Free vibration analysis of transmission lines based on the dynamic stiffness method

An improved mathematical model used to study the coupling characteristic of the multi-span transmission lines is developed. Based on the solution method for single-span cable, an expression for the dynamic stiffness of two-span transmission lines with an arbitrary inclination angle is formulated. The continuity of displacements and forces at a suspension point is used to derive the dynamic stiffness. Interactions between insulator strings and adjacent spans are accommodated. Considering the infinite dynamic stiffness corresponds to the natural frequencies of the transmission lines, the finite-element method (FEM) is employed to assess the validity of the dynamic stiffness. In the numerical investigation, attention is focused on the effect of the inclination angles, Irvine parameter, insulator string length and damping parameter. In addition, the modal function corresponding to the natural frequencies is derived. Then, the results of comprehensive parametric studies are presented and discussed. Special attention is paid to the effect of the Irvine parameter and damping parameter on the in-plane modal shapes. Finally, according to the theoretical model of two-span transmission lines, the generalized dynamic stiffness of transmission lines with an arbitrary number of spans and inclination angles is derived. The method can be used as the basis of the vibration analysis on a wide variety of multi-span transmission lines.


Recommendation?
Major revision is needed (please make suggestions in comments)

Comments to the Author(s)
The paper presents some analytical derivations to theoretically analyze the vibration response of transmission lines. The subject of the paper is of interest because of its significant applicative perspectives. The paper is well organized: the novel aspects of the study are clearly outlined in the introduction, and theoretical results are validated against numerical simulations in the subsequent sections; the influence of relevant parameters is also investigated. However, the last part of the paper discussing the extension of the method to multi-span conductors lacks of a validation example. In view of possible publication, the following comments have to be duly considered and the paper revised accordingly: • A comparison of the analytical results with those obtained by a numerical simulation must be added for the multi-span case; • A careful language editing is necessary: typos and misprints are disseminated throughout the manuscript; • Symbols must be checked (see, for instance, lines 154, 173 and 203); • The English language must be carefully revised, possibly with the aid of a native English speaker, because of the presence of several errors; • Dimensions of all figures must be increased, and image resolution must be improved.

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation?
Major revision is needed (please make suggestions in comments) There are many other mistakes in the usage of the English language which must be rectified. At present the paper is not up to the standard of the Royal Society.

01-Nov-2018
Dear Dr Liu, The editors assigned to your paper ("Free vibration analysis of transmission lines based on the dynamic stiffness method") have now received comments from reviewers. We would like you to revise your paper in accordance with the referee and Associate Editor suggestions which can be found below (not including confidential reports to the Editor). Please note this decision does not guarantee eventual acceptance.
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Comments to Author:
Reviewers' Comments to Author: Reviewer: 1 Comments to the Author(s) In this paper, the dynamic stiffness of a two-span transmission line considering the interaction between the adjacent spans is studied, and the theoretical formulas for dynamic characteristics of the line are presented. The theory is verified through the comparison of theoretical solution and that by means of numerical method, and parameter study is also carried out. The method is extended to a multi-span line at the end. The topic is very interesting and the results are helpful for the understanding of free vibration behavior of multi-span transmission lines. The reviewer recommends the paper be published after some revision. Some comments are as the follows: (1) Page 2, line 41-42: the expression "since galloping is a motion of the cable very close to a free vibration" is unsuitable because galloping is actually a self-excited vibration.
(2) What are the relative errors between the theoretical results (frequency and natural mode) and those determined by the numerical method in Fig.10? Fig. 2, the force on the right acting on the suspension insulator string should be H2 instead of H1? (4) In Eqs. (3a) and (3b), what are the definitions of the symbols le1 and le2 ? (5) Line246: T should be T1 ? (6) Line254: "1th" should be deleted. (7) In the finite element modeling, how many truss elements are used to simulate the suspension insulator string? Each porcelain insulator is simulated by one truss element? (8) In Fig.4, what is Ks ? is Kd the same as Kdyn in Eq.(24)? (9) Lines 528-531, the subscripts i and j should be consistent? (10) Line571: Kx should be Kdyn? (11) In Section 3, the authors presented the dynamic stiffness of multi-span line. It is suggested that one or two examples be added to demonstrate the validation of the theoretical solution. (12) The sub-titles should be modified because the paper investigates not only the dynamic stiffness but also the free vibration characteristics of the transmission lines. It is suggested that Section 2 be divided into two sections, one discussing the dynamic stiffness and the other the free vibration characteristics. (13) The English expression should be improved.

Reviewer: 2
Comments to the Author(s) The paper presents some analytical derivations to theoretically analyze the vibration response of transmission lines. The subject of the paper is of interest because of its significant applicative perspectives. The paper is well organized: the novel aspects of the study are clearly outlined in the introduction, and theoretical results are validated against numerical simulations in the subsequent sections; the influence of relevant parameters is also investigated. However, the last part of the paper discussing the extension of the method to multi-span conductors lacks of a validation example. In view of possible publication, the following comments have to be duly considered and the paper revised accordingly: • A comparison of the analytical results with those obtained by a numerical simulation must be added for the multi-span case; • A careful language editing is necessary: typos and misprints are disseminated throughout the manuscript; • Symbols must be checked (see, for instance, lines 154, 173 and 203); • The English language must be carefully revised, possibly with the aid of a native English speaker, because of the presence of several errors; • Dimensions of all figures must be increased, and image resolution must be improved.

Reviewer: 3
Comments to the Author(s) The paper develops a dynamic stiffness model for multi-span transmission lines by including the insulator strings and inclination angle of the cables. Basically the methodology of a single span cable is extended to two-span and multi-span transmission systems. It is admitted that such an extension is helpful and far from being trivial, but the main criticism of the paper arises from the fact that the authors have not researched the effect of damping in any meaningful way. The paper cannot be accepted if damping is not given due recognition in an effective and rigorous manner, both in the theory and the results. The effect of damping on mode shapes must be elucidated. The discussion of results needs considerable sharpening. There are many presentational issues, particularly on the usage of the English language. Some, but not all, of them are as follows

Comments to the Author(s)
The Authors revised the paper following the reviewers' comments. The current version of the paper can be accepted for publication.

Comments to the Author(s)
In the initial submission, damping was completely left out. As a consequence of my comment on the original manuscript, asking the authors to include damping, as it turned out, the revised version shows that the effects of damping could be quite pronounced. Lessons can be learned from this. The authors are requested to take cognizance of the fact that in an analysis of this type, the effects of damping cannot be ignored. Also, it is noted with some dismay that research papers from some of the leading researchers in the dynamic stiffness method have not been quoted in the list of references. This is most unusual.

12-Feb-2019
Dear Dr Liu: On behalf of the Editors, I am pleased to inform you that your Manuscript RSOS-181354.R1 entitled "Free vibration analysis of transmission lines based on the dynamic stiffness method" has been accepted for publication in Royal Society Open Science subject to minor revision in accordance with the referee suggestions. Please find the referees' comments at the end of this email.
The reviewers and Subject Editor have recommended publication, but also suggest some minor revisions to your manuscript. Therefore, I invite you to respond to the comments and revise your manuscript.
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• Data accessibility It is a condition of publication that all supporting data are made available either as supplementary information or preferably in a suitable permanent repository. The data accessibility section should state where the article's supporting data can be accessed. This section should also include details, where possible of where to access other relevant research materials such as statistical tools, protocols, software etc can be accessed. If the data has been deposited in an external repository this section should list the database, accession number and link to the DOI for all data from the article that has been made publicly available. Data sets that have been deposited in an external repository and have a DOI should also be appropriately cited in the manuscript and included in the reference list.
If you wish to submit your supporting data or code to Dryad (http://datadryad.org/), or modify your current submission to dryad, please use the following link: http://datadryad.org/submit?journalID=RSOS&manu=RSOS-181354.R1 • Competing interests Please declare any financial or non-financial competing interests, or state that you have no competing interests.
• Authors' contributions All submissions, other than those with a single author, must include an Authors' Contributions section which individually lists the specific contribution of each author. The list of Authors should meet all of the following criteria; 1) substantial contributions to conception and design, or acquisition of data, or analysis and interpretation of data; 2) drafting the article or revising it critically for important intellectual content; and 3) final approval of the version to be published.
All contributors who do not meet all of these criteria should be included in the acknowledgements.
We suggest the following format: AB carried out the molecular lab work, participated in data analysis, carried out sequence alignments, participated in the design of the study and drafted the manuscript; CD carried out the statistical analyses; EF collected field data; GH conceived of the study, designed the study, coordinated the study and helped draft the manuscript. All authors gave final approval for publication.
• Acknowledgements Please acknowledge anyone who contributed to the study but did not meet the authorship criteria.
• Funding statement Please list the source of funding for each author.
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Kind regards, Andrew Dunn Royal Society Open Science Editorial Office Royal Society Open Science openscience@royalsociety.org on behalf of Prof R. Kerry Rowe (Subject Editor) openscience@royalsociety.org Associate Editor Comments to Author: Thank you for the revision. As you'll see, the reviewers are more satisfied your manuscript is ready for acceptance. However, one of the reviewers notes your literature review does not include a number of key papers --unfortunately, they have not provided more details, but it would be worth exploring the literature one more time to include additional context to support your work.
Reviewer comments to Author: Reviewer: 3 Comments to the Author(s) In the initial submission, damping was completely left out. As a consequence of my comment on the original manuscript, asking the authors to include damping, as it turned out, the revised version shows that the effects of damping could be quite pronounced. Lessons can be learned from this. The authors are requested to take cognizance of the fact that in an analysis of this type, the effects of damping cannot be ignored. Also, it is noted with some dismay that research papers from some of the leading researchers in the dynamic stiffness method have not been quoted in the list of references. This is most unusual.

Reviewer: 2
Comments to the Author(s) The Authors revised the paper following the reviewers' comments. The current version of the paper can be accepted for publication.