Size-dependent response of electrically pre-deformed micro-plates under mechanical shock incorporating the effect of packaging, a frequency-domain analysis

Complementing the time-history solution by the frequency-domain analysis can provide a better understanding of micro-structures dynamics. Nevertheless, there exist no research studies in the literature dealing with the dynamics of plate-type MEMS undergoing mechanical shock via the frequency-domain analysis. Therefore, the main object of the present work is to introduce a frequency-domain analysis for predicting the size-dependent behavior of packaged plate-type MEMS experiencing shock environments. To this end, employing the Hamilton principle, the coupled governing equations of motion are obtained based on the modified couple stress geometric nonlinear thin plate theory and reduced to a system of initial value problems utilizing the Galerkin weighted residual method. The accuracy of the present model is validated by available results in the literature as well as those obtained through three-dimensional finite element simulations performed in COMSOL Multiphysics commercial software. The behaviors of the micro-plate and its package under mechanical shock are then investigated through the use of the linear shock spectrum together with the backbone curves of the system for the first time. Employing this approach, the critical shock durations, at which the influence of mechanical shock becomes extreme, as well as the internal resonances of the system, are obtained. The results reveal that decreasing the shock duration does not increase the influence of mechanical shock all the time. In addition, it is found that the occurrence of the one-to-one internal resonance, which can drastically increase the micro-plate deflections and provide an unexpected pull-in behavior, is not possible except for cases with very small micro-plate to package mass ratios. Furthermore, it is observed that the occurrence of the one-to-three internal resonance cannot put the micro-plate into a dangerous situation.