The Influence Mechanism and Identification Method of the First Four Harmonics of the Mass Defect of Hemispherical Resonators

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript investigates the influence of the first four harmonic components of mass defects on the dynamic characteristics (quality factor and frequency split) of hemispherical resonators. These are critical components in Hemispherical Resonator Gyroscopes (HRGs). The authors present a shell-rod coupling vibration model that captures the effects of mass error and derive analytical expressions. They do this by linking the harmonic mass defects to resonator performance. The theoretical findings are validated through finite element simulations, and an experimental identification system is proposed that synchronously measures displacements across multiple points on the shell rod. The method promises improved testing efficiency and identification accuracy. I believe the shell-rod coupling vibration model offers a new framework for characterizing the impact of harmonic mass defects. The analytical predictions, simulation and experimental results are in good agreement. The figures and tables are necessary, and they are very beneficial. Experimental setup is also presented well and clear. The authors could clarify what “first four harmonic components” precisely refer to, perhaps via a brief equation or illustration. The concluding paragraph mentions future work such as “distribution cloud map” and “joint tuning.” I would recommend the authors consider briefly explaining the feasibility or preliminary steps toward these goals. The literature is not presented extensively. It could be improved. The similarity rate might look high at first glance. But I checked the report in detail, and found no indicator of any issue. I recommend Accept with minor revisions.

Author Response

Comment 1: " The concluding paragraph mentions future work such as “distribution cloud map” and “joint tuning.” I would recommend the authors consider briefly explaining the feasibility or preliminary steps toward these goals. "

Response:
We have added a brief explanation regarding the feasibility of the proposed future work ('distributed cloud map' and 'joint tuning') as suggested by the reviewer.

 

Comment 2: " The literature is not presented extensively. It could be improved. "

Response:
We have expanded the literature review section to provide a more comprehensive analysis of relevant works.

Reviewer 2 Report

Comments and Suggestions for Authors

The article deals with developing an analytical approach to study the impact of eigenfrequencies on hemispherical resonators. The title is essential, especially in inertial navigation systems and space technologies.

The proposed integration is useful, but not new from a theoretical point of view. It rather extends existing models.

Moreover, due to the extreme significance of the problem studied, the following issues should be addressed:

1. The Abstract is poor. It should be extended by quantitative indicators.
2. In the Literature Review, a state-of-the-art analysis of machine learning and ANN approaches should be added.
3. Formulas (1) and (2) are well-known from the theory of elasticity. So, please add references. Also, please add explanations, e.g., Cauchy’s equations in spherical coordinates (1) and Hooke’s law in the inverse form (2).
4. Formula (3) is the Kirchhoff–Love hypothesis. This should also be added to the text. Moreover, this hypothesis is only accurate for thin shells. So, this simplification should be substantiated, e.g., by the h/R ratio and the assumption that bending energy prevails over tensile energy.
5. The novelty lies mainly in combining the analysis of the four harmonics into a single model and attempting to identify them simultaneously. Simultaneously, it is helpful and practical, but has limited theoretical innovation.
6. Due to the hardware complexity, the actual integration of the beat frequency approach for gyroscope calibration remains quite questionable. A vacuum environment and high accuracy requirements are also needed for such systems.
7. Please indicate the novelty more clearly.
8. It was not proven why the first four harmonics are critical (neither less nor more).
9. It is recommended that the approach’s advantages be substantiated by considering case studies.
10. It is worth visualizing the main advantages of the proposed approach compared to existing ones.
11. Conclusions are too general due to a lack of quantitative indicators to highlight the comparative benefits of the proposed approach.
12. The English language requires a minor correction due to syntax errors, direct translation from the Chinese language, repetition of phrases, etc.

Author Response

Comment 1: " The Abstract is poor. It should be extended by quantitative indicators."

Response:
We have enhanced the Abstract with quantitative indicators as suggested.

 

Comment 2: " In the Literature Review, a state-of-the-art analysis of machine learning and ANN approaches should be added."

Response:
We appreciate the reviewer's suggestion regarding machine learning (ML) and artificial neural network (ANN) approaches. However, our work focuses specifically on physical modeling, theoretical analysis and identification of mass defects rather than data-driven methods. To our knowledge, no existing literature in related field has applied ML/ANN to this problem due to lack of training data. We have therefore maintained the current scope of our literature review.

 

Comment 3: " Formulas (1) and (2) are well-known from the theory of elasticity. So, please add references. Also, please add explanations, e.g., Cauchy’s equations in spherical coordinates (1) and Hooke’s law in the inverse form (2)."

Response:

We have added references to formulas (1) and (2) and included explanations for them.

 

Comment 4: " Formula (3) is the Kirchhoff–Love hypothesis. This should also be added to the text. Moreover, this hypothesis is only accurate for thin shells. So, this simplification should be substantiated, e.g., by the h/R ratio and the assumption that bending energy prevails over tensile energy."

Response:

We have clarified in the text that Equation (3) is derived from the Kirchhoff–Love hypothesis, valid for thin shells under the condition of small deformations, where bending energy dominates tensile energy. Additionally, we explicitly state that Lord Rayleigh’s inextensional theory applies, as the middle surface strains are negligible.

 

Comment 5: " The novelty lies mainly in combining the analysis of the four harmonics into a single model and attempting to identify them simultaneously. Simultaneously, it is helpful and practical, but has limited theoretical innovation."

Response:

We appreciate the reviewer’s insightful assessment of our work.  We agree that the theoretical innovation of this study is somewhat limited, as its primary contribution lies in the methodological advancement of integrating the analysis of the first four harmonics into a unified framework. 

 

Comment 6: " Due to the hardware complexity, the actual integration of the beat frequency approach for gyroscope calibration remains quite questionable. A vacuum environment and high accuracy requirements are also needed for such systems."

Response:

We appreciate the reviewer's valid concerns regarding system implementation challenges. Our experimental configuration was specifically designed to meet the stringent requirements of the beat frequency method, as documented in Section 4.1 of the manuscript. The system's capabilities include:

Ultra-high vacuum environment: Maintained at 1×10⁻⁴ Pa;

Advanced vibration isolation: Electronically-controlled active isolation platform (WAVE Duo100);

Precision motion control: High-precision turntable (FMSR100V) with±0.01° bidirectional repeatability and <0.005° RMS motion error;

Nanometer-scale metrology: Laser Doppler vibrometer (Polytec Vibroflex) with <0.1 nm displacement resolution.

While we acknowledge the system's complexity, all components are commercially available and the configuration has proven reproducible in our lab. 

 

Comment 7: " Please indicate the novelty more clearly."

Response:

We thank the reviewer for the opportunity to clarify our core innovations:

• Characterization model for hemispherical resonators with mass defects is developed.
• Explicit formulas for support loss, frequency split are derived.
• Finite-element simulations with perfectly matched layers to corroborate the analytical model.
• Innovative beam-split optical system that streamlines displacement metrology, enabling synchronous tri-directional displacement measurements and the synchronous identification of the first four harmonic components.

 

Comment 8: " It was not proven why the first four harmonics are critical (neither less nor more)."

Response:

We have addressed the importance of the first four harmonic errors in the Introduction section. The first four harmonic components of mass imbalance have the most significant impact on the resonator's performance, while higher-order harmonics exhibit relatively smaller effects. This point has also been well documented in references [26-28].

 

Comment 9: " It is recommended that the approach’s advantages be substantiated by considering case studies."

Response:

In previous studies, multiple sensors (e.g., laser vibrometers) were required to measure vibrations in different directions, leading to more complex procedures. Switching or repositioning these sensors could also introduce disturbance errors. In contrast, the proposed mirror-based beam-splitting optical setup expands the laser path of a single sensor, enabling simultaneous multi-directional displacement measurements at multiple locations on the shell-rod structure. This approach improves testing efficiency while avoiding disturbance errors caused by non-synchronous measurements, thereby enhancing identification accuracy.

 

Comment 10: " It is worth visualizing the main advantages of the proposed approach compared to existing ones."

Response:

We have visualized the key advantages of our proposed method through a comparative analysis in Table 3, which clearly highlights the superior performance of our approach compared to conventional methods.

 

Comment 11: " Conclusions are too general due to a lack of quantitative indicators to highlight the comparative benefits of the proposed approach."

Response:

We sincerely appreciate the reviewer's constructive suggestion regarding quantitative comparisons. However, we have chosen to retain the original conclusions for the following reasons:

1. i) The core innovation lies in the design of the mirror optical system that expands the sensor's laser path, enabling synchronous multi-directional displacement measurements at multiple locations on the shell-rod structure - rather than being an improvement upon existing methods.
2. ii) The novel nature of this experimental setup makes it difficult to quantify using conventional performance metrics.

 

Comment 12: " The English language requires a minor correction due to syntax errors, direct translation from the Chinese language, repetition of phrases, etc."

Response:

We have carefully revised the manuscript to address grammatical errors, awkward phrasing from direct Chinese-to-English translation, and repetitive expressions. The language has been thoroughly polished to improve clarity and readability.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have provided a revised version of the manuscript prepared considering most (but not all) of the reviewer's comments and suggestions. As a result, the Abstract has been clarified by quantitative indicators. The Literature Review has also been slightly extended. The governing equations and their limitation have been explained accordingly. The research methodology and results have been improved significantly. The Conclusions have been rewritten. So, the article can be recommended for publication. The English language has also been improved.

However, please pay attention that the novety of "Finite-element simulations with perfectly matched layers" (as answered on the 7th comment) is doubtful.

Answer to the question 8 is quite unacceptable because it only based on the references. However, the authors could provide more deep analysis, e.g., discussing about first 4 modes as follows:

• the main mode (n = 1-2), which consists of two orthogonal forms (two natural frequencies that are almost the same);
• to simulate accuracy and sensitivity, another pair of modes (n = 3-4) is added. These additional modes are considered for the effect of asymmetry, mode coupling, damping and perturbation effects (temperature drift, fabrication imperfection), and modal stability analysis. For example, in the Modal Reduction / Modal Superposition method, 4 modes provide a compromise between accuracy and computational efficiency. More modes are more accurate, but more difficult to calculate, so the minimum that ensures adequate system behavior is chosen. Nevertheless, if any pair of modes has frequency splitting or shape distortion, this indicates a defect or inaccuracy in manufacturing.

Hope this recommendation will allows authors to improve their article during a final proofreading.