Generalized Orthogonality Condition for Beams with Intermediate Lumped Masses Subjected to Axial Force
Bibliographic record
Abstract
The orthogonality of the modes of vibration of a distributed parameter system plays an important role in the study of the dynamic behavior of that system. The definition of orthogonality for beams with classical boundary conditions is well known. However, it has been shown that the exact mode shapes of beams that carry one or more attached lumped masses are not orthogonal to each other under the classic condition. In this paper, the effect of axial force on the orthogonality condition of the exact mode shapes of beams with several attached lumped masses, as well as translational and rotational springs, is investigated. It has been shown that, in contrast to the mode shapes themselves, the orthogonality condition remains unchanged when an axial force is applied. Furthermore, the generalized orthogonality condition is employed to study the dynamic behavior of a beam—mass system under different boundary conditions. It has been shown that for the precise investigation of the dynamic response, application of exact mode shapes is not adequate enough, and the orthogonality condition associated with the problem must also be used. Finally, a discussion of the effect of the orthogonality condition on the damping matrices is presented. It has been shown that using the exact mode shapes and generalized orthogonality condition may result in non-modal and non-diagonal damping matrices, which, in turn, may increase the computation time.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.003 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 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 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".