Dynamic evolution of a primary resonance MEMS resonator under prebuckling pattern

This paper analytically investigates the dynamic evolution of the primary frequency response of a prebuckling microbeam-based resonator with symmetry. A doubly clamped straight microbeam actuated by two symmetric stationary electrodes is simplified as a time-varying capacitor model for qualitative analysis purpose. Nonlinearities induced by the midplane stretching of the microbeam and the electrostatic force are considered. During solution procedure, electrostatic force holds its original form without any Taylor series expansion, and only one assumption with a small ratio of AC to DC voltage is introduced. The average equation, frequency response, backbone curve and stability condition are determined, respectively, based on the method of multiple scales combined with homotopy concept. Results demonstrate for the first time that the frequency response includes two types of branches, namely low- energy branch and high-energy branch. As the increase ion AC excitation amplitude, both branches close to each other along the backbone curve until they intersect. Further analyses are then performed to investigate the details of the backbone curve and the frequency response equation. Analytical formulas to determine the hardening and softening switches of the frequency response and the intersection condition of the low- and high-energy branches are both deduced and examined in depth. Primary frequency response properties in pull-in and secondary pull-in case are classified and depicted through theoretical predictions via the method of multiple scales and then verified through numerical results via the finite difference method combined with Floquet theory. Finally, a specific case study based on equivalent lumped parameters via Galerkin method is presented. Excellent agreements between theoretical predictions and simulation results illustrate the effectiveness of the whole analyses.