A Trimming Strategy for Mass Defects in Hemispherical Resonators Based on Multi-Harmonic Analysis

Abstract

This study investigates the impact of etching trimming parameters on the multiple harmonics of the mass distribution in hemispherical resonators and proposes a novel 1st harmonic trimming scheme. As mass balancing technology advances, the extension of identification and trimming from frequency split to multiple harmonics remains a challenge. Initially, a multi-harmonic identification scheme based on spurious mode detection was established, considering the influence of the first three harmonics of the mass distribution on the dynamic characteristics of hemispherical resonators. Finite element method modeling and analysis revealed that common structural geometric errors significantly introduce the 1st harmonic. By integrating a rectangular pulse function into the mass distribution function to simulate etching grooves, spectral analysis revealed that groove depth and width determine the amplitude and gradient of introduced harmonics. This research introduces an innovative discrete trimming scheme aimed at addressing the frequency split and mode mismatch issues associated with traditional single-point trimming of the 1st harmonic. By decomposing the trimming task into primary and auxiliary etching grooves, the 4th harmonic introduced by the primary etching is compensated by the secondary 4th harmonic introduced by the auxiliary etching, achieving decoupling of the 1st harmonic from frequency split during the trimming process. The scheme was verified through finite element simulations and experimental testing. Results demonstrate that, for a similar reduction in the 1st harmonic, the variation in frequency split during the discrete trimming process is only 11% of that observed in single-point trimming, facilitating efficient and low-damage trimming of the 1st harmonic.

Keywords

• frequency split
• harmonic
• hemispherical resonator
• ion beam etching
• mass defect