A Procedure for Solving Under-actuated Optimal Attenuation of Vibrations by means of Symbolic Operators
Bibliographic record
Abstract
The problem of optimal attenuation of vibrations with the number of controlled modes greater than the number of independent discrete actuators is analyzed. The modal variables are coupled via non-holonomic constraints that are satisfied with the help of time-dependent Lagrange's multipliers. A given configuration of actuators is characterized by a normalized matrix of constraints with constant coefficients. The coefficients are used to evaluate controllability and overall effectiveness of the actuators as configured. For the quadratic performance index the optimality equations are derived from Pontryagin's principle in a compact form containing time derivatives of the controlled modal variables and Lagrange's multipliers. The problem is formulated as a two-points-boundary-value problem for the modes involved. The equations are solved automatically by applying symbolic differential operators and standard mathematical software. The proposed procedure consists of structural and control phases, and permits determining the optimal histograms of forces in each actuator and the expected response of the structure, including the active damping ratios. It also provides hints for adjusting the configuration of actuators to improve their performance in the case where some of the modes are too difficult to control. Numerical examples illustrate the procedure.
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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 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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".