A Novel Multi-Objective Dynamic Reliability Optimization Approach for a Planetary Gear Transmission Mechanism

Comments 1: Specifying the standard used in gear strength calculations.

Response 1: Thanks for your comment and we have specified the standard used in gear strength calculations and listed the references. (please see line 198 and line 204 page 7 and the references 25-26)

Comments 2: In case of using ISO 6336 or DIN 3990 then the coefficients KA, KV, ..., ZH, ZE, etc. should be renamed to factors, i.e. Application factor, Dynamic factor, Zone factor, Elasticity factor.

Response 2: Thanks for your advice and we have renamed the factors K_A, K_V, ..., Z_H, Z_E, etc. (please see Table 1 and lines 174 and 177 page 6)

Comments 3:  If the "Precision Level 6" in table 2 refers to the precision of the gears then the standard notation should be "Gear Accuracy Grade" whose value again depends on the standard used.

Response 3: Thanks for your advice and we have renamed it and presents the standard used. (please see Table 2 and line page)

Comments 1:  I am not sure what qualifies this paper for the journal Axioms. This aspects needs to be addressed directly in the paper and it must be clear to an interested reader.

Response 1: Thank you for pointing this out. We agree with this comment.

After re-examining the positioning and objectives of the journal Axioms, we realized that our paper was indeed highly relevant to the scope of the journal. We contributed to the special issue of Reliability and Risk of Complex Systems: Modelling, Analysis and Optimization. In our study, the reliability analysis and optimization of the planetary gear transmission mechanism with multiple meshing gear pairs are investigated. First, the calculation method of dynamic reliability is proposed considering the strength degradation. Second, a multi-objective optimization mathematical model was established. Third, the MOEA/D algorithm with good performance in the field of multi-objective optimization was selected. Finally, an improved strategy was proposed to improve population diversity and avoid falling into local optimal. Compared with the results of the previous design scheme, the proposed method is helpful to realize the lightweight of the gear and prolong the service life. Provide guidance on the gear design and maintenance process.

In order to better demonstrate the correlation between this paper and Axioms, we made the following additions:

In the Introduction, we have highlighted the shortcomings of existing research in reliability design and optimization, and highlight the contribution of this paper (please see lines 94-99 page 2).

In the Model construction, we have introduced the computational difficulties of the model. (please see lines 178-183 page 6)

Comments 2: The quality of English language us generally ok, but some improvements are needed. For instance, some sentences need to be reformulated. A couple of examples are given below:

“For a PGTM, there are many design parameters, only optimizing the tooth number, modulus and tooth width will lead to incomplete performance optimization.” (this one should be divided into 2 sentences.)

“The random forest model consists of four main parameters of, the number of trees (?_??????????), the maximum depth of the trees (???_????ℎ), the minimum number of samples required to split an internal node (???_???????_?????), the minimum number of samples required to be at a leaf node (???_???????_????).”

Response 2: Thanks for your comment and we have revised these sentences and checked the manuscript with the help of a native speaker and revised the whole. (please see lines 144-145 page 5, lines 255-257 page 9, and lines 77-80 page 2, lines 246-248 page 8 etc.)

Comments 3: Some recent work related to planetary gears should be mentioned in the introduction, such as:

https://doi.org/10.22055/jacm.2024.46280.4490

https://doi.org/10.22190/FUME191129013T

Similarly, a recent work on application of particle swarm algorithm for multi-objective optimization of engineering structures is worth of being included in the literature overview:

https://casopisi.junis.ni.ac.rs/index.php/FUMechEng/article/view/10976

Response 3: Thanks for your comment and we have added these literatures. (please see lines 38-43 page 1, lines 91-93 page 2 and the references [5,6,24])

Comments 4: The authors set the objective to maximize transmission efficiency and reliability while minimizing volume. Why exactly those objectives have been chose? Also, this seto ob objectives has been transformed to minimizations as given by Eq. (6). This should be explicitly emphasized in the text.

Response 4: Thanks for your comment. We have explained why we choose those objectives. (please see lines 149-153 page 5). And we have emphasized why and how to transform the objectives to minimizations. (please see lines 160-165 page 5 and lines 1-2 page 6)

Comments 5: It is important to explain in the text clearly all the terms used in the equations. The authors have provided Table 1, which serves as nomenclature, but it still seems that certain terms have not been explained. This is also valid for the objective functions, which are simply denoted as f1, f2 and f3, without specifying which one is which. It seems that the order of giving those functions in equations does not correspond to the order in which they are mentioned prior to Eq. (2)

Response 5: Thanks for your comment. We have adjusted the order of f1,f2,f3 in paragraphs and to match the order of Equations, and added undefined terms in Table 1, such as  and . (please see Table 1)

Comments 6:  What exactly is means by “dynamic reliability”? Do the authors refer to reliability due to dynamic loads? In line 194, the authors write “unstable loads”. What is meant by that? I still assume those are dynamic loads, which is not the same as unstable loads.

Response 6: Thanks for your questions and comments. The dynamic reliability here refers to that the gear is exposed to dynamic loads and other environments, which causes the strength of the gear to decline over time. The reliability also declines with the degradation of strength, so it is defined as the dynamic reliability. Unlike static reliability, dynamic reliability emphasizes how the system changes over time. In this study, a method for calculating residual strength is proposed. (please see lines 125-127 page 3, lines 208-211 page 7,Equations 24-26 and lines 222-227 ). And we have renamed “unstable loads” to “dynamic loads”. (please see line 210 page 7).

Comments 7:  How was the selection of multi-objective evolutionary algorithms done? Why those and not some other algorithms?

Response 7: Thanks for your comment. According to the previous studies, multi-objective evolutionary algorithms based on decomposition (MOEA/D) have been widely applied to multi-objective optimization problems due to its strong search capability, high robustness, and scalability [33-35]. Differential evolution algorithm and decomposition based multi-objective evolution algorithm have stable performance in the multi-objective optimization problems, especially for discrete problems. When exploring pareto solutions, they have better development performance and convergence performance. Therefore, we choose the basic MOEA/D [33] and differential evolution algorithm (DE) which performs well in multi-objective optimization problems[38] as comparison. And we have explained why we choose these algorithms in the manuscript. (please see lines 308-310 page 11 and lines 505-510 page 19)

Comments 8: Number of diagrams use a too small font size. The readability is strongly deteriorated and needs to be improved. Some diagrams miss denotation of axis. Some other diagrams (where applicable) miss units.

Response 8: Thanks for your comment, we have modified these diagrams. (please see Figure 9 and Figure 13).

Comments 9: The conclusions mention some limitations of the suggested approach and mention that those should be addressed in the future work. I suggest to add a couple of sentences mentioning the ideas on how those limitations could be addressed, by which methods (without going into any further details).

Response 9: Thanks for your comment. We have added a couple of sentences mentioning the ideas on how those limitations could be addressed. (For example, the dynamic stress of the gear during operation can be simulated by finite element analysis. Then statistical methods, such as rain flow counting can be used to analyze load spectrum data.) (please see lines 549-552 page 20)