# Natural Frequencies and Modes of Electrostatically Actuated Curved Bell-Shaped Microplates
## 1 Introduction  
Bistable micro and nanostructures such as initially curved beams, plates, or spherical caps manifest several unique features making them attractive for implementation in micro- and nanoelectromechanical systems (MEMS/NEMS). When subject to a quasi-statically increasing loading, these devices manifest abrupt, commonly referred as a snap-through (ST), transition between two different stable configurations once the loads exceed a certain critical ST value. The decrease of the force below another, snap-back (SB) or release (R), critical value is followed by the snapping of the device back to its first, associated with the smaller deflection, stable configuration.The intrinsic hysteresis of the loading-unloading cycle (since SB and ST forces and deflections are different) is exploited in electrical and optical switches, micro- and nanomechanical nonvolatile memories, or MEMS/NEMS logic elements [1; 2; 3]. The bistable devices, which manifest latching and are locked in the switched position at zero loading and therefore zero power, are especially suitable for implementation in autonomous systems and attract significant interest [4; 5; 6]. Since the ST or SB transitions are followed by the high-frequency vibrations of the structure, bistable devices are implemented in energy harvesters for the conversion of the low-frequency aperiodic inertial inputs into oscillatory outputs at a higher, well-defined, frequency [7]. In the sensors arena, the extremely high sensitivity of the bistable devices in the vicinity of the critical ST and SB configurations lies in the foundations of threshold accelerometers, inertial switches, gas, pressure, and flow sensors [1; 2; 8; 9; 10], as well as event-based wake-upsensors for Internet of Things (IoT) applications [11; 12]. Wideband natural frequencies tunability of bistable devices in the configurations close to the ST or SB points is exploit to increase the sensitivity of resonant sensors [13; 14]. It should be noted that the behavior of conventional macro-scale structures prone to snap-through buckling is a long-standing topic in structural mechanics [15; 16; 17]. In these structures, the appearance of bistability is related to the geometric nonlinearities or/and the nonlinear material behavior. What distinguishes micro- and nanodevices is that they require actuation and, as a result, are loaded by forces of a non-mechanical nature, such as magnetic [18] or (the most widely used) electrostatic. The intrinsic nonlinearities of these forces may result in additional instabilities not observed in the structures loaded by prescribed, configuration-independent, forces. For example, devices actuated by electrostatic forces are prone to the so-called pull-in (PI) instability when the deformable element collapses toward the actuating electrode at the operating voltage above a certain critical value. The combination of nonlinearities associated with the electrostatic forces with the geometric nonlinearities of the bistable structures may result in more rich behaviors and open new possibilities for the designers of microsensors. Among electrostatically actuated bistable structures, double-clamped initially curved beams (arches) operated by a gap-closing electrode are the most intensively researched. The beam-type devices are distinguished by simplicity, robustness, manufacturability, small footprint, and the ability to be downscaled to a nanosize. Many interesting effects such as consecutive ST and PI instabilities, [19; 20], symmetry breaking, multistability [21], high-frequency tunability, veering, and internal and parametric resonances [22; 23; 24] were reported, just to name a few. Much less attention was paid to the two- and three-dimensional micro-scale structures, such as plates or shells operated by electrostatic forces. Initially flat plates were considered by several authors [25; 26; 27], and pull-in behavior and nonlinear resonant responses were investigated both theoretically and experimentally.