Vibrations of narrow microbeams predeformed by an electric field

Vibrations of a fixed-fixed narrow microbeam electrostatically actuated by applying a voltage difference to it and a parallel rigid conductor are analyzed. For gaps between the two conductors that are comparable to the beam's thickness, the fundamental frequency of the beam may first increase with increasing applied voltage, before suddenly dropping at the pull-in voltage. Available models fail to accurately describe this behavior of the frequency versus voltage diagram for narrow microbeams, that results from a combination of strain-hardening and electrostatic softening effects. A distributed electromechanical model, that accounts for electrostatic fringing fields, finite deflections and residual stresses, is proposed. A recent estimate of the electrostatic force incorporating fringing fields due to both finite width and finite thickness of the microbeam is employed. The lowest frequency is extracted with a simple and computationally efficient one degree-of-freedom model obtained by approximating the deflection field with the static deflection of a fixed-fixed microbeam loaded by a uniformly distributed force. The model's predictions are in good agreement with those from three-dimensional finite-element simulations.