TY - JOUR
T1 - Constructing Separable Non-$2\pi$-Periodic Solutions to the Navier-Lamé Equation in Cylindrical Coordinates Using the Buchwald Representation: Theory and Applications
AU - Sakhr , Jamal
AU - A. Chronik , Blaine
JO - Advances in Applied Mathematics and Mechanics
VL - 3
SP - 694
EP - 728
PY - 2020
DA - 2020/04
SN - 12
DO - http://doi.org/10.4208/aamm.OA-2019-0128
UR - https://global-sci.org/intro/article_detail/aamm/16420.html
KW - Navier-Lamé equation, cylindrical coordinates, Buchwald representation, exact solutions, $2\pi$-aperiodicity.
AB - <p style="text-align: justify;">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.</p>