Nonlinear Dynamics of Spring Softening and Hardening in Folded-MEMS Comb Drive Resonators
Bibliographic record
Abstract
This paper studies analytically and numerically the spring softening and hardening phenomena that occur in electrostatically actuated microelectromechanical systems comb drive resonators utilizing folded suspension beams. An analytical expression for the electrostatic force generated between the combs of the rotor and the stator is derived and takes into account both the transverse and longitudinal capacitances present. After formulating the problem, the resulting stiff differential equations are solved analytically using the method of multiple scales, and a closed-form solution is obtained. Furthermore, the nonlinear boundary value problem that describes the dynamics of inextensional spring beams is solved using straightforward perturbation to obtain the linear and nonlinear spring constants of the beam. The analytical solution is verified numerically using a Matlab/Simulink environment, and the results from both analyses exhibit excellent agreement. Stability analysis based on phase plane trajectory is also presented and fully explains previously reported empirical results that lacked sufficient theoretical description. Finally, the proposed solutions are, once again, verified with previously published measurement results. The closed-form solutions provided are easy to apply and enable predicting the actual behavior of resonators and gyroscopes with similar structures.$\hfill$[2011-0007]
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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 | 0.001 |
| 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".