Nonlinear Longitudinal Vibration Solutions of an Elastic Rod
Bibliographic record
Abstract
Nonlinear longitudinal vibration of an elastic rod is studied. The motion of a uniform elastic rod is described by a nonlinear partial differential equation, which has a cubic nonlinear term and a Winkler elastic force that acts along the longitudinal axis of the rod. Galerkin method is used to develop the nonlinear differential equation of elastic rod, which resembles similarity with the Duffing equation. Three different types of robust analytical methods are chosen to solve the nonlinear differential equation and obtain the natural frequency of the system. These are the Homotopy analysis method (HAM), Energy balance method (EBM) and Hamiltonian approach (HA). Subsequently, the analytical results are compared with the numerical solution of the exact equation in order to evaluate the correctness of the applied approaches. Moreover, the effects of the constant coefficients of the elastic force on the ratio of the nonlinear to the linear frequencies are studied. The singular points of the nonlinear differential equation of the elastic rod are extracted and the Jacobian matrix is constructed to recognize their types. Finally, phase-plane trajectories of the system are constructed in order to verify the results obtained from the Jacobian matrix.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| 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 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 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 teacher head, 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".