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We propose a sub-1 Hz resonance frequency MEMS resonator that can be used for seismometers. The low resonance frequency is achieved by an electrically tunable spring with an ultra-small spring constant. Generally, it is difficult to electrically fine-tune the resonance frequency at a near-zero spring constant because the frequency shift per voltage will diverge at the limit of zero spring constant. To circumvent this issue, we propose a multi-step electrical tuning method. We show by simulations that the resonance frequency can be tuned by 0.008 Hz/mV even in the sub-1 Hz region. The small spring constant, however, reduces the shock robustness and dynamic range of the seismometer. To prevent this, we employ a force-balanced method in which the mass displacement is nulled by the feedback force. We show that the displacement can be obtained from the voltage that generates the feedback force.

Mode-localization is a promising method to realize high sensitivity sensors, especially in the field of MEMS. Since these sensors monitor amplitude change of weakly coupled resonators, it is important to grasp condition that induces multi-valued amplitude-frequency curve. In this paper, we provide an efficient tool to characterize the nonlinear behavior of the weakly coupled resonators. To analyze the nonlinearity, we solve a two-degrees-of-freedom (2-DoF) coupled equation of motion with nonlinear spring terms. Two approximations are employed to solve the equation; Krylov-Bogoliubov averaging method and approximation based on eigenmode amplitude-ratio at the resonances. As a result, we obtain two decoupled Duffing-like amplitude-frequency equations. We show that nonlinearity of the system is described by factors contained in the equations. The factors can be explicitly written in terms of basic parameters of the system, including coupling spring constant and nonlinear terms. Thus, instead of relying on numerical calculations, we can find parameter condition that brings about multi-valued amplitude-frequency curve. This method can also be utilized to find a condition that eliminates the nonlinearity. As an example, we apply this method to a weakly coupled resonator which uses parallel plate electrode as a coupling spring. We demonstrate the effectiveness and validity of this method by comparing the result with FEM simulations. The methodology and results presented here are general one and can be applied to various systems described by nonlinear coupled resonators.

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