## 4 Metallic MEMS Cantilevers: Nonlinear Dynamics of the Pull-In Behavior  
The goal of our paper is to study the nonlinear dynamics of the pull-in effect of metallic MEMS cantilevers as a function of different cantilever geometries (thickness, length) and material parameters (mass density, Young's modulus). This section presents the influence of different parameters on the pull-in voltage and actuation speed of the MEMS cantilevers,  
Figure 3: Performance of the step response for a second-order system (completely redrawn using the idea of [37]). which are made of different metals. At the beginning of Section 4, a distinct separation of Sections 2 and 3 versus Section 4 is prepended. In Sections 2 and 3, an idealized parallel plate actuator is considered:  
* It is a 1D problem. * An idealized spiral spring is considered, revealing a linear stress-strain relation (Hooke's law, i.e., linear force-displacement relation). * The electrostatic force depends on the square of the displacement and the problem can be treated on the basis of analytical formula. In Section 4, a metallic MEMS cantilever is considered:  
* It is a more complex 3D problem. * The electrical field is varying in 3D, i.e., it is a 3D vector field. Close to the surfaces of the electrodes, the electrical field is perpendicular to the surfaces. The local electrostatic forces vary with \(x\), \(y\) and \(z\). * The electrical capacity is varying with the voltage in a nonlinear way. Thus, also the charging of the cantilever/base plate cannot be any longer treated as a simple \(RC\) element. The voltage built up at the MEMS cantilever is not any more simply exponential. * A non-homogenous charge distribution is existing, and it is varying dynamically. * The description of the elastic behavior is 3D and based on continuum mechanics. * Stray fields in the \(y\)-direction and \(x\)-direction are considered. * Nearly all relations are nonlinear. * The problem has to be treated by a self-consistent dynamic electro-mechanical numerical model. COMSOL Multiphysics is used here. First, a 3D representation of the considered MEMS cantilevers is shown in Figure 4. The metallic cantilever is characterized by the length \(l\), thickness \(h\), width \(b\), mass density \(\rho\) and Young's modulus \(E\). In Figure 5, a 2D cross-section of the cantilever taken in the center of the width \(b\) in the \(xz\)-plane is shown, including electrical field lines (blue) and electrostatic forces (red), schematically. The density of the field lines indicates the magnitude of the electrical field. The stray fields in the \(x\)-direction are shown at the right-hand side end of the cantilever. The electrostatic forces are always oriented in the direction of a tangent to the field lines. Thus, the force increases with growing \(x\) at the lower surface of the cantilever (upper electrode) and is oriented vertical to the surface, locally. As a consequence of the bending, the electrical capacitance of the system nonlinearly increases as the voltage increases. Figure 4: Simulation model of electrostatic MEMS cantilever. COMSOL Multiphysics(r) 5.6 is a simulation tool with Multiphysics field coupling including a MEMS module that can be used for the simulations on a microscopic scale. As the cantilever bends, the air gap geometry changes as a function of coordinates and leads to local changes in the electric field between the two electrodes, together with the varying charge distribution. The electrostatic field in the air gap is described by [PERSON]'s equation. The coupled physics is handled self-consistently by the Electromechanics Multiphysics modules. The electrostatic force density that acts on the electrode of the cantilever is simulated by COMSOL [38]:  
\[\mathbf{F_{es}}=-\frac{1}{2}(\mathbf{E\cdot D})\mathbf{n}+(\mathbf{n\cdot E})\mathbf{D} \tag{11}\]  
where \(\mathbf{E}\) and \(\mathbf{D}\) are the electric field and electric flux density vector fields, respectively, and \(\mathbf{n}\) is the outward normal vector of the surface.