DOI: 10.5061/dryad.xwdbrv1q2
title: Synchronization mechanism within the blind zone of the differential resonant accelerometer

description: Within the range of the differential resonant accelerometer, a blind zone
 exists that leads to nonlinearity, noise, and zero output, thereby
 introducing uncertainty and degrading performance. Exploration of the
 underlying synchronization mechanism leads to an understanding of this
 uncertainty. In this study, the Huygens pendulum model was employed to
 explain how both anti-phase and in-phase synchronization coexist and
 transfer between each other, where jumps occur stemming from the different
 configurations of the potential wells of the resonator. Noise increase,
 attributed to phase slip, was observed before and after synchronization,
 potential to be utilized to detect shocks in the surroundings.
 Consequently, the entire process of synchronization within the blind zone
 was verified, and the frequency stability, bias instability and resolution
 of the accelerometer were greatly improved.
description: The experimental platform was equipped with a precise optical
 dividing head (SJJF-1, TIANHE, Shanghai, China)(SJJF-1, TIANHE, Shanghai,
 China), a precise gating ruler, a extra high-precise adjustable
 thermostat-container, two universal frequency counters (KEYSIGHT 53230A),
 a multimeter (KEYSIGHT 34461A), a network analyzer ((Digilent Analog
 Discovery 2), and an oscilloscope (Agilent Technologies DSO5012A). The
 movies were recorded by camera and processed by Adobe Premiere.
 With the adjustment of the optical dividing head to the
 horizontal position through the agency of the calibrator and installation
 of the accelerometer upon the precise optical dividing head, the
 accelerometer was rotated counterclockwise for a round inside the high
 precision adjustable thermostat container at 50 ◦C. The dividing head was
 rotated to 0◦, 90◦, 180◦, and 270◦ position, where the 0◦ position
 corresponds to 0 g input for the accelerometer, 90◦ to 1 g input, 180◦ to
 0 g input, and 270◦ to −1 g input. In the rotary experiment, because the
 sensitive direction of accelerometer is z-axis of the sensitive structure,
 the input acceleration a is calculated a=g*sin(θ) where g is the gravity
 and θ is the angle between the input axis of accelerometer and direction
 of gravity. Frequency output of resonators were recorded by two universal
 frequency counters. Phase of resonators were recorded by an oscilloscope.
 Amplitude-frequency and phase-frequency curves were recorded by a network
 analyzer. Impedance curves of resonators were recorded by a Impedance
 analyzer (KEYSIGHT E4990A). The Allan deviation is widely used for
 modeling random errors in inertial devices, in which the bias instability
 is represented by the slope value of 0 in the dual logarithmic coordinate
 diagram, referring to low-frequency zero bias jitter caused by random
 fluctuations in internal circuits and temperature changes and
 characterizing the variation of zero bias over time. The output
 acceleration density of the resonant accelerometer is measured in same
 period and sampling rate. Movies are about phase
 locking, phase slips, rotary experiment, and variation of phase of
 resonators in shock. Movie S1 is about phase locking
 with resonator 2 leading in anti-phase synchronization at first; Movie S2
 is about phase locking with resonator 1 leading in anti-phase
 synchronization at first; Movie S3 is about phase slips in which phase of
 resonator 1 slides right relative to that of resonator 2; Movie S4 is
 about phase slips in which phase of resonator 1 slides left relative to
 that of resonator 2; Movie S5 is about rotary experiment; Movie S6 is
 about the variation of phase of resonators in shock (in in-phase
 synchronization); Movie S7 is about the variation of phase of resonators
 in shock (resonator 1 leading in anti-phase synchronization); Movie S8 is
 about the variation of phase of resonators in shock (resonator 1 leading
 in in-phase synchronization); Movie S9 is about the variation of phase of
 resonators in shock (resonator 2 leading in anti-phase synchronization);
 Movie S10 is about the variation of phase of resonators in shock
 (resonator 2 leading in in-phase synchronization). All movies were merel
 processed by montage and subtitles.
description: # Synchronization mechanism within the blind zone of the differential
 resonant accelerometer
 [https://doi.org/10.5061/dryad.xwdbrv1q2](https://doi.org/10.5061/dryad.xwdbrv1q2) ## Description of the data and file structure There are three compressed files. *Data of figures.zip* includes data required to draw figures, which are raw and unprocessed. *Movies.zip* were recorded by a camera and edited by Adobe Premiere, including the processes of synchronization, rotation experiments, and shock detection. *Script of drawing.zip* includes scripts to draw figures. ### Files and variables #### File: Data of figures.zip **Description:** This file contains the data required for drawing each figure, and the file name corresponds to the figure caption. Frequency output of resonators were recorded by two universal frequency counters. Amplitude-frequency and phase-frequency curves of resonators were recorded by a network analyzer. Impedance curves of resonators were recorded by a impedance analyzer. Phase of resonators were recorded by an oscilloscope. The following are detailed descriptions of these files. *Figs3A,3B,S12A,S12B,S13.xlsx* illustrates the frequency outputs of resonators 1 and 2 according to acceleration modulation of the differential resonant accelerometer to exhibit the entire process of synchronization and desynchronization (4 jumps and a steering appeared during synchronization). The acceleration was modulated using dividing head, and the frequency outputs of resonators 1 and 2 were recorded by two universal frequency counters. * Sheet 1 is the data in Fig3A. The 1st column is the acceleration, the 2nd column is the corresponding output of resonator 1, and the 3rd column is the corresponding output of resonator 2. * Sheet 2 is the data in Fig3B. The 1st column is the acceleration, the 2nd column is the corresponding output of resonator 1, and the 3rd column is the corresponding output of resonator 2. * Sheet 3 is the data in Fig12A. The 1st column is the acceleration, the 2nd column is the corresponding output of resonator 1, and the 3rd column is the corresponding output of resonator 2. * Sheet 4 is the data in Fig12B. The 1st column is the acceleration, the 2nd column is the corresponding output of resonator 1, and the 3rd column is the corresponding output of resonator 2. * Sheet 5 is the data in Fig13A. The 1st column is the acceleration, the 2nd column is the corresponding output of resonator 1, and the 3rd column is the corresponding output of resonator 2. * Sheet 6 is the data in Fig13B. The 1st column is the acceleration, the 2nd column is the corresponding output of resonator 1, and the 3rd column is the corresponding output of resonator 2. * Sheet 7 is the data in Fig13C. The 1st column is the acceleration, the 2nd column is the corresponding output of resonator 1, and the 3rd column is the corresponding output of resonator 2. * Sheet 8 is the data in Fig13D. The 1st column is the acceleration, the 2nd column is the corresponding output of resonator 1, and the 3rd column is the corresponding output of resonator 2. *Figs3C,3D,3E,3F,S12C,S12D,S12E,S12F.xlsx* illustrates the phase variations of resonators 1 and 2 according to acceleration modulation of the differential resonant accelerometer to exhibit the entire process of synchronization and desynchronization (anti- and in-phase synchronization and phase slips), with the phase of resonator 1 fixed as a reference. The acceleration was modulated using dividing head, and the phase of resonators 1 and 2 were recorded by an oscilloscope. * Sheet 1 is the data in Fig3C. The 1st column is time. The 2nd and 3rd columns is the phase of resonators 1 and 2 when the phase difference is -163°. The 4th and 5th columns is the phase of resonators 1 and 2 when the phase difference is -3°. The 6th and 7th columns is the phase of resonators 1 and 2 when the phase difference is 0°. The 8th and 9th columns is the phase of resonators 1 and 2 when the phase difference is 3°. The 10th and 11th columns is the phase of resonators 1 and 2 when the phase difference is 163°.  * Sheet 2 is the data in Fig3D. The 1st column is time. The 2nd and 3rd columns is the phase of resonators 1 and 2 when the phase difference is -163°. The 4th and 5th columns is the phase of resonators 1 and 2 when the phase difference is 90°. The 6th and 7th columns is the phase of resonators 1 and 2 when the phase difference is 0°. The 8th and 9th columns is the phase of resonators 1 and 2 when the phase difference is 163°.  * Sheet 3 is the data in Fig3E&12F. The 1st column is time. The 2nd and 3rd columns is the phase of resonators 1 and 2 when the phase difference is 163°. The 4th and 5th columns is the phase of resonators 1 and 2 when the phase difference is 163°. The 6th and 7th columns is the phase of resonators 1 and 2 when the phase difference is 163°. The 8th and 9th columns is the phase of resonators 1 and 2 when the phase difference is 163°.  * Sheet 4 is the data in Fig12C. The 1st column is time. The 2nd and 3rd columns is the phase of resonators 1 and 2 when the phase difference is 163°. The 4th and 5th columns is the phase of resonators 1 and 2 when the phase difference is 3°. The 6th and 7th columns is the phase of resonators 1 and 2 when the phase difference is 0°. The 8th and 9th columns is the phase of resonators 1 and 2 when the phase difference is -3°. The 10th and 11th columns is the phase of resonators 1 and 2 when the phase difference is -163°.  * Sheet 5 is the data in Fig12D. The 1st column is time. The 2nd and 3rd columns is the phase of resonators 1 and 2 when the phase difference is 163°. The 4th and 5th columns is the phase of resonators 1 and 2 when the phase difference is -90°. The 6th and 7th columns is the phase of resonators 1 and 2 when the phase difference is 0°. The 8th and 9th columns is the phase of resonators 1 and 2 when the phase difference is -163°.  * Sheet 6 is the data in Fig3F&12E. The 1st column is time. The 2nd and 3rd columns is the phase of resonators 1 and 2 when the phase difference is 163°. The 4th and 5th columns is the phase of resonators 1 and 2 when the phase difference is 163°. The 6th and 7th columns is the phase of resonators 1 and 2 when the phase difference is 163°. The 8th and 9th columns is the phase of resonators 1 and 2 when the phase difference is 163°. *Figs4A,4B,4C,4D&S15.xlsx* illustrates the amplitude-frequency and phase-frequency curves, frequency output, and quality factor of the observed resonator modulated by the other resonator. The amplitude-frequency and phase-frequency curves of resonators 1 and 2 were recorded by a network analyzer, and the quality factor of resonators 1 and 2 were calculated by corresponding impedance curves recorded by a impedance analyzer.  * Sheet 1 is the data in Fig4A&15A. Each value represents the normalized amplitude of resonator 1, modulated by self-sustained resonator 2. The abscissa is rotary degree from 263° to 272°, and the ordinate is sweeping frequency from 34812 to 34920 Hz. * Sheet 2 is the data in Fig4B&15B. Each value represents the normalized phase of resonator 1, modulated by self-sustained resonator 2. The abscissa is rotary degree from 263° to 272°, and the ordinate is sweeping frequency from 34812 to 34920 Hz. * Sheet 3 is thedata in Fig4C&15C. Each value represents the normalized amplitude of resonator 2, modulated by self-sustained resonator 1. The abscissa is rotary degree from 263° to 272°, and the ordinate is sweeping frequency from 34812 to 34920 Hz. * Sheet 4 is the data in Fig4D&15D. Each value represents the normalized phase of resonator 2, modulated by self-sustained resonator 1. The abscissa is rotary degree from 263° to 272°, and the ordinate is sweeping frequency from 34812 to 34920 Hz. *Figs4E,S14.xlsx* illustrates the frequency output of resonators 1 and 2 modulated by sweeping frequency of the other resonator imposed by a network analyzer, which demonstrated synchronization and phase slips before and after synchronization. * Sheet 1 is the data in Fig4E. The 1st column is the sweeping frequency of resonator 1, the 2nd column is the corresponding output of resonator 2. The 3rd column is the sweeping frequency of resonator 2, the 4th column is the corresponding output of resonator 1. * Sheet 2 is the data in FigS14. The 1st column is the sweeping frequency of resonator 1, the 2nd column is the corresponding output of resonator 2. The 3rd column is the sweeping frequency of resonator 2, the 4th column is the corresponding output of resonator 1. *Figs4F,S26.xlsx* illustrates the quality factor of the observed resonator modulated by the other resonator, calculated by corresponding impedance curves recorded by a impedance analyzer.  * Sheet 1 is the data in Fig4F and FigS26. The1st column is the rotary degree. The 2nd column is the dynamic resistance, the 3rd column is the dynamic capacitance, the 4th column is the dynamic inductance, the 5th column is the static capacitance, and the 6th column is the quality factor of resonator 1. The 7th column is the dynamic resistance, the 8th column is the dynamic capacitance, the 9th column is the dynamic inductance, the 10th column is the static capacitance, and the 11th column is the quality factor of resonator 2. *Figs5A,5B,5C,5D,5E,5F,S29,S30.xlsx* illustrates the comparisons including frequency output of resonators in synchronization and desynchronization, performances of accelerometer in synchronization and desynchronization, and performances of accelerometers among state-of-the-art counterparts. The frequency output of resonators 1 and 2 were recorded by two universal frequency counters, such that BI and resolution could be calculated, respectively. * Sheet 1 is the data in Fig5A. The 1st column is time. The 2nd column is the differential output of acceleration. * Sheet 2 is the data in Fig5B. The 1st column is time. The 2nd column is the differential output of acceleration. * Sheet 3 is the data in Fig5C. The 1st column is time. The 2nd column is the differential output of acceleration. * Sheet 4 is the data in Fig5D. The 1st column is time. The 2nd column is the differential output of acceleration. * Sheet 5 is the data in Fig5E. The 1st column is time. The 2nd column is the normal output. The 3rd column is time. The 4th column is the synchronized output. * Sheet 6 is the data in Fig5F. The 1st column is time. The 2nd column is the normal output. The 3rd column is time. The 4th column is the synchronized output. * Sheet 7 is the data in FigsS28&S29. The 1st column is time. The 2nd column is the differential output of acceleration. *Figs. 6.xlsx* illustrates a novel mechanism to detect subtle shocks based on vulnerable synchronization caused by phase slips. The frequency output of resonators 1 and 2 were recorded by two universal frequency counters when a strike was appliedin proximity to the fixture. * Sheet 1 is the data in Fig6B. The 1st column is time. The 2nd column is the output of resonator 1. The 3rd column is the output of resonator 2. * Sheet 2 is the data in Fig6C. The 1st column is time. The 2nd column is the output of resonator 1. The 3rd column is the output of resonator 2. * Sheet 3 is the data in Fig6D. The 1st column is time. The 2nd column is the output of resonator 1. The 3rd column is the output of resonator 2. * Sheet 4 is the data in Fig6E. The 1st column is time. The 2nd column is the output of resonator 1. The 3rd column is the output of resonator 2. * Sheet 5 is the data in Fig6F. The 1st column is time. The 2nd column is the output of resonator 1. The 3rd column is the output of resonator 2. *Fig. S11.xlsx* illustrates the unitary output of two differential resonators of accelerometer in clockwise rotary and anticlockwise rotary. * Sheet 1 is the data in FigS11A. The 1st column is the sampled time, the 2nd column is the corresponding output of resonator 1, and the 3rd column is the corresponding output of resonator 2. * Sheet 2 is the data in FigS11B. The 1st column is the sampled time, the 2nd column is the corresponding output of resonator 1, and the 3rd column is the corresponding output of resonator 2. *Figs. S16-S19.xlsx* illustrates the amplitude-frequency curves of one resonator, modulated by sweeping frequency and interacted with the other self-sustained resonator, recorded by a network analyzer. * Sheet 1 is the data in FigS16. The 1st column is the sweeping frequency, the 2nd to 10th columns are amplitude of resonator 1. * Sheet 2 is the data in FigS17. The 1st column is the sweeping frequency, the 2nd to 10th columns are phase of resonator 1. * Sheet 3 is the data in FigS18. The 1st column is the sweeping frequency, the 2nd to 10th columns are amplitude of resonator 2. * Sheet 4 is the data in FigS19. The 1st column is the sweeping frequency, the 2nd to 10th columns are phase of resonator 2. *Figs. S20.xlsx* illustrates the impedance curves of resonator modulated by sweeping frequency by network analyzer and interacted with the other self-sustained resonator, recorded by a Impedance Analyzer. * Sheet 1 is the data in FigS20A. Each value represents the G(S) of resonator 1, modulated by self-sustained resonator 2. The abscissa is rotary degree from 0° to 10°, and the ordinate is sweeping frequency from 34800 to 34920 Hz. * Sheet 2 is the data in FigS20B. Each value represents the G(S) of resonator 2, modulated by self-sustained resonator 1. The abscissa is rotary degree from 0° to 10°, and the ordinate is sweeping frequency from 34800 to 34920 Hz. * Sheet 3 is the data in FigS20C. Each value represents the B(S) of resonator 1, modulated by self-sustained resonator 2. The abscissa is rotary degree from 0° to 10°, and the ordinate is sweeping frequency from 34800 to 34920 Hz. * Sheet 4 is the data in FigS20D. Each value represents the B(S) of resonator 2, modulated by self-sustained resonator 1. The abscissa is rotary degree from 0° to 10°, and the ordinate is sweeping frequency from 34800 to 34920 Hz. *Figs. S21.xlsx* illustrates the impedance curves of resonator modulated by sweeping frequency by network analyzer and interacted with the other self-sustained resonator, recorded by a Impedance Analyzer. * Sheet 1 is the data in FigS21A. Each value represents the G(S) of resonator 1, modulated by self-sustained resonator 2. The abscissa is rotary degree from -10° to 10°, and the ordinate is sweeping frequency from 34800 to 34920 Hz. * Sheet 2 is the data in FigS21B. Each value represents the G(S) of resonator 2, modulated by self-sustained resonator 1. The abscissa is rotary degree from -10° to 10°, and the ordinate is sweeping frequency from 34800 to 34920 Hz. * Sheet 3 is the data in FigS21C. Each value represents the B(S) of resonator 1, modulated by self-sustained resonator 2. The abscissa is rotary degree from -10° to 10°, and the ordinate is sweeping frequency from 34800 to 34920 Hz. * Sheet 4 is the data in FigS21D. Each value represents the B(S) of resonator 2, modulated by self-sustained resonator 1. The abscissa is rotary degree from -10° to 10°, and the ordinate is sweeping frequency from 34800 to 34920 Hz. *Figs. S22-25.xlsx* illustrates the impedance curves of resonator modulated by sweeping frequency by network analyzer and interacted with the other self-sustained resonator, recorded by a Impedance Analyzer. * Sheet 1 is the data in FigS22. The 1st column is the sweeping frequency, the 2nd to 10th columns are G(S) of resonator 1. * Sheet 2 is the data in FigS23. The 1st column is the sweeping frequency, the 2nd to 10th columns are G(S) of resonator 2. * Sheet 3 is the data in FigS24. The 1st column is the sweeping frequency, the 2nd to 10th columns are B(S) of resonator 1. * Sheet 4 is the data in FigS25. The 1st column is the sweeping frequency, the 2nd to 10th columns are B(S) of resonator 2. #### File: Movies.zip **Description:** Movies S1 to S4 are about the phase locking in either anti- or in-phase synchronization  and phase slips. The size of Movies S5 to S10 are too big to be uploaded to the submission system such that they are uploaded in this dataset. Movie S1: Phase locking with resonator 2 leading in anti-phase synchronization at first. Phase slips, anti-phase synchronization with resonator 2 leading, in-phase synchronization with resonator 2 leading, in-phase synchronization, in-phase synchronization with resonator 1 leading, anti-phase synchronization with resonator 1 leading, phase slips occurred in turn. Movie S2: Phase locking with resonator 1 leading in anti-phase synchronization at first. Phase slips, anti-phase synchronization with resonator 1 leading, in-phase synchronization with resonator 1 leading, in-phase synchronization, in-phase synchronization with resonator 2 leading, anti-phase synchronization with resonator 2 leading, phase slips occurred in turn. Movie S3: Phase slips in which phase of resonator 1 slides right relative to that of resonator 2. Movie S4: Phase slips in which phase of resonator 1 slides left relative to that of resonator 2. Movie S5: Rotary experiment to record frequency output of resonators 1 and 2. Movie S6: The variation of phase of resonators in shock (in in-phase synchronization). Movie S7: The variation of phase of resonators in shock (resonator 1 leading in anti-phase synchronization). When a shock occurred from surroundings, phase difference between resonators 1 and 2 changed. Movie S8: The variation of phase of resonators in shock (resonator 1 leading in in-phase synchronization). When a shock occurred from surroundings, phase difference between resonators 1 and 2 changed. Movie S9: The variation of phase of resonators in shock (resonator 2 leading in anti-phase synchronization). When a shock occurred from surroundings, phase difference between resonators 1 and 2 changed. Movie S10: The variation of phase of resonators in shock (resonator 2 leading in in-phase synchronization). When a shock occurred from surroundings, phase difference between resonators 1 and 2 changed. #### File: Script of drawing.zip **Description:** There is no code or scripts related to simulation or calculation, but scripts to draw figures. The file name corresponds to the figure caption. *FigS9A.m* and *FigS9B.m* is the script to draw Fig. S9, which is about the synchronization range behavior about the relationship among phase delay, quality factor, and synchronization range and the relationship between phase delay and synchronization range with quality factor varying. This script should be run by Matlab 2020a (or updated version). * In the script of FigS9A: Variable *x* is the phase delay; Variable *y* is the quality factor; Variable *L* is the normalized synchronization range. * In the script of FigS9B: Variable x is the phase delay; Variable *l* is the normalized synchronization range. *FigS10A.m* and *FigS10B.m* is the script to draw Fig. S10, which is about  the relationship between quality factor and frequency. This script should be run by Matlab 2020a (or updated version). * In the script of FigS10A: Variable *m* is the frequency; Variable *l* is the quality factor influenced by air loss. - In the script of FigS10B: Variable *m* is the frequency; Variable *l* is the quality factor influenced by thermoelastic loss. ## Code/software **MATLAB 2022a** (or updated version) is required to open all *.m* files.\ **Microsoft Excel** can be used to view all *.xlsx* files. **\ Any Video viewer** can be used to view all *.mp4* files.