Stabilization of the side pull-in bifurcation of electrostatic comb drive actuators using oscillatory open-loop control

Constant-gap electrostatic actuators like transverse comb drives exhibit the side pull-in instability, which may occur with respect to translational, rotational, or deformational degrees of freedom. Techniques that are successful in delaying the related pull-in instability for voltage-controlled gap-closing actuators-including charge control and charge or position feedback-are ineffective against side pull-in. Although closed-loop stabilization techniques do not work, side pull-in can be significantly delayed using open-loop oscillatory techniques similar to parametric resonance. This research considers the translational planar dynamics of an electrostatic comb drive, in which side pull-in appears as a subcritical pitchfork bifurcation. For small lateral perturbations and negligible damping, the lateral dynamics are approximated by a decoupled second-order linear model in the form of Hill's equation. Classical results of Floquet theory provide a stability diagram, from which is obtained an open-loop oscillatory drive voltage signal that stabilizes the lateral dynamics well beyond the pull-in voltage. We take a heuristic approach to adapt the stability analysis developed for the translational rigid body dynamics to describe the dynamics of a deformed electrode. For that we map the properties of the cantilever on to those of the translational rigid body. We show that the dynamics of the model with the rotational degree of freedom can be put into the form of the translational dynamics. Numerical and finite element simulations show that the linear stability analysis remains valid when the simplifying assumptions are relaxed. To our knowledge, this is the first demonstration of stabilization of side pull-in without adding sensors, actuators, or closed-loop control logic. To our knowledge it is also the first application of open-loop oscillatory control to an electrostatic MEMS model using a single control voltage.