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The measurement of resonator’s frequency splitting is a critical issue in vibratory gyroscopes, which would be elaborately treated in practical applications. The high-precision measurement of frequency splitting plays a significant role in frequency tuning control. A novel time-domain method of frequency splitting measurement for hemispherical resonator based on the standing wave swing effect was proposed. The frequency splitting value of the resonator can be directly obtained by taking the reciprocal of the one cycle time of standing wave swings, rather than through the frequency difference between two resonant modes. To begin with, the method was analyzed theoretically, and the measurement resolution and accuracy of the method were researched in detail. Simulation and experimental results showed that the frequency splitting value can be effectively obtained by measuring the period of the standing wave swings, improving the fine measurement resolution and high accuracy. The frequency splitting of lower than 0.007 Hz has to be effectively obtained in the experiment. It is found that the measurement error is a small proportional part of frequency splitting value, so the measurement accuracy is very high when the frequency splitting is very low. Therefore, this time-domain method would contribute to the measurement of ultralow-frequency splitting for high-Q resonators.

Hemispherical resonant gyroscope (HRG) is a kind of vibrating gyroscope without high-speed rotor and movable parts, which is in operation based on the Coriolis effect. HRG offers many advantages which includes long running time, high accuracy, small structure, and short start-up time [^{7}) and the frequency splitting is very low, it is a great challenge to obtain the measurement, because the frequency splitting value is sufficiently low to reduce the gyro drift.

Choi and Kim [

Measuring each resonant mode’s frequency separately was the main methods have been previously used. The resolution of frequency splitting suffers from the limitations of the

In this paper, the time-domain measurement method of low-frequency splitting for a hemispherical resonator is demonstrated and analyzed in detail, which differs from our previous work in the analysis in detail and the further research of the measurement resolution and accuracy by the comparisons with the traditional AFR method and FFT analysis. A complete set of theoretical analysis and simulations of the time-domain measurement method based on the standing wave swing effect was investigated. The measurement resolution and accuracy of this method is researched by comparative experiments with the AFR method and FFT analysis. The experimental results show that the measurement method has a high-frequency resolution and accuracy. And the method is very suitable for the measurement of low frequency splitting for high-

An ideal hemispherical resonator has a completely axisymmetric structure (as shown in Figure

Basic schematic of the hemispherical resonator and vibration mode. (a) Typical structure. (b)

Due to the processing shape and position deviations of the hemispherical resonator, such as circularity and coaxiality, and the anisotropy of the circumferential density and Young’s modulus of the resonator material, the two modes of the harmonic oscillator will develop into two 45° natural axes, the nature frequencies corresponding to the second-order bending modes of the two different natural axes reach the maximum and minimum, respectively (as shown in Figure

Schematic of the vibration mode of hemispherical resonator.

The two-dimensional dynamic model of an ideal axisymmetric hemispherical resonator could be presented as

where

where

Considering the damping issue of the vibration, the two-dimensional dynamic model of the hemispherical resonator can be expressed as

where

And the vibration of two modes is as follows:

When the external angular velocity is ignored, the vibration of two modes would be exponentially decayed.

For an incompletely axisymmetric hemispherical resonator, the two

where

Thus, the vibration at the azimuth

It can be seen from the formula above, the

The

where

According to Matveev’s analysis, standing waves would be destroyed over time under the influence of frequency splitting. And the resonator’s vibration could be expressed as the superposition of two orthogonal vibration waves, which is

where

where

And the resonator’s vibration is rewritten as

It can be derived that the vibrating energy at the position

It can be obtained from (

Thus, the frequency splitting of the hemispherical resonator can be calculated by measuring the period time of the resonator’s vibrating energy.

Take the partial differential of the vibrating energy, which is

The azimuth of standing wave is presented as

It can be concluded from (

Without external interference, the vibration of hemispherical resonator tends to ring down freely through the impact of vibration damping. It implies that the

Formula (

From (

Formula (

From the formula above, the frequency splitting is much more accessible when the stand wave swing amplitude reaches a maximum. In this azimuth of the hemispherical resonator, the expression for vibration energy is

Let the initial time be zero, and the time will be

where

The detection of effective vibration is limited by the resolution of the detection devices. It will be barely detected when the vibration signal is less than the noise of device.

The resolution of the detection device is recorded as

which can be reduced to

One conclusion can be drawn that under a certain resolution of the vibration detection, the resolution of frequency splitting is higher as the

The time-domain measurement method of frequency splitting based on the standing wave swing effect is simulated in MATLAB based on the above analysis. In the simulations, the natural frequency of the resonator is 5000 Hz. The initial vibration amplitude is 5

Simulation results of the vibrating energy.

An experimental apparatus based on a laser Doppler vibrometer (LDV) is set up (as shown in Figure

Picture of experimental setup.

As shown in Figure

As a first step, adjust the exciting and measuring directions of the hemispherical resonator. Typically, the exciting point, measuring point, and center of resonator are in the same axial line for simplicity.

Evacuate the vacuum chamber to a certain pressure. The resonator’s ^{-5} Pa to make sure the resonator get higher

Strike the resonator in any initial orientation (set as 0°) to excite the initial vibration by the striking hammer in a vacuum chamber which is controlled by a pulse switch outside the chamber. Then, record the vibration signal of the resonator through a LDV.

The vibration signals are processed by filtering and fitting to get the energy exchange period time induced by the standing wave swing effect. The frequency splitting value could be obtained by taking the reciprocal value of the period time (

In this section, the resonator’s frequency splitting is measured based on the method mentioned above. And the measurement resolution and accuracy of the method emphatically analyzed in detail. At last, this method is compared with the traditional AFR method and the FFT analysis.

The frequency splitting of two typical hemispherical resonators is measured using the method mentioned above. As reported in ref. [

The presented method is confirmed by comparing the simulation results and the experimental results. The parameters obtained in the experiment (

The comparison between simulation and experimental results. (a) 1# resonator. (b) 2# resonator.

According to the analysis of effect of

The simulation result is shown in Figure ^{-4} Hz if the

The simulation results of the measurement resolution.

Experimental results of the resonator with lower frequency splitting.

The measurement method based on the standing wave swing effect is essentially converting measuring the natural frequency of each mode into a period time based on the resonator’s inherent characteristics. As can be seen from the experimental results, the measurement error of this method is mainly on account of the reading error at the measurement time. The vibrating period time can be expressed by

where

where

It can be seen that the measurement error is a small proportion of the frequency splitting value. The smaller the frequency splitting, the higher the measurement accuracy is. In this paper, the frequency splitting value is calculated by selecting different swing period of 1# and 2# resonators as shown in Figure

Experimental results of the resonator.

No. | Period time (s) | |||
---|---|---|---|---|

1# | 2.967 | 19.350 | 16.383 | 0.0610 |

19.350 | 35.740 | 16.390 | 0.0610 | |

35.740 | 52.390/ | 16.650 | 0.0601 | |

2# | 1.603 | 4.656 | 3.053 | 0.3275 |

4.656 | 7.707 | 3.051 | 0.3278 | |

7.707 | 10.750 | 3.043 | 0.3286 | |

10.750 | 13.810 | 3.060 | 0.3268 |

As reported, the measurement method of frequency splitting based on AFR is widely used in many areas. However, its measurement accuracy is not enough owing to the resolution of the frequency is limited by the sweeping step of hardware. In the paper, the frequency splitting of hemispherical resonators 1# and 2# was measured based on AFR method for comparison, by the experimental setup reported in the previous article [

The experimental results are shown in Figure

The measurement results of frequency splitting base on AFR. (a) Resonator 2#. (b) Resonator 1#.

The vibration signal of the resonator could also be analyzed through fast Fourier transform (FFT), as shown in Figure

The FFT analysis results of vibrating signals. (a) 1# resonator. (b) 2# resonator.

The time-domain measurement method for low-frequency splitting of hemispherical resonators is researched in detail in the paper. The frequency splitting value of the resonator can be directly obtained without calculating the frequency difference between the two resonance modes. The experimental results reveal that the proposed method can effectively obtain a frequency lower than 0.007 Hz. And based on the proven evaluation, the resolution of frequency splitting value could reach to the level of 10^{-4} Hz if the

The datasets used in the experiments and discussed in the paper will be available if requited.

The authors declare that they have no conflicts of interest.

This work was supported by the National Key Research and Development Program of China (Grant No. 2017YFB1104700) and the Program of Shanghai Academic/Technology Research Leader under Project (Grant No. 18XD1421700).