# Natural Frequencies and Modes of Electrostatically Actuated Curved Bell-Shaped Microplates
## 3 Results and Discussion
### Eigenvalue Analysis  
The fundamental mode profile and its variation with the increasing initial elevation-to-thickness ratio is presented in Figure 4, which shows the normalized displacements \(v(r)\) and \(u_{d}(r)\) from the equilibrium state (rather than the full elevation \(w(r)\) and \(u(r)\) (see Figure 2). The results indicate that the fundamental mode profile is strongly affected by the plate's initial curvature, quantified by the center point initial elevation, \(h_{0}\). The fundamental out-of-plane mode-displacement profile is similar to that of the initially flat plate only for slightly curved plates with an elevation-to-thickness ratio smaller than 2.5. For the plates of a higher curvature, the fundamental mode shapes strongly deviate from that of a flat circular plate. FE maps of the first three axisymmetric modes for plates of various values of the elevation-to-thickness ratio \(h_{0}/d\) are depicted in Figure 5. In the examined examples, the third axisymmetric mode is less sensitive to the effect of the initial curvature.  
Figure 3: Convergence of the quasi-static response of a curved plate with \(h_{0}/d=3.75\). Relative (with respect to the corresponding FE simulation results) errors in the non-dimensional snap-through limit point location, \(w\), (blue line) and in the non-dimensional mechanical force, \(\beta_{M}\), (red line) are shown.  
The linear eigenvalue analysis based on the ROM is carried out by solving the eigenvalue problem (Equations (16) and (17)) with \(\beta=0\), zero static axial deflection \(p_{i}^{\mathrm{s}}=0\), and static elevation, which equals the initial as-fabricated plate profile, \(q_{1}^{\mathrm{s}}=h_{0}\). The latter results, in the form of the fundamental frequency, \(\omega\), versus the initial elevation-to-thickness ratio, are presented in Figure 6, together with the reference eigenfrequencies obtained by  
Figure 4: Cross-section profile of the fundamental vibrational mode for initially curved circular plates. The normalized, non-dimensional displacements from the equilibrium state are shown (**a**) out-of-plane \(v(r)\), (**b**) in-plane \(u_{d}(r)\)  
Figure 5: First three axisymmetric vibrational modes of the plate for different initial elevation-to-thickness ratios. The normalized, non-dimensional out-of-plane displacements \(v(r)\) from the equilibrium state normalized with respect to the maximum modal amplitude are shown.  
the FE analysis (purple \(\times\) markers). In accordance with the corresponding error analysis (shown in the inset), the three DOF ROM predict the frequency of the fundamental vibration mode with an error of \(~{}10\%\) for the plate's elevation-to-thickness ratio of up to \(h_{0}/d=5\). Higher elevation-to-thickness ratios require the use of at least five DOF to achieve an accuracy of less than \(10\%\), while errors of less than \(3\%\) are achieved with ten DOF. The results for the number of DOF between six and nine are not presented in the figure and fall between those obtained for the five and 10 DOF models. The higher required number of DOF for more curved plates can be attributed to the fact that, as the plate curvature increases, its fundamental mode deviates more significantly from that of a flat plate.