Vibrations of parallel arrays of electrostatically actuated microplates

We study small vibrations of a parallel array of microplates predeformed by an electric field. We derive closed-form expressions for the array's modal properties, including resonance frequencies, mode shapes, and modal dampings. Determined expressions depend on the direct current voltage applied across adjacent microplates, on the number of microplates comprising the array, on the boundary conditions imposed on each microplate, and on the geometry and material properties of the array. We also propose a continuum approximation of the array, where the discrete set of microplates' deflections is replaced by a continuous field that represents the envelope of the microplates' displacements. In the continuum approximation, the set of partial differential equations describing the system's dynamics is replaced by a sole partial differential equation. Predictions of the continuum approximation are in very good agreement with the exact solution and provide a new perspective in analyzing complex microsystems.