  Electrostatic actuators are simple but important switching devices for MEMS
applications. Due to the difficulties associated with the electrostatic
nonlinearity, precise mathematical description is often hard to obtain for the
dynamics of these actuators. Here we present two sharp theorems concerning the
dynamics of an undamped electrostatic actuator with one-degree of freedom,
subject to linear and nonlinear elastic forces, respectively. We prove that
both situations are characterized by the onset of one-stagnation-point periodic
response below a well-defined pull-in voltage and a finite-time touch-down or
collapse of the actuator above this pull-in voltage. In the linear-force
situation, the stagnation level, pull-in voltage, and pull-in coordinate of the
movable electrode may all be determined explicitly, following the recent work
of Leus and Elata based on numerics. Furthermore, in the nonlinear-force
situation, the stagnation level, pull-in voltage, and pull-in coordinate may be
described completely in terms of the electrostatic and mechanical parameters of
the model so that they approach those in the linear-force situation
monotonically in the zero nonlinear-force limit.
