# Natural Frequencies and Modes of Electrostatically Actuated Curved Bell-Shaped Microplates
## 2 Materials and Methods  
We consider an initially curved bell-shaped circular plate clamped at its outer circumference (Figure 1). The plate is assumed to be made of homogeneous isotropic linearly elastic material, with Young modulus \(E\) and Poisson ratio \(\
u\). The thickness \(\hat{d}\) of the plate is assumed to be small compared to the plate radius \(R\). The initial shape of the plate is described by the function \(\bar{w}_{0}(\hat{r})=\hat{h}_{0}z_{0}(\hat{r})\), where \(\hat{h}_{0}\) is the elevation of the plate central point above its clamped edge and \(z_{0}(\hat{r},\theta)\) is a non-dimensional function such that \(\max\limits_{\hat{r}\in[R]}(z_{0}(\hat{r}))=1\). Since the device is assumed to be fabricated by the self-molding stamping approach [41], the plate is assumed to be bell-shaped and stress-free in the initial as-fabricated configuration. The plate is actuated by a transverse-distributed electrostatic force (electrostatic pressure) generated by a gap-closing flat electrode spanning the entire plate area and located at a distance \(\delta_{0}\) from the plate outer boundary. The plate is considered in the framework of the Kirchoff hypothesis combined with the nonlinear Foppl [PERSON] shallow plate theory; the in-plane inertia is neglected. The axisymmetric motion of the plate is described by the following system of coupled non-dimensional nonlinear partial differential equations [45], completed by the clamped boundary conditions  
\[\frac{\alpha^{2}}{2}(1-\
u)\Bigg{(}\bigg{(}\frac{dw}{dr}\bigg{)}^{ 2}-\bigg{(}\frac{dw_{0}}{dr}\bigg{)}^{2}\Bigg{)}r\\ +\bigg{(}\alpha^{2}\bigg{(}\frac{dw}{dr}\frac{d^{2}w}{dr^{2}}- \frac{dw_{0}}{dr}\frac{d^{2}w_{0}}{dr^{2}}\bigg{)}+\frac{d^{2}u}{dr^{2}}\bigg{)} r^{2}-u+\frac{du}{dr}r=0 \tag{1}\]  
\[\frac{d^{2}w}{dt^{2}}+\bigg{(}\frac{1}{r}\frac{d}{dr}\bigg{(}r\frac {d}{dr}\bigg{)}\bigg{)}\bigg{(}\frac{1}{r}\frac{d}{dr}\bigg{(}r\frac{d(w-w_{0 })}{dr}\bigg{)}\bigg{)}=\\ \frac{1}{r}\frac{d}{dr}\bigg{(}r\Bigg{(}\gamma\bigg{(}\frac{du}{ dr}+\frac{\alpha^{2}}{2}\bigg{(}\bigg{(}\frac{dw}{dr}\bigg{)}^{2}-\bigg{(}\frac{dw_{0 }}{dr}\bigg{)}^{2}\bigg{)}+\
u\frac{u}{r}\bigg{)}\bigg{)}\frac{dw}{dr}\bigg{)} -f \tag{2}\]  
Here \(u(r,t)\) is the non-dimensional radial displacement and \(w(r,t)\) is the non-dimensional location (z coordinate) of the plate above its outer clamped boundary. The non-dimensional electrostatic force is described using the simplest parallel capacitor formula such that \(f=f_{e}=\beta/(1+w)^{2}\), where \(\beta\) is the voltage parameter. In the case of actuation by a prescribed mechanical, configuration-independent, force, \(f\) is replaced by the mechanical load parameter \(\beta_{M}\). The non-dimensional parameters used in the development are defined in (Table 1), where \(D=E\vec{d}^{3}/12(1-\
u^{2})\) is the bending stiffness, \(\epsilon_{0}=8.85410^{-12}\mathrm{F/m}\) is the permittivity of vacuum, and \(V\) is the voltage difference between the plate and the electrode. To investigate the small-amplitude undamped vibrations of the plate around an equilibrium corresponding to a specific transverse load, the radial displacement \(u\) and the elevation \(w\) are decomposed into the static and the dynamic parts  
\[u(r,t)=u_{\mathrm{s}}(r)+u_{d}(r,t),\qquad w(r,t)=w_{\mathrm{s}}(r)+\
u(r,t) \tag{3}\]  
The static part of the response is governed by the nonlinear equilibrium equations, namely Equation (1) and the static counterpart of Equation (2), with \(u\) and \(w\) replaced by \(u_{\mathrm{s}}\) and \(w_{\mathrm{s}}\), respectively,  
Figure 1: Model of an initially curved bell-shaped clamped circular plate. (**a**) Geometry and the coordinate system \(\hat{r},\theta,z\). (**b**) Model of the plate electrostatically actuated by a gap-closing electrode. Initial configuration is depicted in solid black line; the deformed state is shown in the dashed gray line. Positive directions of the lateral deflection and of the distributed force are shown.