## 4 Metallic MEMS Cantilevers: Nonlinear Dynamics of the Pull-In Behavior
### Simulation of Pull-In Voltages and Displacements for Electrostatic Actuations  
First, stationary simulations are performed for increasing applied voltages. In the voltage-displacement relations, the increase in voltage is interrupted when the pull-in is reached (pull-in voltage \(V_{\text{pi}}\)). In Figure 7, the displacement-voltage curve for an Al MEMS cantilever with \(l\) = 100 \(\upmu\)m, \(h\) = 1 \(\upmu\)m, \(\rho\) = 2700 kg/m\({}^{3}\) and \(E\) = 70 GPa is shown. As above, the displacement \(w_{\text{z}}\) of the cantilever tip is considered in the \(z\)-direction. The simulation stops when the voltage approaches 24.41 V and some irregular variations occur (magnified curve area shown in the inset). However, finally, the cantilever free end (tip) reveals a sudden strong displacement through the entire gap, leading to a contact of the cantilever tip to the lower electrode (snap-down). Beyond the pull-in point A, the elastic counterforces are no longer able to balance the electrostatic attraction forces. Therefore, we define the pull-in voltage \(V_{\text{pi}}\) by the voltage value at the end of the curve still allowing force equilibria between attraction and repulsion before the abrupt change into the non-equilibrium condition occurs. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Length (\(l\)) & 50–400 \(\upmu\)m \\ Thickness (\(h\)) & 1–2 \(\upmu\)m \\ Width (\(b\)) & 20 \(\upmu\)m \\ Air gap distance (\(d\)) & 3 \(\upmu\)m \\ Young’s modulus (\(E\)) & 35–411 GPa \\ Mass density (\(\rho\)) & 2700–19,350 kg/m\({}^{3}\) \\ Poisson ratio (\(v\)) & 0.21–0.44 \\ Specific electrical resistivity (\(\rho_{\text{R}}\)) & 1.587–42 \(\times\) 10\({}^{-8}\)\(\Omega\cdot\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical and material parameters of the considered metallic MEMS cantilevers. Figure 6: Young’s modulus density chart for different metals (completely redrawn using ideas of Reference A [42]). In order to study the influences of several parameters on \(V_{\rm pi}\), four groups with varying \(l\), \(h\), \(\rho\) and \(E\) are considered. The pull-in voltages \(V_{\rm pi}\) and the corresponding pull-in displacements \(w_{\rm pi}\) of the four groups are shown in Table 2. Group 1 and 2 are considering Al MEMS cantilevers with \(h\) = 1 \(\upmu\)m and \(h\) = 2 \(\upmu\)m, respectively. Group 3 considers a halved value of \(E\) (35 MPa) compared to Al (left, vertical downward-pointing arrow in Figure 6). Group 4 considers a doubled value of \(\rho\) (5400 kg/m\({}^{3}\)) compared to Al (left point of the horizontal arrow in Figure 6). The results illustrate that the \(V_{\rm pi}\) values are significantly different, but the \(w_{\rm pi}\) values remain close to each other, which means \(l\), \(h\) and \(E\) have a huge influence on \(V_{\rm pi}\) but little impact on \(w_{\rm pi}\). In addition, comparing the results of Group 1 and 4, \(\rho\) has no significant effect on \(V_{\rm pi}\). The mean values of \(w_{pi}\) of three groups are 1.34 \(\upmu\)m, 1.36 \(\upmu\)m, 1.34 \(\upmu\)m and 1.30 \(\upmu\)m, which translate to 44.7%, 45.3%, 44.7% and 43.3% of the \(d\), respectively. It indicates that there is only a very weak relationship between the \(w_{pi}\) and these parameters of the cantilever. Based on the data from these groups of Table 2, further displacement curves of the free ends as a function of voltage are shown in Figure 8 in order to illustrate the influence of different parameters on the \(V_{\rm pi}\). The Al MEMS cantilever with parameters \(l\) = 100 \(\upmu\)m, \(h\) = 1 \(\upmu\)m, and \(E\) = 70 GPa (case 1 in blue color) is set to a reference. Within each group, we have different cases. Based on the structure of case 1, case 2 with doubled \(h\) = 2 \(\upmu\)m,