Vibration of Thick Circular Disks and Shells of Revolution
Bibliographic record
Abstract
A fully numerical and consistent method using the three-dimensional theory of elasticity is presented in this paper to study the free vibrations of an axially symmetric solid. The solid is defined in the cylindrical coordinates r,θ,z by a quadrilateral cross section in the r-z plane bounded by four straight and/or curved edges. The cross section is then mapped using the natural coordinates (ξ,η) to simplify the mathematics of the problem. The displacement fields are expressed in terms of the product of two simple algebraic polynomials in ξ and η, respectively. Boundary conditions are enforced in the later part of the solution by simply controlling coefficients of the polynomials. The procedure setup in this paper is such that it was possible to investigate the free axisymmetric and asymmetric vibrations of a wide range of problems, namely; circular disks, cylinders, cones, and spheres with considerable success. The numerical cases include circular disks of uniform as well as varying thickness, conical/cylindrical shells and finally a spherical shell of uniform thickness. Convergence study is also done to examine the accuracy of the results rendered by the present method. The results are compared with the finite element method using the eight-node isoparametric element for the solids of revolution and published data by other researchers.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".