Research paperDesign and modeling of a MEMS accelerometer based on coupled mode-localized nonlinear resonators under electrostatic actuation
Introduction
MEMS accelerometer is one of the earliest commercialized MEMS products and is widely used in the fields of automotive, consumer electronics and aerospace [1, 2]. Practically, exploring effective method to improve the sensitivity of such sensors is in great demand for the field of high precision detection. Common accelerometers can be divided into two types according to the sensing mechanisms, static and dynamic sensors [3], [4], [5]. As a typical dynamic sensor, the resonant accelerometers can measure the acceleration by evaluating the frequency shift induced by inertial force, which have been widely used due to their high sensitivity and quasi-digital signal. Hence, the research topic of achieving high sensitivity has attracted extensive attentions. One effective way is to introduce the leveraging mechanisms to amplify the external inertial force. Comi et al. [6] used a single-stage lever to improve the sensitivity, which was verified through experiments that the sensitivity can be up to 455Hz/g. Ding et al. [7] used a two-stage lever mechanism in the resonant accelerometer for measuring tilt, and the high sensitivity was verified through close-loop and open-loop experiments. Enhancing the accelerometer performances cannot be achieved by increasing the number of amplifying levers indefinitely, otherwise the sensitivity will be decreased [8]. Therefore, the improvement of sensitivity by just using the levering mechanism encounters a bottleneck. Besides, the performance of resonant accelerometer is seriously restricted due to its frequency drift and temperature sensitivity.
Recently, mode-localized sensors composed of two or more resonators coupled to each other allowing spatial energy expansion have been widely reported [9], [10], [11]. Compared with traditional resonant sensors, such sensors introducing mode localization mechanisms possess advantages of high sensitivity and good common-mode rejection [12], [13], [14], [15]. Another key identifier of such sensing method is that the readout metric is basically based on amplitude modulation instead of traditional frequency shift [16, 17]. Spletzer et al. [18] first utilized mode localization in two coupled microcantilevers as a mass sensor. And, by comparing the eigenvalue output with the frequency shift, the sensitivity is improved by two orders of magnitude. Zhang et al. [19] proposed a mode-localized accelerometer composed of two resonators coupled by an elastic beam, and it is experimentally verified that the variation of the amplitude ratio induced by inertial force is more than 300 times higher than the traditional frequency shift. Zhao et al. [20], [21], [22] have also done a series of work on the application of mode localization in mass sensors and inertial sensors. Moreover, the three output metrics of the sensitivity commonly used in mode localization sensors are depicted by eigenstate, amplitude ratio, and amplitude shift, and it was found that the output sensitivity of amplitude ratio is the highest [22].
To further improve the performance of the mode localized sensors, a series of methodologies have been introduced such as adjusting the actuation voltage, varying coupling method and changing the number of coupled resonators [23, 24]. The coupling strength is the most critical factor, which can be modulated by the coupling style involving electrostatic coupling [25] and mechanical coupling [21]. The electrostatic coupling has the benefit in terms of high tunability, and was verified for the first time in [25]. After that, Zhao et al. [26] proposed a three-degree-of-freedom mode-localized sensor using an electrostatic coupling, they demonstrated that this coupling method is more sensitive than other methods through experiments. Lyu et al. [23] proposed a mass sensor design based on a pair of electrostatically coupled clamped-clamped microbeams of unequal length, the sensitivity output under different coupling voltages has been studied, and its maximum can be achieved while tuning the electrostatic coupling. In [27], a mode localization accelerometer with four degrees of freedom and two coupling methods in parallel is proposed, and the open-loop circuit test has been performed to verify that the normalized sensitivity range can reach 2050%. In addition to the high sensitivity of mode-localized sensors, the anti-interference ability to external environmental disturbances is another advantage. Thiruvenkatanathan et al. [28] studied the eigenstates and frequency shift output under different external temperatures and pressure, respectively, and the results showed that eigenstate output has better robustness. In [29], it has been demonstrated that the amplitude ratio output can not only improve the sensitivity but also eliminate the ambient pressure drift. The current research on mode-localized accelerometers is mainly limited to linear vibrations, while to improve the resolution, the vibration amplitude of the resonator must be increased and the nonlinear behavior of the device must be considered.
In this paper, a novel dual proof mass accelerometer is proposed by introducing mode localization in two electrostatically coupled resonators. This structure uses a lever mechanism to transmit the motion of the proof mass in term of axial force applied to the weakly coupled resonators, which causes vibration localization. The Multiphysics model is established while including electrostatic and mechanical nonlinearity. The coupled equations of motion have been solved using Galerkin discretization associated with the method of multiple scales (MMS), and the results have been verified by a computational solving procedure based on the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). The simulations have been performed to investigate the sensitivity changes for the linear and nonlinear cases. Moreover, the nonlinearity functionalization for acceleration sensing utilizing mode localization has been demonstrated to enhance the sensitivity and vibration amplitude. When the drive voltage is increased, not only the sensitivity can be improved but also the resolution can be increased with localized vibrations of the second mode. Finally, the effect of the coupling voltage on the sensor's sensitivity is analyzed in detail, and the mode aliasing is clearly shown through the amplitude-frequency response curve, so that the theoretical optimal coupling voltage can be obtained.
Section snippets
Structure of the accelerometer
The proposed structure of the accelerometer is shown in Fig. 1, and the relevant parameters are listed in Table 1. The two proof masses M are suspended on the substrate by the elastic pin, and two resonators are placed at a distance d from the elastic pin. The two resonators are clamped-clamped beams driven by electrode 1 and electrode 2 respectively, and initially balanced. It is worth noting that the two resonators are coupled with an electrostatic force. When the two proof masses are
Perturbation analysis
In this section, a second-order MMS is adopted to explore the response under different acceleration disturbances. Since the first mode is the dominant mode, only one mode is considered (Nm=1) and the other modes are ignored in the Galerkin discretization. For convenience, in Eq. (8), we replace q1,1, q2,1 by q1, q2, respectively. The method of multiple scales is then used on the resulting equation with an appropriate scaling for the quadratic and cubic nonlinearities. A low bookkeeping
Numerical simulations
The previous section has proved that the method of MMS can accurately analyze the dynamic characteristics of the device. Thus, it will be used to determine the sensitivity of the accelerometer. Due to the mechanical nonlinearities dominating the resonator dynamics compared to electrostatic nonlinearities, the frequency responses exhibit a hardening behavior with strong amplitude dependence in the balanced state. The amplitude frequency response under different actuation Vac1 is shown in Fig. 5,
Conclusions
In this paper, a dual-mass MEMS accelerometer based on the mode localization phenomenon has been designed and modeled. Two separate leveraging mechanisms were introduced to amplify the inertial forces applied axially to two electrostatically coupled identical resonators, thus generating mode localization. The main conclusions are listed as follows: (i) A mathematical model considering the electrostatic and mechanical nonlinearities was established, and the dynamic response was calculated by
Credit author statement
Ming Lyu: Wrote the original draft, Conceptualization, Methodology, Jian Zhao: Conceptualization, Methodology, Writing -review & editing, Najib Kacem: Conceptualization, Writing - review & editing, Bin Tang: Validation, Pengbo Liu: Validation, Jiahao Song: Validation, Formal analysis, Heng Zhong: Validation, formal analysis, Yu Huang: Writing-review and editing. All authors have read and agreed to the published version of the manuscript.
Declaration of Competing Interest
No conflict of interest exists in the submission of this manuscript, and the manuscript is approved by all authors for publication. I would like to declare that the work was original research that has not been published previously, and not under considered for publication elsewhere, in whole or in part. All the authors have approved the manuscript that is enclosed.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. U1930206), Natural Science Foundation Project of Liaoning Province (2019-KF-02-03) and project of State Key Laboratory of Structural Analysis for Industrial Equipment, China (Grant No. GZ19202). This project was also performed in cooperation with EUR EIPHI program, Europe (Contract No. ANR 17-EURE-0002).
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