# Natural Frequencies and Modes of Electrostatically Actuated Curved Bell-Shaped Microplates
## 4 Conclusions  
In this work, we analyze the free vibrations of initially curved bistable bell-shaped circular microplates around the equilibrium states. The initial bell-shaped profile of the plate is chosen since this is the shape that comes out as a result of the mold-less stamping fabrication method suggested in study [41]. The plates are loaded either by uniform  
Figure 13: Convergence of single DOF ROM built using the numerically obtained fundamental mode non-dimensional frequency of the curved plate. (**a**) The reference values (\(\times\) markers) are the non-dimensional frequencies obtained by the FE model, and the RO results for one DOF (\(\circ\) markers). (**b**) Errors with respect to the reference FE results are shown. pressure or by nonlinear deflection-dependent electrostatic force. The multi-DOF reduced-order model of the structure, which is described in the framework of the [PERSON] shallow plate theory, is built and verified using the FE analysis. The RO model is then used to describe the static responses of the plates to the quasi-static loading and then to obtain the frequencies and modes of the free vibrations of the mechanically and electrostatically actuated plates around the equilibrium configurations. Several aspects of the work's contribution can be mentioned. First, we found that the modal shapes of even slightly curved bell-shaped plates differ significantly from that of the initially flat plate. Specifically, the fundamental mode of the curved plate has an annular rather than bell-shaped profile and resembles one of the higher axisymmetric modes of the flat plate. We attribute this result to the fact that the Gaussian curvature of the (axisymmetric) bell-shaped plate changes its sign along the radius of the plate. In the vicinity of the zero Gaussian curvature circular line (the inflection point of the plate initial shape profile), Figure 2 shows that the stiffness of the shell is reduced, which may result in higher modal deflections [48]. We also show that the readily available vibrational modes of the initially flat plates can be reliably used as the base functions for the analysis of shallow-curved plates by means of the ROM. In this context, knowledge of the vibrational mode shapes for the curved plate could be useful for the coarse estimation of the required number of DOF in the RO models exploiting the modes of the flat plate. For example, the fact that the fundamental mode of the slightly curved plate looks similar to the third mode of the flat plate is a clear indication that the single DOF ROM incorporating flat plate modes as the base functions will be not sufficient, and that at least three DOF models should be used. The above observations suggest that the utilization of the curved plate vibrational modes as base functions in the ROM may be beneficial. As a preliminary demonstration of this, the fundamental frequencies of curved plates were recalculated by a single DOF ROM incorporating a base function, which was constructed as a fit to the previously obtained first vibrational mode of the curved plate. One of the important and useful features of the curved plates considered in the present work is their wide-range frequency tunability. This is due to the fact that the natural frequencies are influenced by the equilibrium configurations and, therefore, by loading. This is a very beneficial feature, which can be used in various resonant sensors incorporating a vibrating plate as the sensing element [49; 50]. In these devices, uncertainty in the geometric parameters and material properties often makes the achievement of the required frequency by fabrication to be challenging. Therefore an ability to tune the frequency in the fabricated device is of primary importance. The fact that the fundamental modal amplitudes are higher in an annular region rather than in the center of the plate opens opportunities for more efficient bio- and chemical-mass and gas sensing by locating the functionalized binding sites within the regions of maximal amplitudes.