Periodic solutions and the avoidance of pull--in instability in non--autonomous micro--electro--mechanical systems

TL;DR

This paper analytically and numerically investigates periodic solutions in non-autonomous MEMS devices, showing they can be stable and help avoid pull-in instability, thereby expanding safe operational conditions.

Contribution

It provides analytical existence results for periodic solutions in nonlinear MEMS models and demonstrates their stability and stability islands numerically, including higher-period solutions.

Findings

Stable periodic solutions exist for a range of parameters.

Large stability islands prevent pull-in instability.

Higher-period solutions also provide safe operation zones.

Abstract

We study periodic solutions of a one-degree of freedom micro-electro-mechanical system (MEMS) with a parallel-plate capacitor under $T$--periodic electrostatic forcing. We obtain analytical results concerning the existence of $T-$ periodic solutions of the problem in the case of arbitrary nonlinear restoring force, as well as when the moving plate is attached to a spring fabricated using graphene. We then demonstrate numerically on a $T-$ periodic Poincar{\'e} map of the flow that these solutions are generally locally stable with large "islands" of initial conditions around them, within which the pull-in stability is completely avoided. We also demonstrate graphically on the Poincar{\'e} map that stable periodic solutions with higher period $nT, n>1$ also exist, for wide parameter ranges, with large "islands" of bounded motion around them, within which all initial conditions avoid the pull--in instability, thus helping us significantly increase the domain of safe operation of these MEMS models.