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
3. Pseudo-Spectral Method
The governing equations (7) may be numerically solved through a pseudo-spectral approach [29, 33]. Accordingly, a -degree polynomial function is adopted to approximate the function on the interval . Since a collocation method is adopted, the polynomial will be uniquely determined enforcing (7) at some predetermined points (nodes). This procedure results in a system of nonlinear algebraic equations, which may be solved through standard methods, such as the iterative Newton method. The Jacobian of the system may be supplemented in closed form to the numerical equation solver.
Let be the -degree polynomial approximation of . The collocation set is defined through the first Gauss-Lobatto points where stands for the th degree Legendre polynomial. Equation (7), evaluated at the interior nodes, yields the system of algebraic equationswhich is then supplemented with the boundary conditions (8)The problem is now rewritten in matrix form. To this aim, let the unknown column vector, the vector of the square, and the vector of the th derivativeMaking use of the derivative matrix (see [33, Chap.7]) we haveand (16) and (17) may be rewritten through the derivative matrix aswhere is the identity matrix. Here, is supplemented with the BCs (8); that is, the first row is set to zero apart from the first entry that is set to; the second, the last-but-one and the last rows are replaced by the first and the last rows of , , and , respectively. Then we haveAt the RHS of system (20) we have the zero vector supplemented with the BCs, namely,
4. Numerical Solution
In this section, the pseudo-spectral approximation is compared with a Finite Element solution, both methods employing the same number of degrees of freedom, which corresponds to for the order of the interpolating polynomial. For the FE solution, we introducenodes, each endowed withdegrees of freedom, that is, one translational and the other rotational. In Figure 2, the system of algebraic equations (20) is solved for different values of the driving parameter (voltage difference) with the FE method. It clearly appears that, for , the cantilever dimensionless tip deflection is very close to ; that is, the cantilever is at the verge of pull-in instability. Figure 3 shows the product , which should be constant along the cantilever and equal to , for the FE approximation. It is evident that the quality of the numerical solution rapidly deteriorates near the cantilever tip, which is exactly where best accuracy is demanded to effectively capture the pull-in threshold. Conversely, Figure 4 plots the product for the pseudo-spectral approximation. The comparison between the two plots is a striking example of the effectiveness of this method in this kind of nonlinear problems. Figure 5 plots the slope, bending moment and shearing force along the cantilever beam, and it illustrates that BCs are well captured by the numerical solution, either FE or pseudo-spectral. The plot is obtained by successive differentiation of the displacement field and it is readily available for the spectral method, where polynomial functions are employed. Conversely, obtaining the corresponding curves for the FE approximation needs some extra care for curve fitting of the nodal displacement is first applied, which is then successively differentiated. Figure 6 plots the applied line load density for the FE solution near pull-in, which corresponds to (the negative of) the electrostatic Coulomb force . It is remarkable that the electrostatic force appears highly oscillatory (note that curve-fitting is employed to get a continuous plot), and it attains unphysical negative values. As a comparison, Figure 7 describes the same behavior for , that is well below the threshold value for pull-in. It appears that the electrostatic force is rather poorly approximated by the method, even far from instability. On the contrary, Figures 8 and 9 illustrate the same graphs as obtained from the pseudo-spectral method. The smoothness of the solution is remarkable. The reason for the superior performance of the pseudo-spectral approximation seems to lie in the fact that it is more robust in dealing with little deviations of the cantilever tip displacement on the verge of contact. Indeed, it is well known that pseudo-spectral approximation guarantees high precision and exponential convergence (under suitable assumptions, see [29]), and this feature proves important in smoothly approximating the highly nonlinear behavior of the electrostatic force.
5. Conclusions
In this paper, the pseudo-spectral method is adopted to numerically solve the problem of pull-in instability in a cantilever beam. The beam constitutes one armor of a capacitor, the other armor being represented by a grounded flat surface. Although only Coulomb electrostatic force is considered, the method is easily extended to deal with the fringe effect and the Casimir force. Results may be especially relevant for cantilever micro-switches in MEMS. It is shown that the pseudo-spectral method compares very favorably with an equivalent Finite Element approximation, equivalency being constituted by an equal number of degrees of freedom in the methods. In particular, good approximation for the cantilever deflection near its tip is crucial to capturing the threshold voltage on the verge of pull-in. Indeed, poor approximation leads to very unphysical oscillatory attraction/repulsion forces along the cantilever. It is further shown that the oscillatory behavior is not restricted to the near pull-in regime. Finally, it is emphasized that both methods exactly satisfy the boundary conditions (BCs). It is remarked that the present analysis can be extended to incorporate functionally graded cantilevers [34–36] and beam-plates [37, 38] or to include viscoelastic effects [39–42].
Competing Interests
The authors declare that they have no competing interests.