Dynamic Modeling and Coupling Error Analysis of Imperfect Hemispherical Resonators

The hemispherical resonator gyroscope (HRG) represents a new type of inertial sensor. The vibrational properties of the hemispherical resonator, a core component of the HRG, significantly influence the output accuracy of the gyroscope. This article is the first to integrate the effects of residual stress nonuniformity and mass nonuniformity in a unified model for a hemispherical resonator. First, the equation of motion for the resonator with these nonuniformities is derived based on the thin-shell theory and the Lagrange equation. Then, two important vibration parameters, including the frequency split and the azimuth of the stiffness axis, are calculated by a novel method based on the unified model. Next, detailed simulations are conducted, which reveal that the frequency split is directly proportional to the fourth harmonic magnitude of mass or residual stress nonuniformity alone, with the azimuth of the stiffness axis aligning with their initial phase (or differing by 22.5° for shear stress). Under coupled nonuniformities, the frequency split, however, depends nonlinearly on both fourth harmonics, while the azimuth of the stiffness axis exhibits complex nonlinear behavior, with coupling strength critically influenced by their phase difference. Finally, based on the least squares method, an identification scheme for the error parameters associated with residual stress nonuniformity is designed. The experimental results show good consistency with the theoretical predictions, which confirms the correctness of the dynamic model and error analysis. This study addresses the practical limitations of existing models and provides a theoretical foundation and experimental guidance for the optimization of the manufacturing processes of high-precision HRG.