Multiple internal resonance couplings and quasi-periodicity patterns in hybrid-shaped micromachined resonators

Micro- and nano-electromechanical systems may experience internal resonances, which are inherently strengthened by the systems' nonlinearities. In the present paper, we investigate the dynamics of a hybrid resonator, combining straight and initially curved microbeam shapes. The curved part length is tailored to monitor the three lowest natural frequencies and induce simultaneous internal resonances between first and second modes and second and third modes. We examine the nonlinear interaction of dual hardening and softening bending of the fundamental frequency response curves. Due to the specific frequency ratios, different types of subcombination internal resonances emerge, with quasi-periodic energy transfer among the modes. The subcombination may result in frequencies closely-spaced, which leads to quasi-periodic beating among the frequencies involved and, due to the strong nonlinearities, to the emergence of intermodulation products. We analyze the different patterns underlying the quasi-periodicity, which are intrinsically related to the frequency ratios and deeply affected by the nonlinearities.