Constructing Separable Non-$2\pi$-Periodic Solutions to the Navier-Lamé Equation in Cylindrical Coordinates Using the Buchwald Representation: Theory and Applications
Bibliographic record
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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".