We present here a simple analytical model for self-oscillations in nano-electro-mechanical systems. We show that a field emission self-oscillator can be described by a lumped electrical circuit and that this approach is generalizable to other electromechanical oscillator devices. The analytical model is supported by dynamical simulations where the electrostatic parameters are obtained by finite element computations.
Abstract. We study a reaction-diffusion equation in a bounded domain in the plane, which is a mathematical model of an idealized electrostatically actuated microelectromechanical system (MEMS). A relevant feature in these systems is the "pull-in" or "jump-to contact" instability, which arises when applied voltages are increased beyond a critical value. In this situation, there is no longer a steady state configuration of the device where mechanical members of the device remain separate. It may present a limitation on the stable operation regime, as with a micropump, or it may be used to create contact, as with a microvalve. The applied voltage appears in the equation as a parameter. We prove that this parameter controls the dynamics in the sense that before a critical value the solution evolves to a steady state configuration, while for larger values of the parameter, the "pullin" instability or "touchdown" appears. We estimate the touchdown time. In one dimension, we prove that the touchdown is self-similar and determine the asymptotic rate of touchdown. The same type of results are obtained in a disk. We also present numerical simulations in some two-dimensional domains which allow an estimate of the critical voltage and of the touchdown time. This information is relevant in the design of the devices.
In modelling electrostatically actuated micro-and nano-electromechanical systems, researchers have typically relied on a small-aspect ratio to form a leading-order theory. In doing so, small gradient terms are dropped. Although this approximation has been fruitful, its consequences have not been investigated. Here, this approximation is re-examined, and a new theory which includes often neglected small curvature terms is presented. Furthermore, the solution set of the new theory is explored for the unit disk domain and compared to the standard theory. Also, the analytical results are compared to experimental data.
Abstract.The mathematical modeling and analysis of electrostatically actuated micro-and nanoelectromechanical systems (MEMS and NEMS) has typically relied upon simplified electrostatic-field approximations to facilitate the analysis. Usually, the small aspect ratio of typical MEMS and NEMS devices is used to simplify Laplace's equation. Terms small in this aspect ratio are ignored. Unfortunately, such an approximation is not uniformly valid in the spatial variables. Here, this approximation is revisited and a uniformly valid asymptotic theory for a general "drum shaped" electrostatically actuated device is presented. The structure of the solution set for the standard non-uniformly valid theory is reviewed and new numerical results for several domain shapes presented. The effect of retaining typically ignored terms on the solution set of the standard theory is explored.
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