# Natural Frequencies and Modes of Electrostatically Actuated Curved Bell-Shaped Microplates
## 1 Introduction
The frequency tunability of the flat or imperfect plates operated by gap-closing electrodes was studied as well [28]. The natural frequencies and mode shapes of initially flat circular and rectangular micro-plates around electrostatically induced deformed equilibrium positions were studied in [23; 26; 29]. The influence of physical stimuli such as temperature, pressure, and various phenomena typical to systems of micro- and nanoscales, including Casimir and van der Waals force, along with the size effect, were also investigated [30; 31; 32]. The static and dynamic behavior of electrostatically actuated nanoscale structures made of 2D materials, such as graphene or molybdenum disulfide, attracted significant attention from researchers during the last decade [33]. In all these works, the plates were initially flat or of a small, caused by an initial imperfection, curvature, which is not sufficient for bistability. It is not unfounded to expect that electrostatically actuated bistable microplates and shallow microshells may exhibit advantages similar to those of curved beams. However, the works devoted to these structures are sparse. These include the investigations of the static and dynamic behaviors of slightly curved and imperfect microplates by studies [34; 35], which were formulated in the framework of couple-stress theory and studies where electrostatic actuation was taken into account [28; 36; 37]. Initially flat plates pre-loaded by a uniform constant pressure and then loaded by the nonlinear electrostatic force were shown to be bistable in studies [38; 39]. The bistability of the pressurized graphene membrane was investigated in studies [39; 40]. The bistable behavior of electrostatically actuated initially curved circular plates was recently investigated theoretically [6] and later demonstrated experimentally [41]. The theoretical analysis, in the framework of the [PERSON] plate theory, was based on a reduced-order model built using the [PERSON] method. The experiments were carried out using bell-shaped shallow shells, fabricated by a novel self-molding soft punch stamping approach. Note that while the implementation of microshells in MEMS/NEMS-resonant sensors has been demonstrated [42; 43; 44], the fabrication of micro- and nanoscale three-dimensional thin-walled structures is still a challenge. In the present work, we explore the spectral behavior of initially curved circular bistable microplates, actuated by a nonlinear configuration-dependent distributed electro static force provided by a gap-closing electrode. The main focus is on the investigation of the frequencies and associated modes of the free undamped vibrations of the plate around the deformed equilibria of the structure, dictated by the electrostatic force. The vibrations around the configurations corresponding to the voltages lower than the ST values (the first stable branch), as well as post-buckled equilibrium states (the second stable branch), are considered. In contrast to the previously considered devices of this type, the plate analyzed here is bell-shaped in its initial, stress-free, configuration and is also sufficiently curved as to be bistable. The choice of the initial geometry of the structure and its dimensions is motivated by, and is consistent with, the actually fabricated devices previously reported by us [41]. We show that in the bell-shaped plates, the vibrational modes may differ significantly from those of the initially flat or slightly curved imperfect plates, or of the shallow spherical caps. The motivation of the work is two-fold. From one side, understanding of the role of the initial curvature and of the (electrostatic) pre-loading on the natural frequencies and modes of the plate, along with the evaluation of the appropriate geometrical and operational parameters required to achieve the desired response, is a necessary prerequisite for the implementation of bistable microplates in sensors.