Vibrations of Beams and Plates: Review of First Closed-Form Solutions in the Past 250 Years View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

2007

AUTHORS

Isaac Elishakoff

ABSTRACT

This paper is dedicated to derivation of eigenvalues of structures that possess modulus of elasticity and/or material density that vary from point to point. There is a large selection of methods that can deal with such a structures’ vibration spectra. In very rare circumstances one has a possibility to obtain an exact solution, usually in terms of transcendental functions (hypergeometric, Bessel, Lommel, and other special functions). In other cases, one resorts to powerful numerical methods, like the finite element method; finite difference method; Rayleight-Ritz or Galerkin methods; collocation method and others. In such circumstances, seeking to find a closed-form solution may appear to be a hopeless task. In 1759, Leonhard Euler was able to derive some closed-form solutions for buckling of non-uniform columns. However, to the best of our knowledge, no closed-form solutions were available for the vibration problems of non-uniform and inhomogeneous structures, until very recently. In this study some closed-form solutions are reported which we were fortunate to derive over the recent six years. Only a few solutions are discussed, in order to provide the scope of developments. It is hoped that this method, and its generalizations will be further developed, to advance the beautiful world of closed-form solutions, that retain their attractiveness even in the present era of numerical solutions. Parts 1 and 2 are devoted to the beam vibrations, whereas Parts 3–5 deal with vibrations of circular plates. More... »

PAGES

389-453

References to SciGraph publications

Book

TITLE

Mechanical Vibration: Where do we Stand?

ISBN

978-3-211-68586-0
978-3-211-70963-4

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-211-70963-4_15

DOI

http://dx.doi.org/10.1007/978-3-211-70963-4_15

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1016971473


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51 schema:description This paper is dedicated to derivation of eigenvalues of structures that possess modulus of elasticity and/or material density that vary from point to point. There is a large selection of methods that can deal with such a structures’ vibration spectra. In very rare circumstances one has a possibility to obtain an exact solution, usually in terms of transcendental functions (hypergeometric, Bessel, Lommel, and other special functions). In other cases, one resorts to powerful numerical methods, like the finite element method; finite difference method; Rayleight-Ritz or Galerkin methods; collocation method and others. In such circumstances, seeking to find a closed-form solution may appear to be a hopeless task. In 1759, Leonhard Euler was able to derive some closed-form solutions for buckling of non-uniform columns. However, to the best of our knowledge, no closed-form solutions were available for the vibration problems of non-uniform and inhomogeneous structures, until very recently. In this study some closed-form solutions are reported which we were fortunate to derive over the recent six years. Only a few solutions are discussed, in order to provide the scope of developments. It is hoped that this method, and its generalizations will be further developed, to advance the beautiful world of closed-form solutions, that retain their attractiveness even in the present era of numerical solutions. Parts 1 and 2 are devoted to the beam vibrations, whereas Parts 3–5 deal with vibrations of circular plates.
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