DC dynamic pull-in predictions for a generalized clamped–clamped micro-beam based on a continuous model and bifurcation analysis

Abstract

This study is devoted to providing precise predictions of the dc dynamic pull-in voltages of a clamped–clamped micro-beam based on a continuous model. A pull-in phenomenon occurs when the electrostatic force on the micro-beam exceeds the elastic restoring force exerted by beam deformation, leading to contact between the actuated beam and bottom electrode. DC dynamic pull-in means that an instantaneous application of the voltage (a step function such as voltage) is applied. To derive the pull-in voltage, a dynamic model in partial differential equations is established based on the equilibrium among beam flexibility, inertia, residual stress, squeeze film, distributed electrostatic forces and its electrical field fringing effects. The method of Galerkin decomposition is then employed to convert the established system equations into reduced discrete modal equations. Considering lower-order modes and approximating the beam deflection by a different order series, bifurcation based on phase portraits is conducted to derive static and dynamic pull-in voltages. It is found that the static pull-in phenomenon follows dynamic instabilities, and the dc dynamic pull-in voltage is around 91–92% of the static counterpart. However, the derived dynamic pull-in voltage is found to be dependent on the varied beam parameters, different from a fixed predicted value derived in past works, where only lumped models are assumed. Furthermore, accurate closed-form predictions are provided for non-narrow beams. The predictions are finally validated by finite element analysis and available experimental data.