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<p>The frequency locking (FL) mechanism of 1:2:2 internal resonance (IR) under parametric excitation in an electrostatically coupled MEMS resonator has been investigated theoretically, aiming to address the challenge of nonlinear amplitude–frequency dependence, which undermines frequency stability in MEMS devices. Based on Euler–Bernoulli beam theory, the nonlinear governing equations of the coupled resonator system are derived and discretized using the Galerkin method, retaining the first three modes to capture essential modal interactions. The IR response under parametric excitation is systematically analyzed using the rotating wave approximation (RWA), and numerically validated using the harmonic balance method coupled with the asymptotic numerical method (ANM). The analysis reveals that tuning the bias voltage induces both primary and secondary FL phenomena. By exploring the interplay among parametric excitation voltage, coupling voltage, and resonant frequency, the dynamic response is classified into eight distinct phases: (1) no response, (2) uncoupled Mathieu response, (3) first FL, (4) first frequency unlocking (FuL), (5) extended first FL, (6) second FL with first FuL, (7) second FuL, and (8) isolated branch. The corresponding voltage ranges for each phase are quantitatively defined. This research provides a theoretical foundation for enhancing the frequency stability of MEMS resonators and is of practical relevance for the design of high-performance sensing and timing applications, such as mass sensors, accelerometers, and frequency reference devices.</p>

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Frequency locking of 1:2:2 internal resonance in electrostatically coupled MEMS resonators under parametric excitation

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Rongjian Sun,
  • Jian Zhao,
  • Najib Kacem,
  • Pengbo Liu,
  • Zeyuan Dong,
  • Wenxi Sun

摘要

The frequency locking (FL) mechanism of 1:2:2 internal resonance (IR) under parametric excitation in an electrostatically coupled MEMS resonator has been investigated theoretically, aiming to address the challenge of nonlinear amplitude–frequency dependence, which undermines frequency stability in MEMS devices. Based on Euler–Bernoulli beam theory, the nonlinear governing equations of the coupled resonator system are derived and discretized using the Galerkin method, retaining the first three modes to capture essential modal interactions. The IR response under parametric excitation is systematically analyzed using the rotating wave approximation (RWA), and numerically validated using the harmonic balance method coupled with the asymptotic numerical method (ANM). The analysis reveals that tuning the bias voltage induces both primary and secondary FL phenomena. By exploring the interplay among parametric excitation voltage, coupling voltage, and resonant frequency, the dynamic response is classified into eight distinct phases: (1) no response, (2) uncoupled Mathieu response, (3) first FL, (4) first frequency unlocking (FuL), (5) extended first FL, (6) second FL with first FuL, (7) second FuL, and (8) isolated branch. The corresponding voltage ranges for each phase are quantitatively defined. This research provides a theoretical foundation for enhancing the frequency stability of MEMS resonators and is of practical relevance for the design of high-performance sensing and timing applications, such as mass sensors, accelerometers, and frequency reference devices.