Abstract
The recovery of the membrane profile of an electrostatic micro-electro-mechanical system (MEMS) is an important issue, because, when an external electrical voltage is applied, the membrane deforms with the risk of touching the upper plate of the device producing an unwanted electrostatic effect. Therefore, it is important to know whether the movement admits stable equilibrium configurations especially when the membrane is closed to the upper plate. In this framework, this work analyzes the behavior of a two-dimensional (2D) electrostatic circular membrane MEMS device subjected to an external voltage. Specifically, starting from a well-known 2D non-linear second-order differential model in which the electrostatic field in the device is proportional to the mean curvature of the membrane, the stability of the only possible equilibrium configuration is studied. Furthermore, when considering that the membrane is equipped with mechanical inertia and that it must not touch the upper plate of the device, a useful range of possible values has been obtained for the applied voltage. Finally, the paper concludes with some computations regarding the variation of potential energy, identifying some optimal control conditions.
Highlights
In the recent years, micro-electro-mechanical systems (MEMS) with circular membrane have come to be increasingly exploited in various fields, such as thermo-elasticity [1,2,3], microfluidics [4,5], electroelasticity [6,7,8,9,10,11], and, biomedical applications [12,13,14]
We focus our attention on the 2D electrostatic circular membrane MEMS device that was studied in [20], with the aim of studying the critical point and stability, determining the range of possible values of the external electrical voltage applied while taking both the mechanical inertia of the membrane and the risk that the membrane accidentally touches the upper disk into account, because, as highlighted in [24], it has only been dealt with in 1D geometry
It is worth noting that, from an energy point of view, the fact that the area of possible values is limited guarantees us that the membrane MEMS device under study is subject to variations in potential energy of limited amplitude reducing the risk of any damage
Summary
Micro-electro-mechanical systems (MEMS) with circular membrane have come to be increasingly exploited in various fields, such as thermo-elasticity [1,2,3], microfluidics [4,5], electroelasticity [6,7,8,9,10,11], and, biomedical applications [12,13,14]. The analytical model obtained is a non-linear ordinary second-order differential equation with radial symmetry, which, in dimensionless conditions, presents a singularity 1/r, where the independent variable is the profile of the membrane u(r) [20]. For this model, an algebraic condition of existence of the solution obtained by highlighting the fact that the uniqueness of the solution is not guaranteed has been proposed. The numerical recovering of the membrane profile occurs for small displacements, so that the risk of the membrane touching the upper disc does not exist, as observed in Remark 6
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