Delayed pull-in transitions in overdamped MEMS devices

TL;DR

This paper analyzes the slow dynamics of overdamped MEMS devices near pull-in instability, deriving a universal scaling law for pull-in time and providing a practical method to estimate pull-in voltage from time data.

Contribution

It introduces a quantitative analysis of critical slowing down in pull-in dynamics, deriving an inverse square-root scaling law and an analytical expression for pull-in time.

Findings

Pull-in time follows an inverse square-root scaling law near the transition.

The analytical expression accurately predicts observed pull-in times.

The scaling law is applicable across various experimental data sets.

Abstract

We consider the dynamics of overdamped MEMS devices undergoing the pull-in instability. Numerous previous experiments and numerical simulations have shown a significant increase in the pull-in time under DC voltages close to the pull-in voltage. Here the transient dynamics slow down as the device passes through a meta-stable or bottleneck phase, but this slowing down is not well understood quantitatively. Using a lumped parallel-plate model, we perform a detailed analysis of the pull-in dynamics in this regime. We show that the bottleneck phenomenon is a type of critical slowing down arising from the pull-in transition. This allows us to show that the pull-in time obeys an inverse square-root scaling law as the transition is approached; moreover we determine an analytical expression for this pull-in time. We then compare our prediction to a wide range of pull-in time data reported in the literature, showing that the observed slowing down is well captured by our scaling law, which appears to be generic for overdamped pull-in under DC loads. This realization provides a useful design rule with which to tune dynamic response in applications, including state-of-the-art accelerometers and pressure sensors that use pull-in time as a sensing mechanism. We also propose a method to estimate the pull-in voltage based only on data of the pull-in times.