# Natural Frequencies and Modes of Electrostatically Actuated Curved Bell-Shaped Microplates
## 3 Results and Discussion
### Frequency Tuning
#### 3.3.2 Electrostatic Tuning  
In addition to the analysis of the frequency tuning by the mechanical load (Section 3.3.1), electrostatic frequency tuning was studied. The static FE analysis up to the first limit point (ST) was first carried out, followed by the eigenvalue analysis. The equilibrium curve of the electrostatically loaded plate is presented in Figure 9. In this case, similarly to the case of mechanical loading, good agreement between the ROM and FEM predictions is observed for both the equilibrium path (Figure 9) and the tuned fundamental modes (Figure 10).  
Figure 8: Spectral response of mechanically loaded curved plate with \(h_{0}/d=3.75\). The real parts of the eigenvalues associated with Equations (16) and (17) corresponding to the first five axisymmetric modes obtained by the seven DOF ROM and FEM analyses (\(\times\) markers) are shown. Black dashed lines correspond to the critical ST and R deflections.  
Figure 7: Center point non-dimensional deflection \(w_{s}(0)\) of the plate with \(h_{0}/d=3.75\) quasi-statically loaded by the non-dimensional mechanical force \(\beta_{M}\): seven DOF ROM (dashed red line) and FEM (solid blue line). The insets show the fundamental mode of vibrations around the corresponding equilibrium states marked by circles on the equilibrium curves. The modes were obtained by FEM, the dashed line in the insets corresponds to the zero deflection.  
The effect of the (time-independent) electrostatic force on the fundamental mode frequencies of the curved plates of various initial elevation-to-thickness ratios are presented in Figure 11 in their non-dimensional and dimensional forms. The latter clearly reveals that an increased curvature results in an increased stiffness, higher initial frequency, and in a wider frequencies tuning range. As the voltage increases, the stiffness is diminished in a somewhat similar way in all the bistable curved plates, due to the reduction of curvature and the accompanying increase of the in-plane stresses. The sensitivity to the electrostatic loading associated with the voltage change dramatically increases upon approaching the snap-through limit point.  
Figure 10: Spectral response of the electrostatically loaded curved plate with \(h_{0}/d=3.75\) as a function of the static non-dimensional deflection \(w_{S}\). The eigenvalues associated with Equations (16) and (17) corresponding to the first 5 axisymmetric modes obtained by the seven DOF ROM and FEM analysis (\(\times\) markers) are shown. Black dashed line corresponds to the critical ST deflection.  
Figure 9: Center point non-dimensional deflection of the plate with \(h_{0}/d=\)3.75 loaded by the quasi-static non-dimensional electrostatic force: seven DOF ROM (dashed red line) and FEM (solid blue line).  
To demonstrate the influence of the electrostatic force nonlinearity, in addition to the geometric mechanical nonlinearity of the plate, on the frequency tuning, the sensitivity of the electrostatically and mechanically actuated plates are compared in Figure 12. Tuning by forces below the critical ST value is examined. The results show that for the realistic initial distances between the electrode and the plate the influence of the electrostatic nonlinearity on the frequency tuning is minor. Consequently, the simpler analysis of the plate mechanically loaded by a uniform pressure can be used to predict frequency tuning also of the electrostatically actuated curved microplates.