The vibrations of electrically actuated micro- and nano rectangular plates, described by strongly nonlinear PDEs, are considered. The geometric nonlinearity is taken into account within the Berger model. One of the essentially nonlinear effects is the pull-in phenomenon, i.e., the transition from the oscillatory regime to the attracting one. A simple and physically justified algorithm for determining the voltage for which the system collapses is proposed. The algorithm is based on the detection of the voltage that leads to the merging of stable (center) and unstable (saddle) equilibrium points. The model and algorithm were validated by comparing them with other existing results in the literature, which were obtained by using the Galerkin method and FEM. The closeness of these results confirmed the adequacy of the adopted model and the high accuracy of the algorithm used in this paper. The study was conducted for a wide range of frequency changes and amplitude ratios of DC and AC voltages. Along with the determination of the pull-in voltages, the change in displacements over time was also studied. A spectral analysis was performed, which allows us to analyze the relationships between the input frequencies and the response spectra. The presence of the AC can lead to a dramatic decrease of the pull-in values. This is caused by resonances. The resonances arising in the system have a dual character. This can be either a nonlinear resonance caused by force excitation or a parametric resonance. A separate study was conducted to determine the nature of the emerging resonances. This provides useful information in practical situations. Knowing the resonant frequencies allows us to avoid them in operating electromechanical systems. This can be done by changing the frequency of the alternating voltage or by changing the ratio of the parameters of the system itself. The amplitude of the nonlinear resonance caused by force excitation can be reduced by introducing linear or nonlinear damping. Knowledge of resonant frequencies allows us to design new more effective electromechanical systems.