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

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.