Constructing Separable Non-$2\pi$-Periodic Solutions to the Navier-Lamé Equation in Cylindrical Coordinates Using the Buchwald Representation: Theory and Applications
Year: 2020
Author: Jamal Sakhr, Blaine A. Chronik
Advances in Applied Mathematics and Mechanics, Vol. 12 (2020), Iss. 3 : pp. 694–728
Abstract
In a previous paper (Adv. Appl. Math. Mech., 10 (2018), pp. 1025-1056), we used the Buchwald representation to construct several families of separable cylindrical solutions to the Navier-Lamé equation; these solutions had the property of being $2\pi$-periodic in the circumferential coordinate. In this paper, we extend the analysis and obtain the complementary set of separable solutions whose circumferential parts are elementary $2\pi$-aperiodic functions. Collectively, we construct eighteen distinct families of separable solutions; in each case, the circumferential part of the solution is one of three elementary $2\pi$-aperiodic functions. These solutions are useful for solving a wide variety of dynamical problems that involve cylindrical geometries and for which $2\pi$-periodicity in the angular coordinate is incompatible with the given boundary conditions. As illustrative examples, we show how the obtained solutions can be used to solve certain forced-vibration problems involving open cylindrical shells and open solid cylinders where (by virtue of the boundary conditions) $2\pi$-periodicity in the angular coordinate is inappropriate. As an addendum to our prior work, we also include an illustrative example of a certain type of asymmetric problem that can be solved using the particular $2\pi$-periodic subsolutions that ensue when there is no explicit dependence on the circumferential coordinate.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2019-0128
Advances in Applied Mathematics and Mechanics, Vol. 12 (2020), Iss. 3 : pp. 694–728
Published online: 2020-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 35
Keywords: Navier-Lamé equation cylindrical coordinates Buchwald representation exact solutions $2\pi$-aperiodicity.
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