Free Vibration Analysis of a Simply Supported Axial Functionally Graded Beam Using the Rayleigh Method
Bibliographic record
Abstract
This study aims to make investigation for the free vibration problem of simply supported axial-FGB by applying the Rayleigh method.The model of (power law) is adopted to describe the change in physical mechanical and properties through the axial direction.The computer program is built in this work.The accuracy of Rayleigh method is checked by comparison of the results of Rayleigh method with the results available in literatures and a very good accuracy was found.The combined effects of power-law index, modulus-ratio and density-ratio on the fundamental frequency and mode shapes of axial-FGBs are investigated.The results explain that the non-dimensional frequency parameter at any modulus-ratio is approximately constant when the power-law index increases.When the power-law index is smaller than (1), the effect of modulus-ratio is greater than the effect of density-ratio.While the effect of modulus-ratio is smaller than the effect of density-ratio when the power-law index is greater than (1).Also, when density-ratio equals (1), the dimensionless deflection (or amplitude) increases with reducing of power-law index (β).When the modulus-ratio is smaller than (1), the dimensionless deflection increases with increasing the power-law index (β).From results, the Rayleigh method is able to calculate the natural frequency and mode shape of simply supported axial-FGM beam.
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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 | 0.000 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".