## 4 Metallic MEMS Cantilevers: Nonlinear Dynamics of the Pull-In Behavior
### Simulation of Dynamic Properties  
The required time to reach a certain final situation after the electrostatic actuation of the MEMS cantilevers includes two aspects: the temporal charging of the electrodes \(t_{\mathrm{c}}\) and the mechanical response time \(t_{mr}\). A simple calculation of the electrical charging time of an Al cantilever with l = 100 \(\upmu\)m, b = 20 \(\upmu\)m and h = 1 \(\upmu\)m which was kept static in a first step (capacitance kept constant) reveals charging times in the fs range. In the next step, the cantilever was slowly actuated, which is leading to a variation of the capacitance. This actuation was performed via a quasi-static variation of the voltages. A corresponding rise in capacitance from 8.6 to only 10.5 fF was observed (Figure 10a). In the third step of fast actuation, the charging time is found to increase but still remains in the fs range. That means that \(t_{\mathrm{c}}\) can be neglected, since it is very short compared to \(t_{mr}\). Figure 9: The eigenfunctions of the lowest four modes having the lowest eigenfrequencies: (**a**) 1 st order mode, (**b**) 2 nd order mode, (**c**) 3 rd order mode, (**d**) 4 th order mode. Next, a moving cantilever under a step function of voltage 22.2 V is considered, and the temporal variation of capacitance is simulated by COMSOL and found to be nonlinear (Figure 10b). Due to the temporal bending of the cantilever, the capacitance is varying with the voltage in a nonlinear way. The required time to reach a certain final situation is denoted \(t_{\text{a}}\) (actuation time). In our case, the final situation is specified that the tip is reaching the lower electrode (snap-down and pull-in is finished). Next, the dynamic behavior, i.e., the temporal response curves of this Al MEMS cantilever after applying a step-function at \(t=0\), is simulated. For voltages steps ranging from 22 to 23 V, the results are shown in Figures 11 and 12. The slopes of the curves represent the movement speed of the cantilever tips. Finally, when the voltage is less than or equal to 22.195 V, the system reveals a damped oscillation tending to a stable state. However, when the voltage is higher than 22.196 V, the free-end will snap down and will be attached to the lower electrode. Here, the value of dynamic pull-in voltage is between about 22.196 and 23.0 V and can be represented by \(V_{\text{pi}}\)', which is smaller than the static pull-in voltage \(V_{\text{pi}}\) with a value of 24.41 V (shown in Table 2). The reason is that when the displacement \(w_{\text{z}}\) is inside the overshoot (see Figure 3) of the step response, the voltage required to generate the same electrostatic force decreases. Figure 11: Temporal response of the Al MEMS cantilever tip for different applied voltages; \(l=100\) μm, \(b=20\) μm and \(h=1\) μm. Figure 10: (**a**) Capacitance as a function of actuation voltage, revealing nonlinear rising. (**b**) Capacitance as a function of time, also revealing nonlinear rising. A voltage step function is considered, with a step from zero to 22.2 V occurring at \(t=0\). The simulation includes temporally rising voltage as well as the bending dynamics of the cantilever. The inset of Figure 11 indicates that when the applied voltage is at 22.2 V, the movements show retardation effects in the pull-in process. At the beginning, the free-end acts fast, a bit later, the moving speed slows down, and finally, the pull-in is completed by a rapid motion. In the meantime, as the voltage increases, the time required for the pull-in to occur becomes shorter, and the retardation effects become weaker. This is because the cantilevers always tend to oscillate, and as the \(w_{\mathrm{z}}\) approaches the peak value of oscillation, the velocity of the free-end decreases, while the \(w_{\mathrm{z}}\) still raises, causing a continuous increase in the \(F_{\mathrm{e}}\). When the cantilever tip passes the critical point, the increasing speed of \(F_{\mathrm{e}}\) is faster than for \(F_{\mathrm{elast}}\), so the pull-in action completes quickly.