Constructing Separable Non-$2\pi$-Periodic Solutions to the Navier-Lam\'{e} Equation in Cylindrical Coordinates Using the Buchwald Representation: Theory and Applications
| Autor: | Sakhr, Jamal, Chronik, Blaine A. |
|---|---|
| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Advances in Applied Mathematics and Mechanics, Volume 12, Issue 3 (2020), pp. 694-728 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.4208/aamm.OA-2019-0128 |
| Popis: | 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-Lam\'{e} 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. Comment: arXiv admin note: text overlap with arXiv:1704.06669 |
| Databáze: | arXiv |
| Externí odkaz: |
|