# Natural Frequencies and Modes of Electrostatically Actuated Curved Bell-Shaped Microplates
## 3 Results and Discussion
### Curved Plate Vibrational Modes as Base Functions  
In the previous sections, it was found that the curved plate vibrational modes are significantly different from those of a flat plate. As a result, more than five DOF were needed in order to obtain an accuracy of less than 10% of the ROM analysis, employing the vibrational modes of initially flat plate as base functions. Here, we show that the efficiency of the ROM can be improved by utilizing the vibrational modes of curved plates as base  
Figure 11: Sensitivity of the fundamental frequency to the tuning voltage for plates of various initial elevation-to-thickness ratios. The plate radius is \(700\,\upmu\)m, the thickness is \(0.4\,\upmu\)m, and the gap is \(5\,\upmu\)m. (**a**) Non-dimensional graph, (**b**) dimensional graph.  
Figure 12: Sensitivity of the fundamental mode non-dimensional frequency to the tuning electrostatic (solid red line) and mechanical (dashed blue line) loads for the plate with initial elevation-to-thickness of 3.75. The plate radius is \(700\,\upmu\)m, thickness is \(0.4\,\upmu\)m, and the gap is \(5\,\upmu\)m; \(\beta_{ST}\) denotes the non-dimensional critical ST values of the electrostatic or mechanical loading parameter, \(\omega_{0}\) is the unloaded plate non-dimensional frequency. The relative difference between the normalized frequencies is presented in the inset.  
functions. As a primary demonstration of this option, we obtain the fundamental mode frequency of the unloaded plate using the single DOF RO model. To this end, the free vibrations (Equations (16) and (17)) are modified as follows. The static in-plane deflection is set to zero, and out-of-plane deflection is taken to be equal the initial elevation of the unloaded plate, i.e., \(w_{s}=w_{0}\). Recall that in Equations (16) and (17), both the initial elevation, Equation (12), and the dynamic out-of-plane displacements were represented using the out-of-plane modes \(\Phi_{n}\) of a flat plate. Since now the initial shape and the deflection are represented by different functions, to avoid confusion, \(\Phi_{1}(r)\) in the notation for the initial shape was replaced by \(\Psi_{1}(r)\), such that \(w_{0}(r)\approx h_{0}\Psi_{1}(r)\).  
\[\alpha^{2}(1-v)q_{1}h_{0}\int_{0}^{1}\frac{d\[PERSON]_{1}}{dr}\frac{d \[PERSON]_{1}}{dr}r^{2}H_{1}rdr+\alpha^{2}q_{1}h_{0}\int_{0}^{1}\frac{d\[PERSON]_{1}}{dr} \frac{d^{2}\[PERSON]_{1}}{dr^{2}}r^{3}H_{1}rdr+\\ \alpha^{2}q_{1}h_{0}\int_{0}^{1}\left(\frac{d^{2}\[PERSON]_{1}}{dr^{2 }}\frac{d\[PERSON]_{1}}{dr}\right)r^{3}H_{1}rdr+p_{1}\int_{0}^{1}\left(\frac{d^{2} H_{1}}{dr^{2}}r^{3}-H_{1}r+\frac{dH_{1}}{dr}r^{2}\right)H_{1}rdr=0 \tag{18}\]  
\[-\omega_{1}^{2}q_{1}\int_{0}^{1}\[PERSON]_{1}\[PERSON]_{1}rdr+q_{1}\int_{0}^{1}\left( \frac{d^{4}(\[PERSON]_{1})}{dr^{4}}+\frac{2}{r}\frac{d^{3}(\[PERSON]_{1})}{dr^{3}}-\frac {1}{r^{2}}\frac{d^{2}(\[PERSON]_{1})}{dr^{2}}+\frac{1}{r^{3}}\frac{d\[PERSON]_{1}}{dr} \right)\[PERSON]_{1}rdr-\\ \gamma\alpha^{2}h_{0}^{2}q_{1}\int_{0}^{1}\frac{1}{r}\frac{d}{dr }\left(r\left(\frac{d\[PERSON]_{1}}{dr}\right)^{2}\frac{d\[PERSON]_{1}}{dr}\right)\[PERSON]_{ 1}rdr-\gamma h_{0}p_{1}\int_{0}^{1}\frac{1}{r}\frac{d}{dr}\left(\frac{d\[PERSON]_{ 1}}{dr}\left(r\frac{dH_{1}}{dr}+vH_{1}\right)\right)\[PERSON]_{1}rdr=0 \tag{19}\]  
The mode shapes \(\Phi_{1}\) and \(H_{1}\) are obtained by fitting the FE solution. Ten Bessel and modified Bessel functions are used as the fitting functions. As expected, the solution of Equations (18) and (19) for each elevation are in good agreement with the FE fundamental mode frequency. The discrepancy between the two increases with the elevation due to lower fit accuracy mainly in the center and at the edge. The error is less than 2%, even at a large elevation of \(h0/d=30\), (Figure 13).