A Variational Formulation for Coupled Single-Walled Carbon Nanotubes Undergoing Vibrations in the Presence of an Axial Magnetic Field
Bibliographic record
Abstract
A variational formulation is presented for a system consisting of two coupled singlewalled carbon nanotubes subjected to forced vibrations, longitudinal magnetic field, and axial compression, based on the nonlocal elasticity theory.The variational principle for the double nanotube system is derived, followed by the application of Hamilton's principle to express kinetic and potential energies.Subsequently, the time-independent scenario is investigated, and governing equations for the freely vibrating system are provided.The variational formulation for this case is established, and expressions for Rayleigh quotients concerning the vibration frequency and buckling load are derived.The Rayleigh quotient for the frequency demonstrates that the magnetic field increases the vibration frequency of the coupled nanotube system.Nonlocal effects appear in both the numerator and the denominator of the Rayleigh quotient, influencing the frequency increase or decrease depending on the relative values of various problem parameters.In contrast, the magnetic field reduces the buckling load, as evidenced by its negative contribution to the numerator of the Rayleigh quotient for buckling.The effect of the nonlocal parameter on buckling, however, cannot be inferred directly from the Rayleigh quotient.In this study, involving a system of two coupled partial differential equations, it is crucial to derive variationally consistent boundary conditions.Utilizing the formulated variational principle, variationally consistent natural boundary conditions are established in terms of moment and shear force expressions.It is revealed that the Pasternak interlayer between the nanotubes results in coupled boundary conditions when a shear force and/or a moment is specified at the boundaries of the nanotube system.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 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".