## 4 Metallic MEMS Cantilevers: Nonlinear Dynamics of the Pull-In Behavior
### Simulation of Dynamic Properties
Similar studies have been reported in Ref [43], focusing on transition to a chaotic behavior for specific voltages. Different authors use different wordings such as chaos, irregularities and instabilities [43; 44]. It seems that we can also confirm these findings in a voltage range between 22.199 and 22.2 V for the considered Al cantilever. However, this is not the scope of our paper. In Figure 12, the retardation is even more pronounced. For voltages between 22.196 and 22.199 V, the pull-in occurs much later after following another oscillation period. Most probably, the pull-in is initiated by a larger peak value at about 22 \(\upmu\)s compared to the case at 22.2 V, revealing an ongoing damped oscillation. Based on the temporal response of an Al MEMS cantilever with a longer length \(l=400\)\(\upmu\)m, Figure 13 shows the retardation effects in a more pronounced way. In this case, the pull-in voltages are smaller than in the previous figures, but when there is a small change in the voltage, the actuation time of cantilever shifts considerably. As the value of voltage changes from 1.4065 to 1.41 V, the actuation time difference is about 60 \(\upmu\)s.  
Next, for the Al MEMS cantilevers with the parameters \(E=70\) GPa, \(\rho=2700\) kg/m\({}^{3}\), \(h=1\)-2 \(\upmu\)m, and \(l=50\)-400 \(\upmu\)m, the influence of length \(l\) and thickness \(h\) on actuation time \(t_{\mathrm{a}}\) are shown in Figure 14. The actuation time \(t_{\mathrm{a}}\) decreases as \(l\) decreases and \(h\) increases. Among these three points A (\(h=1\)\(\upmu\)m, \(l=400\)\(\upmu\)m), B (\(h=1\)\(\upmu\)m, \(l=200\)\(\upmu\)m) and C (\(h=2\)\(\upmu\)m, \(l=400\)\(\upmu\)m), the point A has the highest actuation time \(t_{\mathrm{a}A}\). From point A to B, the length is halved and the actuation time decreases by 75%, and from point A to C, the thickness is doubled and the actuation time reduces by 44%. Thus, the length \(l\) has a greater impact on actuation time than the thickness \(h\). Figure 12: Additional temporal response curves of the same Al MEMS cantilever tip for different applied voltages. Figure 15 depicts the influence of variations in mass density \(\rho\) and [PERSON]'s modulus \(E\) on actuation time \(t_{\rm a}\) for the MEMS cantilevers with the parameters \(l=100\)\(\mu\)m, \(h=1\)\(\mu\)m, \(\rho=5400\)-10800 kg/m\({}^{3}\) and \(E=70\)-210 GPa. In three points D (\(\rho=10800\) kg/m\({}^{3}\), \(E=70\) GPa), E (\(\rho=5400\) kg/m\({}^{3}\), \(E=70\) GPa) and F (\(\rho=10800\) kg/m\({}^{3}\), \(E=140\) GPa) selected in Figure 15, the maximum actuation time \(t_{\rm aD}\) occurs in point D. From point D to E, the mass density is halved, and the actuation time decreases by 24%. From point D to F, the Young's modulus is doubled and the actuation time reduces by 21%, which means the impact on actuation time from density and Young's modulus are relatively close to each other. Combining Figures 14 and 15, as the parameters \(l\), \(h\), \(\rho\) and \(E\) vary regularly (doubled and halved), the reduction ratios of \(t_{\rm a}\) are 75%, 44%, 24% and 21%, respectively. The results show that the primary factors affecting actuation time are length, then thickness, and finally mass density and Young's modulus (in this sequence decreasing influence). This is because when the parameters \(h\) and \(E\) increase, while \(l\) and \(\rho\) decrease, the resonance frequency \(f_{1}\) increases and the actuation time \(t_{\rm a}\) is decreasing. Figure 14: Variation of the actuation time \(t_{\rm a}\) with length \(l\) and thickness \(h\) for Al MEMS cantilevers. Figure 13: Temporal response of a longer Al MEMS cantilever tip for different applied voltages; \(l=400\)\(\mu\)m, \(b=20\)\(\mu\)m and \(h=1\)\(\mu\)m.  
The results are partly surprising concerning increasing thickness, since doubling the thickness doubles the mass. The negative influence of growing mass (inertia) is however overcompensated by the counter-running positive effect of growing thickness coming from much higher resonance frequencies.