Research on the Dynamic Response of a Cracked-Spur Gear System with Parameter Uncertainty
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsPlease refer to the attachment “machines-3556513-Review-R1”for details
Comments for author File: 
Comments.pdf
Author Response
Response to editor and reviewers
Dear editor and reviewers,
Thank you for your letter and for the thoughtful and constructive comments provided by the reviewers. These comments have been highly valuable in helping us improve the quality and clarity of our manuscript, as well as guiding the development of our research. We have carefully revised the manuscript based on all the suggestions, including improvements to grammar, spelling, and sentence structure. For your convenience, the revised manuscript has been edited using the track changes feature, and detailed point-by-point responses are provided below.
Comment 1: Fig. 4 and Fig. 5 aim to demonstrate the consistency between the potential energy method and the finite element method. However, a visual comparison alone is insufficient to rigorously establish their agreement. It is recommended to incorporate quantitative metrics to enhance the verification of their agreement.
Response: Thank you for your valuable comment. To enhance the rigor of the comparison between the potential energy method and the finite element method, we have incorporated quantitative metrics for the analysis of mesh stiffness. The specific results are presented in Table 1, which provides a more robust validation of the consistency between the two methods. The modification is as follows:
Table. 2 Comparison between FEM results and analytical results under different crack conditions
|
Condition |
FEM (108 N/m) |
Proposed methed (108 N/m) |
Difference (%) |
|
Health |
2.832 |
2.695 |
4.8 |
|
q=2mm, =45° |
2.761 |
2.644 |
4.2 |
|
q=3mm, =45° |
2.662 |
2.548 |
4.2 |
|
q=4mm, =45° |
2.554 |
2.407 |
5.7 |
|
q=3mm, =30° |
2.745 |
2.632 |
4.1 |
|
q=3mm, =60° |
2.576 |
2.426 |
5.8 |
Table. 2 presents the average values and comparative results obtained from the FEM and analytical methods under various crack conditions. The mesh stiffness over a single tooth engagement cycle is calculated using a weighted averaging approach. The close agreement between the two sets of results confirms the accuracy and reliability of the proposed model.
Comment 2: The Chebyshev polynomial method is employed to address parameter uncertainty. However, can the errors introduced by this approach be considered negligible? Please provide clarification with supporting analysis or references.
Response: Thank you for your valuable comment. We have added relevant formulas in the manuscript to illustrate the potential approximation error introduced by the Chebyshev polynomial method. The analysis demonstrates that the error remains sufficiently small under the conditions considered in this study and can be regarded as negligible to a certain extent. This supports the reliability and applicability of the Chebyshev approach for handling parameter uncertainty in the proposed gear system model. The modification is as follows:
|
(40) |
It can be seen from the above expression that the error decreases with increasing order n, and can be neglected to some extent.
Comment 3: You mentioned that the theoretical model was validated through experiments. Were any noise suppression measures implemented during the experiment? Please elaborate on this aspect.
Response: Thank you for your valuable comment.During the experiment, a relatively controlled and stable environment was selected to minimize external disturbances and human-induced interference. In addition, shock absorption devices were installed on the test bench to suppress mechanical vibrations and noise. Multiple measurements were conducted throughout the experiment to ensure the repeatability and reliability of the results.
Final Remarks:
We sincerely appreciate the reviewers’ time and effort in evaluating our manuscript. We believe that the revisions made in response to the comments have significantly improved the quality of the paper. We hope that the revised version meets the requirements for publication in your esteemed journal.
Thank you again for your consideration.
Best regards,
Yang Yang
On behalf of all authors
Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsHello. Congratulations on the results obtained. For a better understanding of the article, please take into account the following observations and recommendations.
Percent match: 34% iThenticate report. Seems a bit high to me, I'm not sure what the magazine's policy is on this.
To increase interest in reading the article, the abstract must also include some concrete results.
Congratulations on the way you focused the information from the bibliographic references. At the end of this chapter you described the structure of the article starting with section 2 (section 1 is Introduction???).
In Figure 2, please explain the variable used XQ in relation to XF.
You need to put the dimensions hB, hC on figure 2.
Please tell me if you have used any mesh validation method.
I recommend introducing an execution drawing for the 2 gears.
I recommend providing results obtained directly from the FEM software.
The crack depth is set to 3 mm but nothing is known about the size of the tooth so that we can form an overall picture.
Author Response
Response to editor and reviewers
Dear editor and reviewers,
Thank you for your letter and for the thoughtful and constructive comments provided by the reviewers. These comments have been highly valuable in helping us improve the quality and clarity of our manuscript, as well as guiding the development of our research. We have carefully revised the manuscript based on all the suggestions, including improvements to grammar, spelling, and sentence structure. For your convenience, the revised manuscript has been edited using the track changes feature, and detailed point-by-point responses are provided below.
Comment 1: Percent match: 34% iThenticate report. Seems a bit high to me, I'm not sure what the magazine's policy is on this.
Response: Thank you for your valuable comment. We acknowledge that the similarity index reported by iThenticate is 34%. However, after carefully reviewing the report, we found that the majority of the matched content originates from commonly used technical terminology, standard methodological descriptions, and citations of well-established theories or equations. We have revised the manuscript accordingly to reduce the similarity and improve originality where possible.
Comment 2: To increase interest in reading the article, the abstract must also include some concrete results.
Response: Thank you for your valuable comment. We have revised the abstract and incorporated several key results from the study to enhance its informativeness and increase reader interest. The modification is as follows:
In this paper, the time-varying mesh stiffness (TVMS) of the gear is meticulously derived using the potential energy method (PEM), and an analytical expression for it is obtained. Subsequently, cal-culations are performed to determine the effects of crack depth and crack angle on the TVMS. The validation is carried out using the finite element method (FEM). Then a discussion is carried out on the dynamic characteristics of the spur gear system with a crack. Moreover, uncertainty is an objective reality in gear systems, arising from various factors such as material properties and the working environment. To enable a more reasonable evaluation of the dynamic characteristics of the spur gear system, this paper presents a deviation of an uncertainty interval analysis method based on Che-byshev polynomials. A dynamic model of the spur gear system with uncertain parameters is then proposed. The dynamic response of the gear transmission system with these uncertain parameters is investigated in detail. Additionally, a further examination is conducted on the interval response of the uncertain gear system with root cracks. The experimental results confirm the inherent presence of uncertainty in the gear system and validate the effectiveness of the proposed uncertainty analysis method.
Comment 3: Congratulations on the way you focused the information from the bibliographic references. At the end of this chapter you described the structure of the article starting with section 2 (section 1 is Introduction???).
Response: Thank you for your valuable comment. You are correct in pointing out that Section 1 corresponds to the Introduction. We have accordingly revised the relevant paragraph to explicitly incorporate Section 1 in the overall outline of the article’s structure. The modification is as follows:
The structure of this paper is organized as follows: In Section 1, the research on the TVMS of gear pairs and the studies on gear systems with uncertainty factors are introduced.
Comment 4: In Figure 2, please explain the variable used XQ in relation to XF.
Response: Thank you for your valuable suggestion. The symbol ‘Q’ has been revised to ‘E’ to maintain consistent and continuous numbering.
Comment 5: You need to put the dimensions hB, hC on figure 2.
Response: Thank you for your valuable suggestion. The parameters hB and hC have been added to Figure 2. The modification is as follows:
Figure 2. Schematic diagram of a cracked gear tooth model.
Comment 6: Please tell me if you have used any mesh validation method.
Response: Thank you for your valuable suggestion. Yes, a mesh independence verification was conducted. In the finite element analysis, several mesh densities were tested. The mesh presented in the manuscript was found to be sufficiently accurate for the purposes of the analysis.
Comment 7: I recommend introducing an execution drawing for the 2 gears.
Response: Thank you for your valuable suggestion. Figure 6 presents a simplified schematic of the gear system. In the experiment, due to the enclosed nature of the gearbox, only a top view of the gear pair meshing is available and illustrated.
Comment 8: I recommend providing results obtained directly from the FEM software.
Response: Thank you for your valuable suggestion. As previously stated in the manuscript, a static analysis approach was used to obtain the angular displacement . Since the focus of the study is on stiffness validation, was subsequently converted into stiffness values. The results obtained from the FEM calculations are presented in Figures 4(b) and 5(b).
Comment 9: The crack depth is set to 3 mm but nothing is known about the size of the tooth so that we can form an overall picture.
Response: Thank you for your valuable suggestion. For the cracked tooth, when the crack angle is 45° and the crack root is located exactly at the centerline of the gear, the corresponding crack depth q is approximately 5.4 mm. In the manuscript, the stiffness values for crack depths of q=2 mm, q=3mm q=4mm are provided. According to the Dynamic Transmission Error (DTE) results, a crack depth of 3 mm does not yet induce significant periodic impacts on the system, indicating that it represents a moderate level of damage. Therefore, this crack depth was selected for further analysis in the subsequent study.
Final Remarks:
We sincerely appreciate the reviewers’ time and effort in evaluating our manuscript. We believe that the revisions made in response to the comments have significantly improved the quality of the paper. We hope that the revised version meets the requirements for publication in your esteemed journal.
Thank you again for your consideration.
Best regards,
Yang Yang
On behalf of all authors
Author Response File: Author Response.pdf
