Free Vibration Characteristics Analysis of Metal-Rubber Cylindrical Shells Based on Viscoelastic Theory
Round 1
Reviewer 1 Report
- What is the subject of the research mentioned ?
- Why is it important to study the vibration characteristics of metal rubber cylindrical shells (MRCS)?
- Which methods were used to investigate the dynamic characteristics of MRCS in this research?
- How was the correctness of the proposed model verified?
- What were the main findings of the research?
- How did the Pasternak elastic foundation affect the natural frequency and modal loss factor of MRCS?
- What was the similarity between the effect of Pasternak elastic foundation and the artificial spring technique on higher order vibrations?
Author Response
- What is the subject of the research mentioned ?
Author response: We express our gratitude to the reviewers for their valuable questions, which helped to clarify the main idea of this paper. The objective of this study is to examine the vibration characteristics of a cylindrical shell made of metal-rubber and to analyze how boundary conditions and preload states impact these characteristics. The abstract and introduction of the article provide an overview of the topic, highlighted in red for emphasis and clarity.
- Why is it important to study the vibration characteristics of metal rubber cylindrical shells (MRCS)?
Author response: We would like to express our appreciation to the reviewers for their valuable and constructive questions, which contribute to the research background of this paper. Our belief is that investigating the vibration characteristics of metal-rubber cylindrical shells can serve as a useful reference for designers of metal-cylindrical shells. This study can enhance the rationality of the design process, and prevent abnormal vibrations caused by design errors in metal-rubber cylindrical shells. Hence, studying the vibration characteristics of metal-rubber cylindrical shells is crucial.
- Which methods were used to investigate the dynamic characteristics of MRCS in this research?
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Author response: We extend our gratitude to the reviewers for their inquiries. This paper presents a study based on Rayleigh's method, first-order shear deformation theory (FSDT), and Schmidt's orthogonal polynomials, to establish the kinetic equations of metal-rubber cylindrical shells (MRCS). Firstly, we calculate the kinetic and potential energies of the MRCS using FSDT and Schmidt orthogonal polynomials. The kinetic equations of MRCS are obtained by applying Rayleigh's method. The nontrivial solutions of the kinetic equations are then used to calculate the intrinsic frequencies and modal loss factors of MRCS. Additionally, the mechanical properties of the metal-rubber are obtained through the spiral-angle cone model (IHPM). The second part of the paper provides a detailed explanation of the calculation method.
- How was the correctness of the proposed model verified?
Author response: Thanks to the reviewers for their questions. The accuracy of the model is crucial for this study. In section 3.1, the accuracy of the model is verified using the method of literature comparison. The results are presented in Table 1, which shows that the computational results of this paper are in high agreement with the results of the literature [30]. Hence, the calculation results of this paper are correct.
- What were the main findings of the research?
Author response: We express our appreciation to the reviewers for their helpful questions, which aided us in summarizing the paper. The primary conclusions of this study are as follows:
- 金属橡胶圆柱壳（MRCS）的固有频率和模态损耗因子受两端边界条件的影响，而两端边界条件不同。具体而言，两端的固定（C-C）边界条件导致最高的固有频率和模态损耗因子，而两端的自由（F-F）边界条件导致最低的固有频率和模态损耗因子。
- 帕斯捷尔纳克弹性基础对MRCS固有频率和模态损耗因子的影响随轴向半波数m的变化而变化。对于m<30，MRCS的固有频率随帕斯捷尔纳克弹性基础的刚度而增加，而模态损耗因子随帕斯捷尔纳克弹性基础的刚度而减小。当m>30时，MRCS的固有频率仅在帕斯捷尔纳克弹性基础刚度的有效范围内变化。当帕斯捷尔纳克弹性基础刚度低于或高于有效范围时，MRCS固有的频率和模态损耗因子保持不变。
- MRCS的内在频率随预位移的增加而增大，而MRCS的模态损耗因子随预位移的增加而减小。
上述结论在论文的结论部分提出，并以红色突出显示，以便于参考。
- 帕斯捷尔纳克弹性基础如何影响MRCS的固有频率和模态损耗因子？
作者回应：我们感谢审稿人的宝贵反馈和有见地的问题。我们的研究结果表明，帕斯捷尔纳克弹性基础对MRCS固有频率的影响取决于轴向半波数m。对于m < 30，MRCS 的固有频率随着 Pasternak 弹性基础刚度的增加而增加。然而，对于30< m<35，MRCS的固有频率仅在Pasternak弹性基础刚度的有效范围内变化，而在此范围之外，MRCS的固有频率几乎保持不变。此外，我们观察到MRCS的模态损耗因子随着Pasternak弹性基础刚度的增加而降低。这些结果在论文的讨论部分详细呈现，并用红色标记以便于识别。
- 帕斯捷尔纳克弹性基础和人工弹簧技术对高阶振动的影响有什么相似之处？
作者回应：我们感谢审稿人对本文的评论和宝贵意见。本文提出了人工弹簧技术，假设一组可调刚度弹簧支撑壳体，并调整弹簧的刚度值以实现不同的边界条件。人造弹簧对固有频率的影响具有有效范围。如果人造弹簧的刚度低于有效范围，则固有频率与无约束情况下的固有频率相似，并保持恒定。当人造弹簧的刚度高于有效范围时，固有频率与固定约束工况相似，并保持恒定。在有效范围内，随着刚度的增加，固有频率从无约束情况增加到固定约束情况。当轴向半波数为30 < m<35时，结果表明，帕斯捷尔纳克弹性基础对固有频率的影响与人工弹簧技术相同。
Reviewer 2 Report
The authors should add in the title viscoelastic.
The authors should determine pecisely the ranges of the validity of the relations
- you eliminated the thickness effects although you write about thick shells
-FSDT is not valid for thick structures - is it possible tou use it
The explanation of the above problems should be discussed in the review of the literature. A lot of those problems were presented in the Soviet Union literature.
Author Response
- The authors should add in the title viscoelastic.
Author response: We would like to thank the reviewers for reviewing the article and providing constructive comments, which have helped us a lot in our writing. As suggested by the reviewers, we have added "Viscoelasticity" to the title and marked it in red. The revised title is: "Free vibration characteristics analysis of Metal-Rubber cylindrical shells based on viscoelastic theory".
- The authors should determine pecisely the ranges of the validity of the relations。
Author response: We would like to express our gratitude to the reviewers for taking the time to review our manuscript and for providing us with positive feedback. We wholeheartedly agree with the suggestions made by the reviewers, as we believe that they have significantly contributed to the improvement of our paper. In this paper, the range of validity of the model is determined by the strain-based inclined helix pyramid (IHP) model, which is valid when the strain ε ≤ 2.5%. To illustrate this point, we have made certain modifications to our paper. Specifically, we have used the ratio of the pre-displacement to the thickness of the metal-rubber cylindrical shell (MRCS) δ/h as a measure of the preload state. By doing so, we were able to demonstrate that our research work was conducted within the validity range of the model. Please refer to the red-marked section in Section 3.2 of our paper for further details.
- you eliminated the thickness effects although you write about thick shells, FSDT is not valid for thick structures - is it possible tou use it？
Author response: Firstly, we would like to express our gratitude to the reviewers for their valuable comments and for reviewing our article. We have carefully considered the concerns raised and provided responses below, and we hope that the reviewers will find them satisfactory. Regarding the object of study, we focus on a medium thickness cylindrical shell in this paper. Specifically, the thickness range of 0the cylindrical shell is 0.05 < h/R < 0.167, and previous studies in the literature have demonstrated that FSDT theory is applicable in this range[1]. As such, FSDT theory is also used by many scholars to investigate the vibration characteristics of medium-thickness cylindrical shells [2-4]. Furthermore, we would like to explain why the effect of thickness is neglected in this paper in favor of adopting FSDT theory. This is because we have used a 17th order polynomial for the fitting of volume fraction, which would result in more than millions of computational terms if we were to use HSDT. As a result, the computational cost of using HSDT would be substantial and not practical for our study. Therefore, we have opted to use FSDT theory in our analysis. Thank you once again for your review, and please let us know if you have any further questions or concerns.
- The explanation of the above problems should be discussed in the review of the literature. A lot of those problems were presented in the Soviet Union literature.
Author response: We appreciate this suggestion. As per the reviewers' feedback, we have revised the paper's introduction and highlighted the sections of interest in red. Furthermore, we have cited the book "Design of Metal-Rubber Construction," which is a collaborative textbook between Samara Aviation University and Harbin Institute of Technology, Russia. This book contains numerous Soviet texts that are relevant to our paper. We hope that our revisions and citation will satisfy the reviewers.
References
- Reddy, J.N. Mechanics of laminated composite plates and shells: theory and analysis. CRC press: 2003, ISBN 0203502809.
- Van Dung, D.; Chan, D.Q. Analytical investigation on mechanical buckling of FGM truncated conical shells reinforced by orthogonal stiffeners based on FSDT. Compos Struct 2017, 159, 827-841, doi:https://doi.org/10.1016/j.compstruct.2016.10.006.
- Tornabene, F.; Liverani, A.; Caligiana, G. General anisotropic doubly-curved shell theory: A differential quadrature solution for free vibrations of shells and panels of revolution with a free-form meridian. J Sound Vib 2012, 331, 4848-4869, doi:https://doi.org/10.1016/j.jsv.2012.05.036.
- Madani, H.; Hosseini, H.; Shokravi, M. Differential cubature method for vibration analysis of embedded FG-CNT-reinforced piezoelectric cylindrical shells subjected to uniform and non-uniform temperature distributions. Steel Compos Struct 2016, 22, 889-913, doi:10.12989/scs.2016.22.4.889.
Reviewer 3 Report
Authors are encouraged to improve their work based on the following comments:
1. Symbols in formula (1) need to be consistent with symbols in formulas (6) and (16)
2. In formula (10): The expression for calculating Q11 and Q33 involving imaginary numbers i, it is necessary to cite references and explain symbols.
3. Some formulas need to cite references
4. Symbols should be explained after formulas
5. Figure 6b, what does the minimum value of the blue line (F-F) represent?
6. Further clarification of the physical significance of the results is needed
7. The introduction should state more clearly the purpose and new point of this work.
8. The authors should review the theories of deformation theory, thereby giving reasons for choosing the research problem. To enrich the introduction section, the authors should refer to some of the following related publications:
- https://doi.org/10.1155/2020/2836763
- https://doi.org/10.15625/2525-2518/58/1/14278
- https://doi.org/10.1080/15397734.2022.2088558
Author Response
- Symbols in formula (1) need to be consistent with symbols in formulas (6) and (16)
Author response: We would like to express our gratitude to the reviewers for their valuable time and efforts in reviewing our article. We appreciate the positive feedback provided by them. We have carefully considered the comments made by the reviewers and have made necessary changes to address their concerns. Specifically, we have corrected Eq. (1) to ensure consistency with the symbols used in Eq. (6) and Eq. (16). We hope that the reviewers find our corrections satisfactory.
- In formula (10): The expression for calculating Q11 and Q33 involving imaginary numbers i, it is necessary to cite references and explain symbols.
Author response: We would like to express our gratitude to the reviewers for their valuable suggestions, which we have carefully considered and implemented in the revised version of our paper. Specifically, we have provided a detailed explanation of the symbols Q11 and Q33 in Eq. (10) and cited relevant literature to clarify the origin of the viscoelastic theory used in our study.
- Some formulas need to cite references。
Author response: We would like to express our gratitude to the reviewers for providing us with their valuable comments on this paper. After carefully considering their feedback, we have made several improvements to the manuscript. Specifically, we have inserted reference citations into some formulas where it was deemed necessary, and highlighted them in red for ease of identification. We sincerely hope that these revisions will meet the expectations of the reviewers.
- Symbols should be explained after formulas
Author response: We express our gratitude to the reviewers for their valuable and constructive feedback on the manuscript. In response to their request, we have carefully reviewed all of the equations in the manuscript. We have provided explanations for the important symbols within the text and have included a parameter table in Appendix A for the remaining symbols. To aid identification, we have highlighted all symbols in red. We trust that these modifications will address the concerns raised by the reviewers.
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- Figure 6b, what does the minimum value of the blue line (F-F) represent?
Author response: We would like to thank the reviewer for their question, and we are pleased to provide the following response: The scenario described by the reviewer is represented by the blue line in Fig. 6b. This line depicts the trend of the modal loss factor of Metal rubber cylindrical shell (MRCS) as a function of the circumferential wave number (n) under free boundary conditions at both ends. As n increases, the modal loss factor of MRCS tends to decrease initially and then increase. The minimum value occurs at n = 8, m = 1, with a value of 0.05797. This represents the lowest percentage of dissipated energy absorbed from vibration in the case of n = 8, m = 1, when the MRCS has the lowest damping capacity. We hope that this response sufficiently addresses the reviewer's question and that they find it satisfactory.
- Further clarification of the physical significance of the results is needed
Author response: We have revised Section 3 of the article based on the reviewers' comments, as the physical significance was unclear in the previous version. To address this issue, we have provided a clearer explanation of the physical meaning represented by the results. The revised section is highlighted in red to make it easier for the reviewers to identify the changes. We hope that our corrections will meet the satisfaction of the reviewers.
- The introduction should state more clearly the purpose and new point of this work.
Author response: We express our gratitude to the reviewers for their valuable comments, which have helped us to bring the focus of this paper into sharper relief. In response to their feedback, we have made some revisions to the conclusion of the introduction to more clearly articulate the purpose of the paper and the key ideas that will be explored. The revised section has been highlighted in red for easy identification.
- The authors should review the theories of deformation theory, thereby giving reasons for choosing the research problem. To enrich the introduction section, the authors should refer to some of the following related publications:
- https://doi.org/10.1155/2020/2836763
- https://doi.org/10.15625/2525-2518/58/1/14278
- https://doi.org/10.1080/15397734.2022.2088558
Author response: We express our gratitude to the reviewers for their valuable suggestions. Following their recommendations, we have diligently examined the literature suggested by them and discovered that it has immense reference value. Using this literature as a guide, we have made significant improvements to the introduction section, particularly in the review of deformation theory. To highlight these changes, we have marked the revised sections in red. We sincerely hope that the reviewers will find our revisions satisfactory.
Round 2
Reviewer 3 Report
this verson is good
