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.

Size-dependent free vibration analysis of electrostatically pre-deformed rectangular micro-plates based on the modified couple stress theory

“…It is to be noted that the fringing field does not have a sizable effect especially for the case of wide micro-plates [52,53]. Also, Ω refers to the bottom surface of present micro-plate according to the state of loading shown in Fig.…”

Estimation of oscillation period/switching time for electrostatically actuated microbeam type switches

“…Several researchers have described rigorous approaches towards the dynamic analysis of electrostatically actuated microbeams. Some of the representative approaches include: two degrees of freedom model described by Elata and Bamberger [15], Galerkin method leading to the multimode analysis [16][17][18], finite element analysis [19], reduced order modeling [20,21], nonlinear modal analysis [22], and the application of perturbation method [23]. Comprehensive reviews of the modeling and simulation techniques pertinent to the structural dynamics of microsystems have been presented by Lin and Wang [24] and Rhoads et al [25].…”

Comprehensive Reduced-Order Models of Electrostatically Actuated MEMS Switches and Their Dynamics Including Impact and Bounce

“…The lower limit of voltage when pull-in occurs due to this dynamic effect is called the DC-dynamic pull-in. This is typically around 8% lower than the static pull-in voltage [23]. This is found with this model by a bi-section search method.…”