# Natural Frequencies and Modes of Electrostatically Actuated Curved Bell-Shaped Microplates
## 2 Materials and Methods
\[\frac{\alpha^{2}}{2}(1-\
u)\Bigg{(}\bigg{(}\frac{dw_{s}}{dr}\bigg{)}^{2} -\bigg{(}\frac{dw_{0}}{dr}\bigg{)}^{2}\Bigg{)}r^{2}\\ +\bigg{(}\alpha^{2}\bigg{(}\frac{dw_{s}}{dr}\frac{d^{2}w_{s}}{dr^{2 }}-\frac{dw_{0}}{dr}\frac{d^{2}w_{0}}{dr^{2}}\bigg{)}+\frac{d^{2}u_{s}}{dr^{2}} \bigg{)}r^{3}-u_{s}r+\frac{du_{s}}{dr}r^{2}=0 \tag{4}\]  
\[\frac{d^{4}(w_{s}-w_{0})}{dr^{4}} +\frac{2}{r}\frac{d^{3}(w_{s}-w_{0})}{dr^{3}}-\frac{1}{r^{2}}\frac {d^{2}(w_{s}-w_{0})}{dr^{2}}+\frac{1}{r^{3}}\frac{d(w_{s}-w_{0})}{dr}\] \[-\frac{1}{r}\frac{d}{dr}\bigg{(}r\gamma\bigg{(}\frac{du_{s}}{dr}+ \frac{\alpha^{2}}{2}\bigg{(}\bigg{(}\frac{dw_{s}}{dr}\bigg{)}^{2}-\bigg{(} \frac{dw_{0}}{dr}\bigg{)}^{2}\bigg{)}+\
u\frac{u_{s}}{r}\bigg{)}\frac{dw_{s}}{ dr}\bigg{)}+f=0 \tag{5}\]  
and with the force parameter \(f\) suitably adapted for the specific loading case, either mechanical or electrostatic. The notations, \(u_{d}(r,t),\ v(r,t)\) are, respectively, the time-dependent (dynamic) radial and transverse displacements with respect to the equilibrium (Figure 2). The equations governing the small free vibrations around the forced static equilibrium are obtained by linearizing Equations (1) and (2) around the equilibrium configuration associated with the deflections \(w_{s}\), \(u_{s}\)  
\[\alpha^{2}(1-\
u)\frac{dw_{s}}{dr}\frac{dv}{dr}r^{2}+\bigg{(}\alpha^{2}\bigg{(} \frac{dw_{s}}{dr}\frac{d^{2}v}{dr^{2}}+\frac{dv}{dr}\frac{d^{2}w_{s}}{dr^{2}} \bigg{)}+\frac{d^{2}u_{d}}{dr^{2}}\bigg{)}r^{3}-u_{d}r+\frac{du_{d}}{dr}r^{2}=0 \tag{6}\]  
\[\frac{d^{2}v}{dt^{2}}+\frac{d^{4}v}{dr^{4}}+\frac{2}{r}\frac{d^{3}v}{dr^{3}} -\frac{1}{r^{2}}\frac{d^{2}v}{dr^{2}}+\frac{1}{r^{3}}\frac{dv}{dr}-\frac{1}{ r}\frac{d}{dr}\bigg{(}r\gamma\bigg{(}\frac{du_{d}}{dr}+\alpha^{2}\bigg{(} \frac{dw_{s}}{dr}\frac{dv}{dr}\bigg{)}+\
u\frac{u_{d}}{r}\bigg{)}\frac{dw_{s}} {dr}\bigg{)}\\ -\frac{1}{r}\frac{d}{dr}\bigg{(}r\gamma\bigg{(}\frac{du_{s}}{dr} +\frac{\alpha^{2}}{2}\bigg{(}\bigg{(}\frac{dw_{s}}{dr}\bigg{)}^{2}-\bigg{(} \frac{dw_{0}}{dr}\bigg{)}^{2}\bigg{)}+\
u\frac{u_{s}}{r}\bigg{)}\frac{dv}{dr} \bigg{)}-f_{d}=0 \tag{7}\]  
Here, \(f_{d}=0\) for the case of the mechanical loading and \(f_{d}=2\beta v/(1+w_{s})^{3}\) in the case of the electrostatic loading. Equations (6) and (7) were used for the calculations of the natural modes and frequencies of the plate around the deformed equilibrium. \begin{table}
\begin{tabular}{l l} \hline \(r\stackrel{{\Delta}}{{=}}r/R\) & Radial coordinate \\ \(u\stackrel{{\Delta}}{{=}}\dot{u}/R\) & Radial displacement \\ \(w\stackrel{{\Delta}}{{=}}\dot{w}/g_{0},\ w_{0}=\dot{w}_{0}/g_{0}\) & Plate elevation, initial elevation \\ \(h_{0}\stackrel{{\Delta}}{{=}}\dot{h}_{0}/g_{0}\) & Initial elevation at the center \\ \(d\stackrel{{\Delta}}{{=}}\dot{d}/g_{0}\) & Thickness \\ \(\alpha\stackrel{{\Delta}}{{=}}g_{0}/R\) & Nonlinearity parameter \\ \(\gamma\stackrel{{\Delta}}{{=}}\Big{(}R^{2}\dot{E}\dot{d}\Big{)}/ \big{(}D\big{(}1-\
u^{2}\big{)}\big{)}\) & Membrane load parameter \\ \(\beta=e_{0}V^{2}R^{4}/(2\s_{0}{}^{3}D)\) & Voltage parameter \\ \(\beta_{M}=qR^{4}/g_{0}D\) & Mechanical load parameter \\ \(t=\dot{t}\sqrt{D/\Big{(}\rho\dot{d}R^{4}\Big{)}}\) & Time \\ \hline \end{tabular}
\end{table} 
Table 1: Non-dimensional quantities used in the formulation.