Vibration of beams using novel boundary characteristic orthogonal polynomials satisfying all boundary conditions
Bibliographic record
Abstract
Boundary characteristic orthogonal polynomials proposed by the author in 1985 have been used in the Rayleigh Ritz method extensively in order to obtain natural frequencies of vibrating plates with different boundary conditions. The method used products of the characteristic orthogonal polynomials along the two directions of the plate. The first member of the boundary characteristic orthogonal polynomials set satisfied all the boundary conditions of the vibrating beam, including the natural conditions. However, the higher members of the set satisfied only the geometry boundary conditions. In this study, a modified Gram–Schmidt orthogonalization method is presented where all the members of the orthogonal set of polynomials satisfy all the boundary conditions including the natural boundary conditions. Furthermore, the exact solution of the beam differential equation is expressed in the form of a generalized Fourier series in terms of the set of new boundary characteristic orthogonal polynomials which forms an eigenvalue problem that can provide the natural frequencies and the corresponding normal modes of the beam more accurately.
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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 | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".