Fractional Kelvin–Voigt Model for Beam Vibrations: Numerical Simulations and Approximation Using a Classical Model

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I have completed your review of the paper. The review report is attached.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

No

Author Response

I sincerely appreciate your thorough review and constructive feedback. Please find my detailed responses to all comments and suggestions in the attached PDF document.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The author has proposed a model based on Fractional Kelvin-Voigt to simulate the vibrations of a cantilever beam with a tip mass. The topic is of high interest, and the manuscript is well written with enough details. The results show the accuracy of the proposed model, and the conclusions are validated with the results. Overall, the manuscript has a high merit and could be published given some minor modifications as follows:

1. The abstract can include some quantitative results, such as the proposed model's accuracy compared to other classical and numerical approaches.
2. The proposed model is an analytical/numerical approach for a purely mechanical problem, which is not highly related to the journal "Electronics". The relevance is only mentioned in the first sentence of the introduction, which mentions piezoelectric harvesting. The author may need to expand the literature review to that specific application and focus on the relevance of the vibrations to piezoelectric harvesters, so that it could be more relevant to the journal of choice.
3. In line 102, "Bending moment" is misspelled. Please correct.
4. In Table 1, you may need to add a column showing the discrepancies in the results of the geometric and GA approaches for a better comparison.
5. The author may also mention the computational cost of his approach compared to other classical or numerical approaches, so the readers can have an idea of which approach is more suitable for their resources.
6. Again, in the conclusions section, the author may need to add some quantitative results by showing the accuracy and computational cost compared to other approaches.

Author Response

I sincerely appreciate your thorough review and constructive feedback. Please find my detailed responses to all comments and suggestions in the attached PDF document.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Dar Author,

In my opinion, this is a very good work. The research is relevant, and it is well presented. However, for its better valorization, it could be subject to improvement in the Abstract and Conclusion. The abstract has to reveal the main achievements of the research, so it raises the interest of the readers to read the entire article.  The conclusion is supposed to give a comparison of the two methods used on different criteria, other than the accuracy of the results. It is important to foresee a future continuation of the research to get better results in the proximity of the resonance (𝜔/Ω=1).

Further remarks on the article can be found in the attached file.

Comments for author File: Comments.pdf

Author Response

I sincerely appreciate your thorough review and constructive feedback. Please find my detailed responses to all comments and suggestions in the attached PDF document.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript presents an interesting approach and is generally well-written. I’ve outlined a few suggestions.

1. The Wynn-ε acceleration technique is important to your convergence but is briefly explained (line 154-158). Consider adding a sentence or two to briefly explain its role and why it's especially effective in this problem.
2. Variables like M₁, K₁, z_j⁽¹⁾, m_k⁽¹⁾ are not defined. Please ensure every symbol used is clearly defined at its first appearance.
3. Avoid use “Subfigures labeled (a), (b)” as it does not indicate which specific figure number, try to use "Figure 2(a)", "Figure 2(b)".

Author Response

I sincerely appreciate your thorough review and constructive feedback. Please find my detailed responses to all comments and suggestions in the attached PDF document.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The author revised the paper in accordance with most of the review opinions. However, the latest important articles recommended were still not cited. For instance,  Fractal and Fractional, 2024, 8(2): 111. doi: 10.3390/fractalfract8020111; 
Advances in Continuous and Discrete Models, 2024, 2024(1): 5. doi: 10.1186/s13662
024-03801-y

 

Author Response

Thank you very much for your valuable comments and for pointing out the need to include recent relevant literature. I appreciate the recommendation of the articles Fractal and Fractional, 2024, 8(2):111 (doi:10.3390/fractalfract8020111) and Advances in Continuous and Discrete Models, 2024, 2024(1):5 (doi:10.1186/s13662024-03801-y).

In response, I have added a citation of the first article in the revised manuscript (see lines 52–53, in red) to further strengthen the literature review. I would like to note that, as part of my previous revision, I had already addressed this general remark by enhancing the introduction (lines 47–52), where I included references [12] and [13]: one being a recent monograph on advances in fractional-order derivatives, and the other a fresh review article summarizing key developments in the field.

I hope this clarifies the revisions and meets the expectations.