  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.
