Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The eigenvalue problem for a class of long, thin elastic structures with periodic geometry

Author: Robert E. Miller
Journal: Quart. Appl. Math. 52 (1994), 261-282
MSC: Primary 73B27; Secondary 35B27, 73K05, 73K12, 73K20
DOI: https://doi.org/10.1090/qam/1276237
MathSciNet review: MR1276237
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Abstract: Elastic structures consisting of many thin elements arranged periodically (such as grids and trusses) are common in applications. Using standard numerical techniques such as splines in attempting to analyze these structures leads to serious difficulties due to the complicated geometry. Instead, one can use methods of asymptotic analysis to derive a ``simple'' problem whose solution approximates that of the original problem. In this paper we begin with a linearized elastic system on a three-dimensional domain with two of the dimensions small relative to the third and derive a one-dimensional eigenvalue problem by letting a small parameter tend to zero. The resulting equation has coefficients which vary periodically with the spatial variable, so we let the period tend to zero to obtain the ``homogenized'' equation which has constant coefficients whose values can be easily calculated once the geometry of the structure is specified. We illustrate with several examples.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/qam/1276237
Article copyright: © Copyright 1994 American Mathematical Society

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