Investigation of Non-linear Characteristics of Electrostatically Coupled MEMS Resonator Arrays

This paper presents an investigation of non-linear static and dynamic characteristics of electrostatically coupled n-microbeam MEMS resonator arrays, to understand their collective behavior. We develop a novel reduced-order model using a single-mode Galerkin approximation to derive the governing dynamic equations for an arbitrary number of microbeams, 'n'. Our comprehensive investigation reveals that while the first natural frequency remains constant irrespective of 'n', the system's pull-in voltage converges to an asymptotic value as the array size increases. We systematically explore the intricate influence of inter-beam and electrode-beam gap distances, microbeam thickness variations, and both tensile and compressive axial stresses on natural frequencies, mode shapes, and the operational tuning range. A key finding is that the highest-order mode consistently exhibits all microbeams vibrating in-phase. Furthermore, we demonstrate that strategic non-uniformities in gaps, thickness, or axial stress can be used to significantly tune system stability and dynamic response. The theoretical framework is rigorously validated against Finite Element Analysis (FEA) using COMSOL Multiphysics across various array sizes and complex asymmetric scenarios, confirming the model's accuracy and robustness. This research provides crucial insights for the advanced design and optimization of MEMS devices that leverage complex non-linear and emergent behaviors.