic Operator
Discrepancy

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

Cite this

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title = "Vibrations of thin plates with small clamped patches",
abstract = "Eigenvalues of fourth-order elliptic operators feature prominently in stability analysis of elastic structures. This paper considers out-of-plane modes of vibration of a thin elastic plate perforated by a collection of small clamped patches. As the radius of each patch shrinks to zero, a point constraint eigenvalue problem is derived in which each patch is replaced by a homogeneous Dirichlet condition at its centre. The limiting problem is consequently not the eigenvalue problem with no patches, but a new type of spectral problem. The discrepancy between the eigenvalues of the patch-free and point constraint problems is calculated. The dependence of the point constraint eigenvalues on the location(s) of clamping is studied numerically using techniques from numerical algebraic geometry. The vibrational frequencies are found to depend very sensitively on the number and centre(s) of the clamped patches. For a range of number of punctures, we find spatial clamping patterns that correspond to local maxima of the base vibrational frequency of the plate.",
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Vibrations of thin plates with small clamped patches. / Lindsay, A. E.; Hao, Wenrui; Sommese, A. J.

In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 471, No. 2184, 20150474, 08.12.2015.

Research output: Contribution to journalArticle

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AB - Eigenvalues of fourth-order elliptic operators feature prominently in stability analysis of elastic structures. This paper considers out-of-plane modes of vibration of a thin elastic plate perforated by a collection of small clamped patches. As the radius of each patch shrinks to zero, a point constraint eigenvalue problem is derived in which each patch is replaced by a homogeneous Dirichlet condition at its centre. The limiting problem is consequently not the eigenvalue problem with no patches, but a new type of spectral problem. The discrepancy between the eigenvalues of the patch-free and point constraint problems is calculated. The dependence of the point constraint eigenvalues on the location(s) of clamping is studied numerically using techniques from numerical algebraic geometry. The vibrational frequencies are found to depend very sensitively on the number and centre(s) of the clamped patches. For a range of number of punctures, we find spatial clamping patterns that correspond to local maxima of the base vibrational frequency of the plate.

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