On the dynamics of bistable micro/nano resonators: Analytical solution and nonlinear behavior

• Nonlinear dynamics of an electrostatically actuated arch MEMS resonator is studied.
• We use the Homotopy Analysis Method to analytically derive arch’s frequency response.
• We use numerical simulations to validate the analytical solution.
• We study the nonlinear vibrations of the arch using obtained analytical solution.
• Bistable behaviors of the resonator in forward/backward frequency sweeps are studied.

With the rapid development of micro/nano-electro-mechanical systems (MEMS/NEMS), arch shaped resonators are becoming increasingly attractive for different applications. Nevertheless, the dynamics of bistable resonators is poorly understood, and the conditions for their appropriate performance are not well known. In this paper, an initially curved arch shaped MEMS resonator under combined DC and AC distributed electrostatic actuation is investigated. A reduced order model obtained from first mode Galerkin’s decomposition method is used for numerical and analytical investigations. We have used the Homotopy Analysis Method (HAM) in order to derive analytical solutions both for the amplitude and the temporal average of nonlinear vibrations. The obtained analytical expressions, validated by numerical simulations, are able to capture nonlinear behaviors including softening type vibrations and dynamic snap-through. We have used the derived analytical results in order to study the nonlinear vibrations of the bistable MEMS resonator. According to our results, in the bistability region the overall dynamic response of the system is described by means of a pair of softening type frequency responses merging together in a specific frequency band. The dynamic snap-through is then described by transitions between these two frequency responses, each of which corresponding to one of the stable configurations of the arch. This fresh insight to the problem can be used in the design and optimization of bistable resonators and determination of their sharp roll-off frequencies. A feature that can be implemented in the design of bandpass filters.