Micro and nanoelectromechanical systems M/NEMS have been extensively investigated and exploited in the past few decades for various applications and for probing fundamental physical phenomena. Understanding the linear and nonlinear dynamical behaviors of the movable structures in these systems is crucial for their successful implementation in various novel technologies and to meet the long list of new sophisticated requirements. This paper presents a review for some of the recent topics pertaining to the dynamical behaviors, linear and nonlinear, of M/NEMS resonating structures. First, an overview is presented of the various used dynamical approaches to enhance the sensitivity of resonators for sensing applications. Then a summary is presented of the recent works on the linear and nonlinear mode coupling in M/NEMS resonator. Next, recent research is reviewed on coupled M/NEMS resonators, mechanically and electrically, leading to collective behaviors like mode localization. The final part of the paper discusses analytical approaches that have been developed to better understand and investigate the dynamical behavior of M/NEMS resonators focusing on the perturbation method the multiple time scales.

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Linear and nonlinear dynamics of micro and nano-resonators: Review of recent advances

This paper reports a high-sensitivity resonant electrometer based on the mode localization of two degree-of-freedom weakly coupled resonators (WCRs). When charges are applied to the input electrodes, the effective stiffness of a specific resonator of the WCRs will be perturbed, leading to a drastic change in mode shape owing to the mode localization phenomenon. By measuring the shift of the amplitude ratio, the small charge fluctuation can be accurately sensed. The theoretical mode of the electrometer is established based on the transfer functions of the WCRs. In particular, we establish the design rules of the coupling factor according to the −3-dB bandwidth, amplitude ratio measurement errors, and frequency misalignment between the resonators. The experimental results show that the amplitude ratio-based sensitivity is $\sim 2151$ times higher than the frequency-based sensitivity. The amplitude ratio-based resolution of the electrometer is approximately 1.269 fC . [2016-0119]

A High-Sensitivity Micromechanical Electrometer Based on Mode Localization of Two Degree-of-Freedom Weakly Coupled Resonators

Progressive contact-separate triboelectric nanogenerator based on conductive polyurethane foam regulated with a Bennet doubler conditioning circuit

This paper, for the first time, reports a closed-loop accelerometer based on three degree-of-freedom weakly coupled resonator (WCR). Three resonators are weakly coupled by two bridge-type coupling beams. When acceleration acts on the proof masses, the stiffness perturbation between the two outer resonators will incur the mode localization and cause the shift of amplitude ratio. The accelerometer is tested under open-loop circuit and closed-loop circuit to obtain the frequency responses and transient responses, respectively. The experimental results show that the sensitivity based on the amplitude ratio readout is improved by 348% compared with the 2-DoF WCR accelerometer. The measured signal-to-ratio noise is 124.2 dB and resolution reaches $1.1~\mu \text {g} / \surd \text {Hz}$ , which are significantly improved compared with the previous works. The resolution and signal-to-ratio noise under closed-loop measurement are improved by 46181% and 123.2%, respectively, compared with those under open-loop measurement which demonstrate that the performance of the mode-localized accelerometer can be improved using the closed-loop measurement method.

A Closed-Loop Accelerometer Based on Three Degree-of-Freedom Weakly Coupled Resonator With Self-Elimination of Feedthrough Signal

The mode localization phenomenon of disordered weakly coupled resonators (WCRs) is being used as a novel transduction scheme to further enhance the sensitivity of micromechanical resonant sensors. In this paper, two novel characteristics of mode localization are described. First, we found that the anti-resonance loci behave as a linear function of the stiffness perturbation. The anti-resonance behavior can be regarded as a new manifestation of mode localization in the frequency domain, and mode localization occurs at a deeper level as the anti-resonance approaches closer to the resonance. The anti-resonance loci can be used to identify the symmetry of the WCRs and the locations of the perturbation. Second, by comparing the forced localization responses of the WCRs under both the single-resonator-driven (SRD) scheme and the double-resonator-driven (DRD) scheme, we demonstrated that the DRD scheme extends the linear measurement scale while sacrificing a certain amount of sensitivity. We also demonstrated experimentally that the amplitude ratio-based sensitivity under the DRD scheme is approximately an order of magnitude lower than that under the SRD scheme, that is, the amplitude ratio-based sensitivity is −70.44% (N m−1)−1 under the DRD scheme, while it is −785.6% (N m−1)−1 under the SRD scheme. These characteristics of mode localization are valuable for the design and control of WCR-based sensors. Two sensing mechanisms, newly isolated in wave-trapping microresonators, can enhance the use of such devices in medical diagnostics. The standing waves generated by microresonators shift their resonant frequencies in the presence of foreign molecules. Honglong Chang and colleagues from Northwestern Polytechnical University in China now report that weakly coupled microresonator devices can access discreet energy modes with startling sensitivity properties. Fabrication of silicon-on-insulator devices helped the team to reveal that localized ‘anti-resonance’ minimums in oscillation amplitudes were linearly dependent on stiffness properties—a relationship that can characterize the localized energy modes on various resonators. By using a double-resonator-driven scheme to tune the microresonator device and force the localization modes to emerge, the team could then extend the linear measurement scales to a greater potential range of biosensing applications.

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Characterization of forced localization of disordered weakly coupled micromechanical resonators.

This paper reports an acceleration sensing method based on two weakly coupled resonators (WCRs) using the phenomenon of mode localization. When acceleration acts on the proof masses, differential electrostatic stiffness perturbations will be applied to the WCRs, leading to mode localization, and thus, mode shape changes. Therefore, acceleration can be sensed by measuring the amplitude ratio shift. The proposed mode localization with the differential perturbation method leads to a sensitivity enhancement of a factor of 2 than the common single perturbation method. The theoretical model of the sensitivity, bandwidth, and linearity of the accelerometer is established and verified. The measured relative shift in amplitude ratio ( $\sim 312162$ ppm/g) is 302 times higher than the shift in resonance frequency ( $\sim 1035$ ppm/g) within the measurement range of ±1 g. The measured resolution based on the amplitude ratio is 0.619 mg and the nonlinearity is $\sim 3.5$ % in the open-loop measurement operation. [2015-0247]

An Acceleration Sensing Method Based on the Mode Localization of Weakly Coupled Resonators

This paper reports a resonant MEMS tilt sensor based on mode localization phenomenon for the first time. The tilt sensor consists of two mode localization based accelerometers which are perpendicular to each other. The input tilt angle will cause dramatic changes of the mode shape of both accelerometers. Furthermore, the linear measurement range is extended to a full-range of [−90°, 90°] by alternatively measuring the shifts of the two amplitude ratios with a phase difference of 90°. Current test results show that the relative sensitivity based on the amplitude ratio is averagely 169 times higher than that of the frequency with a non-linearity of less than 4.5% in the range of [−90°, 90°]. The outputs of the input tilt angle (θ) from [90°, 270°] and [−90°, 90°] are axial symmetry about θ = 90°, so it is easy to extend the linear measurement range to 360°.

A mode localization based resonant MEMS tilt sensor with a linear measurement range of 360

This paper reports a high-sensitive MEMS resonant electrometer based on mode localization of two weakly coupled resonators (WCRs). In the electrometer, the small electric charge input will cause a stiffness perturbation to one resonator of the WCRs, leading to a drastic change of the mode shape owing to the mode localization phenomena. By measuring the shift of the amplitude ratio, the small charge fluctuation can be accurately sensed. The experimental results show that with a charge input of 144fC the relative sensitivity based on the amplitude ratio is 663751ppm, which is −2341 times higher than that of the frequency (283.56ppm).

A high-sensitive resonant electrometer based on mode localization of the weakly coupled resonators

The invention discloses a weakly coupled MEMS resonance type accelerometer based on a mode localization effect, and belongs to the field of MEMSs. The accelerometer comprises two movable mass blocks and two single beam resonance systems, wherein the two movable mass blocks are the same completely, and the two single beam resonance systems are connected together via a mechanical coupling beam. According to the weakly coupled MEMS resonance type accelerometer disclosed by the invention, a design of single resonance beams is used, so the mode interference is greatly reduced; a capacitor plate is designed at an inside and an outside of each of the resonance beams, so the differential detection for amplitudes of the single resonance beams can be realized, not only can the intensity of a signal be enhanced, but also the feedthrough capacitance signal interference due to a potential difference between a drive electrode and a detection electrode is eliminated, and the stability and the accuracy of a measurement signal can be greatly improved; as a stiffness adjustment electrode is increased, the flexible selection for a working point and the adjustment for a linearity working range are realized; and by placing an alternating current drive electrode in the middle of the two resonance beams, the frequency components of an output signal are more single, and the detection for the output signal and the design for a closed loop control circuit are enabled to be easier.

Weakly coupled MEMS resonance type accelerometer based on mode localization effect

The invention, which belongs to the MEMS field, discloses a micro-mechanical resonant electrometer with ultra-high sensitivity. The method comprises a mechanical structure design method of an electrometer header and a testing circuit design method. Compared with the prior art using a movable resonator charge input polar plate, the mechanical structure design enables the conversion efficiency between the electrostatic force and the axial stress to be improved; and with a micro-mechanical lever, the electrostatic force increases to improve mechanical sensitivity. When an external charge is inputted, rigidity disturbance on a resonator I increases and thus the localization phenomenon of the mode becomes dramatic. Meanwhile, the two provided resonators with different structures employ differential detection structures; the differential amplification circuit is used for detecting the amplitudes of the resonators and eliminating feed-through signals, so that the signal to noise ratio of the detection signal can be improved. In terms of the testing circuit design, a closed-loop testing plane is employed. To be specific, a signal at a detection electrode passes through a trans-impedance amplifier, a subtracter, a band-pass filter and a comparator and then is loaded at an alternating-current drive electrode to form a closed loop; rectification and filtering are carried out on two paths of outputs of the subtracter and dividing is carried out to obtain a direct-current voltage signal reflecting an amplitude ratio of two resonators. With a closed-loop drive detection circuit, the amplitude and frequency noise of the resonator can be reduced.

Boyang Li

Papers

An Acceleration Sensing Method Based on the Mode Localization of Weakly Coupled Resonators

A mode localization based resonant MEMS tilt sensor with a linear measurement range of 360

A high-sensitive resonant electrometer based on mode localization of the weakly coupled resonators

Weakly coupled MEMS resonance type accelerometer based on mode localization effect

Micro-mechanical resonant electrometer with ultra-high sensitivity