Normal modes of a defected linear system of beaded springs
Bibliographic record
Abstract
A model of a one-dimensional mass-spring chain with mass or spring defects is investigated. With a mass defect, all oscillators except the central one have the same mass, and with a spring defect, all the springs except those connected to the central oscillator have the same stiffness constant. The motion is assumed to be one-dimensional and frictionless, and both ends of the chain are assumed to be fixed. The system vibrational modes are obtained analytically, and it is shown that if the defective mass is lighter than the others, then a high frequency mode appears in which the amplitudes decrease exponentially with the distance from the defect. In this sense, the mode is localized in space. If the defect mass is greater than the others, then there will be no localized mode and all modes are extended throughout the system. Analogously, for some values of the defective spring constant, there may be one or two localized modes. If the two defected spring constants are less than that of the others, there is no localized mode.
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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) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".