Linear calibration models for mass defects of hemispherical resonator

In response to the limitations of nonlinear calibration models for the amplitude and azimuth of mass defects of the hemispherical resonator without metal coating in terms of algorithm convergence, model reliability, identification accuracy, and test rapidity, this paper proposes two linear calibration models for the eigenfrequency axis azimuth and frequency split based on the amplitude of the quadrature wave antinode and phase shift of the two eigenfrequency axes respectively, which can significantly improve test efficiency, identification accuracy, and measurement reliability of mass defects of the resonator. Firstly, the linear relationship between the amplitude of the quadrature wave antinode and the sine of the eigenfrequency axis azimuth in a short period of time is deduced by solving the approximate solution of the dynamic model of the quadrature wave antinode. Secondly, the equal distribution mechanism of initial vibration amplitude along the two eigenfrequency axes is proposed and analyzed based on the radial vibration model of the resonator, and then the biphasic phase-locked algorithm is further proposed to achieve linear characterization of the frequency split based on the vibration phase shift of the two eigenfrequency axes. The two calibration models break through the limitations of traditional nonlinear calibration models that rely on the full-period measurement data of the beat vibration of the resonator to identify the two vibration parameters, and then greatly improves the test efficiency while ensuring the identification accuracy of vibration parameters. In addition, the two linear calibration models can not only solve the degradation problem of the identification accuracy of vibration parameters induced by the eigenfrequency drift and principal wave antinode axis drift that traditional nonlinear calibration models cannot overcome, but also distinguish between high and low eigenfrequency axes for avoiding the reverse mass balance phenomenon caused by traditional nonlinear calibration models. After that, the vibration excitation and detection scheme, the signal modulation and demodulation scheme, and the multi-loop control scheme of the standing wave of the resonator without metal coating are designed respectively, and then the optical test system for identifying mass defects of the resonator is implemented, laying the foundation for automated test and balance of the resonator. Finally, the test results indicate that the two linear calibration models can shorten the test time by 20 times for the resonator with frequency split below 1mHz, and the balance results reveal that, based on the identification results of the two linear calibration models, the frequency split of the resonator can be reduced to 0.3mHz through the ion beam etching process., which indicates that the two linear calibration models have both high test efficiency and high identification accuracy, making it more suitable for batch test and balance of resonators.