Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript focuses on the correlation between the FKV oscillator and the classical harmonic oscillator. The proposed classification criteria and equivalent parameter formulas can be applied to the vibration analysis of viscoelastic structures and material characterization. The research presented in the manuscript has a clear significance, strong practicality, a scientific and rigorous methodology. However, it still needs to clarify the following issues:

• Both the core oscillation classification criteria and equivalent parameter formulas are derived based on data fitting. They have not been deduced from basic principles such as fractional calculus and viscoelastic mechanics, resulting in a lack of physical interpretation.
• The parameter μ is missing in equations (1) and (2), yet this symbol is explained in the subsequent text. These two equations require further verification.
• The format of the references is inconsistent. For instance, the title of Reference 2 is in italics while the journal name is in upright font, which does not conform to the format of other references.
• The in line 164 should be replaced by , and the meaning of latter needs to be explained.
• The author is suggested to carefully checked the equations in lines 175 and 182. Why are they separated?
• Section 2.1 is too long so that it is better to reconstruct this section.
• Two different sets are examined. Please explain the reasons selecting these two groups of data, and what’s the case if changes from 0 to 1?
• The aims and scope of the study should given in the introduction, but not as a section near the middle of the work.

Author Response

The responses to the reviewer’s comments are provided in the attached PDF document.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper aims to provide a set of practical, empirical tools for the analysis of the Fractional Kelvin-Voigt (FKV) oscillator. The work focuses on two primary aspects: 1) establishing a criterion to distinguish between oscillatory and non-oscillatory behavior, and 2) deriving empirical formulas to determine the parameters of a classical harmonic oscillator that best approximates the FKV oscillator. Through numerical simulations, genetic algorithm optimization, and data fitting, the author proposes a series of empirical formulas and validates their effectiveness. The core contribution of the research lies in offering a simplified method for the vibration analysis of complex viscoelastic structures. In general, the paper is well structured and is also well presented. I would like to recommend this paper to be published with considering the following comments:

1. The formulas for the equivalent harmonic oscillator parameters (Eq. 23, 24) and the oscillation classification functions (Eq. 19, 20) proposed in the paper appear to be empirical formulas derived directly from fitting multiple datasets, and they lack a clear physical interpretation. Although the author performed some random sampling validation outside the scope of the original database, this does not fully guarantee the generalizability of these formulas to a broader range of conditions.
2.Figures 6 and 7 present a comparison of curves corresponding to different formulas, but an error analysis is missing. The inclusion of an error analysis would offer a more intuitive comparison and allow for a more direct assessment of the strengths and weaknesses of each formula.
3. The criteria for distinguishing between oscillatory and non-oscillatory behavior lack precision. Classifying the system's behavior is a central concept of this paper, and the author employs a geometric criterion based on the zero-crossings of the solution's components. However, this criterion appears to be imprecise. In the conclusion, the author concedes that this criterion is not entirely reliable, and in the validation section, the misclassification rate near the boundary reaches as high as 30%.

Author Response

The responses to the reviewer’s comments are provided in the attached PDF document.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript presents a well-structured and technically sound investigation into the dynamics of fractional Kelvin-Voigt (FKV) oscillators, with a focus on establishing empirical criteria for oscillatory behavior and deriving equivalent harmonic oscillator parameters. The study addresses a relevant gap in viscoelastic modeling by providing practical tools for approximating fractional systems with classical models, supported by extensive numerical validation. The methodology is rigorous, and the results are clearly presented. Before considering for publication, revisions are still needed to further improve the overall quality of the manuscript.

1. While the switch from Caputo to Riemann-Liouville derivatives due to non-oscillatory initial behavior is justified, the manuscript should explicitly discuss whether this choice affects the generalizability of results to real-world applications where Caputo derivatives are often preferred for physical interpretability.
2. Algorithm 1 uses a tolerance of ε= 0.05 for solution convergence. Please justify this threshold with quantitative error analysis. Additionally, it is better to report computational time savings relative to the Wynn-epsilon method in absolute terms rather than ratios.
3. I suggest to add “International Journal of Mechanical Sciences 301 (2025) 110503” and “Energy 233 (2021) 121146” to the references list.
4. The empirical criterion F(μ, α, ω) < 0 performs well but misclassifies ≈30% of cases near the boundary surface. The authors should quantify this error using standard metrics (such as F1-score) in the extended validation. Also, please discuss how solutions like Figure 3(b), where only one component crosses zero, challenge the binary classification.
5. Formula (24) for ξ_m fits data admirably but lacks physical justification. It is better to relate its functional form to known asymptotic behaviors. Further, please clarify why ξ_m≤0.5 in all valid cases. Is this inherent to FKV dynamics or an artifact of the fitting scope?

Author Response

The responses to the reviewer’s comments are provided in the attached PDF document.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The aim of this paper is to develop practical tools for such analysis by deriving approximate formulas that relate the parameters of an FKV oscillator to those of a best-fitting harmonic oscillator. The Reviewer has some suggestions and clarifying questions to improve the manuscript and make it more readable and understandable for readers:

 

 

1). What are the substantial results of this research (approach, model, method or et cetera)?

 

2). I would like to see a more detailed description of numerical methods and techniques. Which scheme (explicit or implicit) was used, the number and size of cells, the degree of accuracy, etc.?

 

3). Could you please summarize advantages and disadvantages of a fractional Kelvin–Voigt (FKV) oscillator, used in your work with others?

 

4). What about verification?

 

5). Authors are advised to add a few recent papers,  related to the topic, example gratia:

 

Gritsenko, D.; Paoli, R. Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: General Solutions. Appl. Sci. 2020, 10, 9093. https://doi.org/10.3390/app10249093

 

Reviewers comment: minor revision

Author Response

The responses to the reviewer’s comments are provided in the attached PDF document.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have addressed all of issues raised by the reviewer, and therefore I recommend acceptance for publication.