Free Vibration of Spinning Stepped Timoshenko Beams Using Finite Element Method
Bibliographic record
Abstract
Abstract Free lateral vibration of stepped shafts is investigated in this paper using the Timoshenko beam theory and the finite element method. Beam finite elements having two nodes and 16 degrees of freedom were employed to model flexural vibration of a stepped shaft for a total four field variables — two lateral displacements and two bending angles. Within each uniform segment, the stepped shaft is modeled as a substructure for which a system of equations of motion may be easily formulated using the Galerkin method. The global equations of motion for the entire stepped shaft are subsequently formulated by enforcing the displacement continuity and force equilibrium conditions across the interfaces between two adjacent substructures. The second order governing differential equations for a non self-adjoint dynamic system are then reduced to the equivalent first order differential equations for which eigenvalue problem is formulated and solved using the Matlab® program. Values of natural frequencies are in excellent agreement with those available in the literature. Effects of rotational springs attached to the end of a stepped shaft, used to simulate the non-classical boundary constraints of chuck on a work piece in a typical turning process, are also investigated. The bi-orthogonal conditions for modal vectors, which are useful in chatter analysis during turning processes, are given in this paper.
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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) | 0.001 | 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".