Lie group analysis of non-linear dynamic of micro structures under electrostatic field
Bibliographic record
Abstract
This paper presents an analytical solution of nonlinear differential equation of micro-structures subjected to electrostatic fields. The constitutive equation of such a model is a second order differential equation (ODE). The problem is solved when the assumption of linear deflection is considered. However, deflection of micro cantilevers in practical applications is non-linear. Moreover, the constitutive ODE is stiff and various numerical algorithms used to solve it yield non-consistent numerical solutions. A deduction order method - Lie group symmetry is employed to reduce the order of the ODE. Although the resulting first order ODE has no symmetry that would guarantee an explicit close form solution, it enables an analytical formulation for the no-damping assumption only. The restoring force term in the first order ODE reveals the pull-in voltage as expressed in classical MEMS textbooks. It is shown that the numerical solution for the second order ODE and the reduced first order ODE are same. Finding any symmetry other than translation, scaling or rotation will enable the reduction of the first order ODE and thus, the formulation of an analytical solution to this highly non-linear problem.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".