Characterization of Three-Mode Combination Internal Resonances in Electrostatically Actuated Flexible–Flexible Microbeams

Nonlinear intermodal coupling based on internal resonances in MEMS resonators has advanced significantly over the past two decades for various real-world applications. In this study, we demonstrate the existence of various three-mode combination internal resonances between the first five flexural modes of electrostatically actuated flexible–flexible beams and dynamic modal interaction between three modes via internal resonance. We first calculate the natural frequencies of the beam as a function of the stiffnesses of the transverse and rotational springs of the flexible supports, utilizing both analytical formulation and finite element analysis (FEA). Following this, we identify six combination internal resonances among the first five modes and use applied DC voltage to validate the exactness of one commensurable internal resonance condition (⁠$ω2=ω5−ω4$⁠). Subsequently, we studied a detailed forced vibration analysis corresponding to this resonance condition by solving the five-mode coupled governing equations through numerical time integration and the method of multiple scales. The results compellingly exhibit three-mode intermodal coupling among the second, fourth, and fifth modes as a function of excitation amplitude and frequency. Alongside this, intriguing nonlinear phenomena such as threshold behavior, saturation phenomena, and autoparametric instability are observed. Finally, this paper provides a systematic methodology for investigating three-mode combination internal resonances and related nonlinear dynamics, offering significant insights that could be used in observing phonon or mechanical lasing phenomena in MEMS resonators.