Modelling and Simulation in Engineering

Volume 2016 (2016), Article ID 8543616, 6 pages

http://dx.doi.org/10.1155/2016/8543616

## Numerical Investigation of Pull-In Instability in a Micro-Switch MEMS Device through the Pseudo-Spectral Method

Dipartimento di Scienze e Metodi dell’Ingegneria (DISMI), Universitá degli Studi di Modena e Reggio Emilia, Via G. Amendola 2, 42122 Reggio Emilia, Italy

Received 4 December 2015; Revised 10 May 2016; Accepted 5 October 2016

Academic Editor: Julius Kaplunov

Copyright © 2016 P. Di Maida and G. Bianchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A pseudo-spectral approximation is presented to solve the problem of pull-in instability in a cantilever micro-switch. As well known, pull-in instability arises when the acting force reaches a critical threshold beyond which equilibrium is no longer possible. In particular, Coulomb electrostatic force is considered, although the method can be easily generalized to account for fringe as well as Casimir effects. A numerical comparison is presented between a pseudo-spectral and a Finite Element (FE) approximation of the problem, both methods employing the same number of degrees of freedom. It is shown that the pseudo-spectral method appears more effective in accurately approximating the behavior of the cantilever near its tip. This fact is crucial to capturing the threshold voltage on the verge of pull-in. Conversely, the FE approximation presents rapid successions of attracting/repulsing regions along the cantilever, which are not restricted to the near pull-in regime.

#### 1. Introduction

Micro-Electro-Mechanical Systems (MEMS) form a rather diverse and inhomogeneous group of micro-devices aimed at sensing and actuating in a wide array of fields, ranging from mechanical or electronic engineering to chemistry or biology, from micro-mechanics to micro-machining [1–5]. The manufacturing technology is the common standground for such devices, which heavily relies on the different lithographic techniques borrowed from the technology of micro-electronics. Indeed, MEMS devices are mostly obtained from a silicon substrate. It is observed that MEMS are really “systems” in the sense that they are often made up of several functional parts joint together in the device (like piezo- and magneto-sensors [6]). Among MEMS, micro-switch forms a distinct set with great application potential, with special regard to phase shifters and Radio Frequency MEMS (RFMEMS). They are usually gathered in two groups, namely, capacitor and metal-air-metal switches. Besides, they are further divided according to the actuation method: electrostatic, electrothermal, magnetostatic, and piezoelectric among the most common. A study of magnetoelastic actuated micro-switch is given in [7, 8] for the low-frequency asymptotic analysis of energy scavengers. In this paper, we focus attention on the pull-in instability of a capacitor micro-switch actuated by electrostatic Coulomb force. This particular application has received extensive attention in the literature, owing to the importance of pull-in induced failures in applications. A recent review on the subject can be found in [9]. A theoretical analysis of this problem within the static regime is provided in [10] and references therein. Pull-in voltage in cantilever MEMS have been considered in [11–14]. Failure mechanisms of MEMS include cracking [15–18], peeling of the cantilever [19–21], stiction to the substrate [22–24], and temperature [25, 26]. Besides, micropolar theories are often preferred when dealing with micro- and nanodevices to incorporate the scale effect [27, 28]. Spotlight is set on a pseudo-spectral approximation of the problem, which is compared with a Finite Element (FE) solution. Spectral methods belong to the family of Galerkin’s (or Ritz’s) methods [29]. Spectral methods are often divided into two groups, namely,* pseudo-spectral* or* interpolating*. The former group enforces the fulfillment of the differential operator at a set of points termed nodes (this is sometimes also named* orthogonal collocation*). For the latter group, wherein the Galerkin’s method is properly placed, the expansion coefficients are obtained projecting the solution onto the basis set [30].

The paper is structured as follows: Section 2 sets forth the governing equations and the boundary condition for a cantilever. Section 3 introduces the pseudo-spectral method. A numerical comparison with the FE method is illustrated in Section 4. Finally, conclusions are drawn in Section 5.

#### 2. Governing Equations

Let us consider a micro-cantilever switch device subjected to electrostatic attractive force (Figure 1). The micro-cantilever acts as one armor of a capacitor under the electric potential difference . Let denote the distance between the capacitor armors. We consider a plane problem and introduce the transverse displacement for the cantilever. Let us introduce the dimensionless variablesThen the governing equation for the cantilever readswhereandare the Casimir and the electrostatic line-load, beingHere, stands for the electric potential difference acting between the capacitor armors (in the SI this is expressed in volt, i.e., where N stands for newton, m for meter, and C for electric charge, expressed in Coulomb),is the armors width, is the electric permittivity (in vacuum), andis generally a function ofwhich takes into account the fringe effect. For the sake of illustrating the method, we neglect the Casimir force contribution and assume independent of. Then, we can rewrite the governing equation (2) aswhere prime denotes differentiation with respect to and the following driving parameter is obtained:Under the attractive electrostatic force, it is and the boundary conditions (BCs) for the cantilever readLet us define ; thus (4) further reduces towith and the BCsIt is observed that, integrating and making use of the last BC, it may be deduced that which shows that the shearing force is generally positive and it is zero only at . The same argument may be applied to infer that is generally negative, apart from the point where it is zero, and that is generally negative, although it vanishes at . Consequently, is a* monotonic decreasing* function of and . The nonlinear fourth order ODE (7) may be integrated once [31, §4.2.1] to giveIt is observed that, in the case , (10) falls into the Emdem-Fowler class of nonlinear ODEs, which, in special cases, may admit closed form solutions [31, 32]. Evaluation atand making use of the BCs (8) givewhich shows that the situation is not relevant in this problem. Besides, it followsand evaluation at lends Consideration of the sign for and yields the inequality