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4 - Vibrations of Rectangular Plates

Published online by Cambridge University Press:  08 January 2010

Marco Amabili
Marco Amabili
Affiliation:
Università degli Studi, Parma
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Summary

Introduction

Flat rectangular plates with restrained normal displacement at the four edges exhibit a strong hardening-type nonlinearity for vibration amplitude of the order of the plate thickness. In order to have a behavior correctly described by a linear theory, the vibration amplitude of thin plates must be of the order of 1/10 of the thickness, or smaller. In-plane constraints largely enhance the nonlinear behavior; consequently, in-plane stretching is produced for large-amplitude deflection, differently from what is stated for linear theory.

Rectangular plates with normal displacement that is not restrained at all the edges can present a linear behavior for larger vibration amplitude. This is the case of the cantilever plate (clamped at one edge and free on the other three edges). In fact, quite large displacement w can be associated with very small rotations ∂w/∂x and ∂w/∂y for these boundary conditions, so that nonlinear terms in the strains can be neglected.

Geometric imperfections play an important role; they transform the flat plate in a curved panel (even if very shallow), which exhibits an initial weak softening behavior, turning to strong hardening nonlinearity for larger vibration amplitude.

In this chapter, the linear vibrations of simply supported rectangular plates are first addressed; numerical and experimental results are presented. Then, nonlinear forced vibrations of plates with different boundary conditions are studied by using the Lagrange equations of motion and the von Kármán theory. The effect of geometric imperfections is investigated. Numerical and experimental results are presented and satisfactorily compared.

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Nonlinear Vibrations and Stability of Shells and Plates , pp. 120 - 140
Publisher: Cambridge University Press
Print publication year: 2008

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References

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  • Vibrations of Rectangular Plates
  • Marco Amabili, Università degli Studi, Parma
  • Book: Nonlinear Vibrations and Stability of Shells and Plates
  • Online publication: 08 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511619694.006
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  • Vibrations of Rectangular Plates
  • Marco Amabili, Università degli Studi, Parma
  • Book: Nonlinear Vibrations and Stability of Shells and Plates
  • Online publication: 08 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511619694.006
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Vibrations of Rectangular Plates
  • Marco Amabili, Università degli Studi, Parma
  • Book: Nonlinear Vibrations and Stability of Shells and Plates
  • Online publication: 08 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511619694.006
Available formats
×