A CLOSED FORM SOLUTION FOR NON-LINEAR DEFLECTION OF NON-STRAIGHT LUDWICK TYPE BEAMS USING LIE SYMMETRY GROUPS
Bibliographic record
Abstract
Micro Electro Mechanical Systems (MEMS) have found a large range of applications over the recent years.One of the prodigious application of micro-cantilever beams that is in use is represented by AFM probes (Atomic Force Microscopy).The AFM principle is based on the realtime measurement of the deflection of a micro-beam while following a surface profile.Hence, the prior knowledge of the deflection of beams has been of great interest to designers.Although both analytical and numerical solutions have been found for specific type of loads, there is no general solution specifically formulated for micro-cantilever beams that are not geometrically perfectly straight.Hence, the problem has not been specifically considered so far.The current work presents an analytical method based on Lie symmetry groups.The presented method produces an exact analytical solution for the deflection of Ludwick type beams subjected to any point load for nonstraight beams.The Lie symmetry method is used to reduce the order of the Ordinary Differential Equation (ODE) and formulate an analytical solution of the deflection function.The result is compared with an analytical solution for a particular case that is available in the open literature.It was found that the two results coincide.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".