Theoretical analysis and one-step balancing solutions of four types of density harmonic defects in hemispherical resonators

The performance of hemispherical resonators is highly sensitive to circumferential density harmonic defects. Due to a limited understanding of the imbalances induced by the first three harmonic defects and their corresponding balancing theory, typical resonators operating in the second bending mode often balance only the 4th harmonic defect iteratively. This paper studies unbalanced effects caused by density harmonic defects in the n-th bending mode using elastic thin shell theory. It demonstrates that imbalances are related solely to the (n − 1)-th, n-th, (n + 1)-th, and 2n-th density harmonic defects. Furthermore, identification and balancing equations for the four types of harmonic defects are presented. The existence of one-step balancing solutions is examined, and the one-step balancing solutions are investigated. Fixed balancing orientation can linearize the balancing equations, and balancing the four types of harmonic defects requires a minimum of eight positions. Using the commonly employed hemispherical resonator in the second bending mode as an example, this study offers a complete solution with “arbitrary constant” properties. Numerical calculations indicate that the proposed complete solution achieves an accuracy of 10 ng, and uncertainties in the magnitude of corrections do not lead to divergence. The new theory and method eliminate coupling issues inherent in traditional n-symmetry removal algorithms, reduce the number of balancing iterations, and facilitate quick, efficient, and accurate balancing of high-precision hemispherical resonators.