The Extended Galerkin Method for Approximate Solutions of Nonlinear Vibration Equations

Round 1

Reviewer 1 Report

This manuscript deals with an application of Gelerkin method to obtain
the approximate solutions of nonlinear equations.  The authors show a
fundamental formulation of the method and applications to a couple of
typical non-linear equations: Duffing equation and Van der Pol
equation.  The idea of procedure, which is based on asymptotic
expansion analysis of polynominal perturbation, is quite simple, but
the functional of temporal integral over a time period of dynamical
system is a new viewpoint.

In the revised manuscript, the authors clearly describe their original
idea which they insist in.  There is significant originality of the
unique results which readers of the journal would be interested in.

Author Response

The authors are grateful to this reviewer for the support and encouragement of this research.

The paper will be checked and modified further for best presentation of this original study and novel contribution.

The authros thank the reviewer for the positive comments.

Reviewer 2 Report

Comments and Suggestions for Authors

Concerning the Manuscript ID: applsci-1551887 “The Extended Galerkin Method for Approximate Solutions of Nonlinear Vibration Equations”

The article research field is in applied mathematics. I consider the idea very useful for the engineers and designers in the field of nonlinear dynamics so that the article can have a significative impact.

Unfortunately, by my opinion it must be developed and rewrite because in this form is not emphasizing and is not highlighted its full potential.

Concerning the scientific content, I suggest the authors

1. must develop paragraph 1 Introduction for a better understanding of the extended Galerkin Method;
2. must highlight the aim of this article in paragraph 1 Introduction;
3. must make a clear distinction between Galerkin Method and extended  Galerkin Method, therefore paragraph 2 Galerkin formulation must contain two subparagraphs ( 2.1 State of the Art and 2.2 Extended Galerkin Method).
4. must introduce the lines 137 to 140 in the paragraph 1 Introduction where its belong;
5. must introduce 2 new paragraphs Results & Discussions where the authors must emphasize what brings new as accuracy the extended Galerkin Method;
6. must develop paragraph 3 Application examples with an example for a nonlinear equation of vibration of Mathieu-Hill type that describes either a parametric nonlinear dynamic behavior of a thin plate with moderately large deflection or the nonlinear dynamic behavior of a geared system and in the meantime the presentation must be improved with some graphs or numerical results;
7. must develop and highlight in the new paragraph Discussion what brings new this extended Galerkin Method for the designers and engineers.

 

Concerning the respect of the Manuscript Type MDPI journal template, I suggest the authors

1. move the Featured Application (lines 10-13) inside paragraph 1 Introduction;
2. rephrased the English sentences : line 33, lines 42-44, lines 69-72, lines 81-83, lines 129-131, lines 139-143, that are not very clearly;
3. must put the References in the form required by Manuscript Type MDPI journal template because they didn’t respect this form 100%;
4.  must change the use of Eq. along the entire article because this is not allowed by the Manuscript Type MDPI journal template. 

Author Response

The authors are grateful to the reviewer for the careful reading and thoughtful suggestions.  The authors have read the comments and revisions are made to improve the manuscript as suggested.

1. must develop paragraph 1 Introduction for a better understanding of the extended Galerkin Method;

Reply> The suggestion is taken. A paragraph has added to the manuscript to present a simple and essential introduction to the Galerkin method.

1. must highlight the aim of this article in paragraph 1 Introduction;

Reply> Suggestion taken. The objective of this short paper is added in the first paragraph from Lines 52-55.

1. must make a clear distinction between Galerkin Method and extended  Galerkin Method, therefore paragraph 2 Galerkin formulation must contain two subparagraphs ( 2.1 State of the Art and 2.2 Extended Galerkin Method).

Reply> Excellent suggestion. The section is now divided as suggested for better introductions and descriptions.

1. must introduce the lines 137 to 140 in the paragraph 1 Introduction where its belong;

Reply> There is a sentence similar in Introduction already, so this part is slightly modified  and remain here.

1. must introduce 2 new paragraphs Results & Discussions where the authors must emphasize what brings new as accuracy the extended Galerkin Method;

Reply> Good suggestion.  The structure has been modified as suggested.

 

1. must develop paragraph 3 Application examples with an example for a nonlinear equation of vibration of Mathieu-Hill type that describes either a parametric nonlinear dynamic behavior of a thin plate with moderately large deflection or the nonlinear dynamic behavior of a geared system and in the meantime the presentation must be improved with some graphs or numerical results;

Reply> The reviewer has made a very good suggestion for more studies.  As it is known, there are many types of different nonlinear vibration problems, and it is definitely our goal to understand the applicability of the extended Galerkin method in dealing with broad problems from science and engineering.  Two typical examples of different nonlinear equations are solved with satisfactory approximations.  It shows that the extended Galerkin method is valid for these and other problem we have solved in aother paper [11].  If we go to another type of equations from plates, most likely there will be a lengthy derivation and solution procedure which should be better presented in another paper.  It is hard to have such new problems added to this short paper as another example.  For this reason, I have to respectfully decline this suggestion, and hope the review will accept my position for not adding more examples.  The examples presented in this paper are enough to support the objective of this paper.

1. must develop and highlight in the new paragraph Discussion what brings new this extended Galerkin Method for the designers and engineers.

Reply> Very good suggestion.  It is evident that the extended Galerkin method is an efficient and elegant technique to be favored by researchers and engineers for reasons listed in the new Lines 240-243.

Reviewer 3 Report

Please, see the attachment!

Comments for author File: Comments.pdf

Author Response

The paper under review mainly proposes a numerical approach to solve nonlinear equations of vibration for asymptotic solutions. The idea is based on an extension of Galerkin method by adding an integration of time over one period of vibrations. Proposed approach is applied for Duffing and Van der Pol equations.
In addition to the lack of mathematical basis of the proposed methodology, the manuscript does not contain any numerical experiments to illustrate the efficiency of the proposed approach. Therefore, the manuscript as submitted cannot be considered as a full-type article. For these reasons, I do not recommend for the publication.
Below some comments and suggestions can be found.

Reply> The authros appreciate the frank comments from the reviewer.  This is a short paper  to demonstrate the proposed extended Galerkin method with examples.  Because the Galerkin method has been widely accepted and used, there is no need to go through a lengthy process to provide some mathematical reasoning.  The procedure and examples are enough to show the effectiveness of the method, and the procedure also reveals the simplicity.  Further research can be performed upon the acceptance of the method in more applications to nonlinear problems, as we showed in another recent paper [11].  The authors hope this can persuade the reviewer to support the publication of this paper.

ˆ Page 1, line 43: Please, provide necessary references for the usage of an extension of Galerkin method in studying nonlinear vibrations of elastic solids.

Reply>  There are no references on the extended Galerkin method.  Two earlier papers used the same approach but with the integration in the interval [0, T/4].  We believe the integration in the complete period will provide better inclusion of properties of the equation and solution.  The method is new and it is called the extended Galerkin method [21, 22].

ˆ Page 2, line 47: Why an extension of Galerkin method for nonlinear vibration problems can be useful? Please, make a clear discussion by providing necessary references.

Reply> The extension provides correct solutions to nonlinear vibration equations because it is equivalent to the harmonic balance method (HBM).  It was observed in [19] in Line 115, but the procedure is never practiced.  We showed the procedure is correct and efficient with examples presented.

ˆ Page 2, line 53: Typo mistake ˙u, u, ... ¨

Reply> We checked the line and they are the derivatives respect to time t, as explained in Line 61.

ˆ Page 2: Please, make a clear discussion of Galerkin method an extended Galerkin method in terms of theory such as convergence, etc.

Reply> The convergence of Galerkin method is the result of the least square method because of the same fucntion is used as solution and weighting function.  Galerkin method is the foundation of the currently popular finite element method.  The convergence of the extended Galerkin method is guranteed by the correct solution with periodicity.

ˆ Page 3, line 111: Please, use a different notation for the upper bound of index in Eqn. 9 since N denotes the nonlinear problem.

Reply> Thanks for pointing out out this careless writing.  It has been revised as suggested.

ˆ Page 3-4: The approximate displacements are assumed in the differen forms for each example, see Eqns. 9 and 20. How do you set the representation of the solution?

Reply> The general principle is the observation of the structure of solutions.  For Duffing equation, the solution in the series of cosine fuction is enough.  For van der Pol equation, the solution will have both sine and cosine fucntions.  This essential assumption is needed for all approximate techniques.

ˆ Page 4, line 135: ”Although there are differences in comparison with other approximate techniques ...” Please, illustrate the efficiency of the proposed approach numerically or theoretically.

Reply> A simple explantion is that the integration is more efficient because it is a one step process, while the harmonic balance method requires the split and combination of harmonic terms.

Round 2

Reviewer 2 Report

Concerning the Manuscript ID: applsci-1551887 “The Extended Galerkin Method for Approximate Solutions of Nonlinear Vibration Equations” version 2.

 

I thank the authors that they take into consideration the comments and suggestions of the first review report. All the  recommendations and suggestions were satisfied.

Therefore, I recommend the acceptance of the article with minor revision after fine/minor spell check of English language and style .   

 

Author Response

I thank the authors that they take into consideration the comments and suggestions of the first review report. All the  recommendations and suggestions were satisfied.

Therefore, I recommend the acceptance of the article with minor revision after fine/minor spell check of English language and style .   

Reply> The authors are grateful to the reviewers and editor on their carefully crafted comments and suggestions for the improvement of the paper.

The paper has been carefully checked and some corrections are made with spelling and language.  In addition, equations are also modified to be consistent in style.  We hope the revisions will improve the style to meet the requirements of Applied Sciences.

Once again, our sincere thanks to the reviewers.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

This manuscript presents a new method, as an extension of the well known Galerkin Method, to solve nonlinear equations. While, this manuscript fits into the journal's scopus, there are some aspects that could be addressed to improve the overall quality. My comments are arranged from specific/trivial to more general/critical.

1. It would be convenient to previously define the acronyms that appear in equation 18 and line 128 of the manuscript: (KBM, L-P, HAM).
2. In line 168, it would be more accurate to refer to displacement instead of deformation.
3. It is not sufficiently clear to me what the difference and novelty is with respect to the traditional Galerkin method.
4. It would be good if the authors could highlight the advantages of applying this method compared to other methods available in the scientific literature such as those mentioned in the manuscript itself: KBM, HAM, L-P.
5. A more detailed analysis should be included with quantifiable results of percentage errors obtained with the method presented in the manuscript and other existing methods, so that the possible advantages to be clarified in the previous point are made clear with numerical results.

Author Response

Dear Reviewer，

Happy New Year!

Thanks for the reading and comments.  We have revised the manuscript as suggested and our reply can be found in the file.

Regards,

 

Ji Wang

Author Response File: Author Response.doc

Reviewer 2 Report

This manuscript addresses an application of Gelerkin method to obtain
the approximate solutions of nonlinear equations.  The authors show a
fundamental formulation of the method and applications to a couple of
typical non-linear equations: Duffing equation and Van der Pol
equation.  The idea of procedure, which is based on asymptotic
expansion analysis of polynominal perturbation, is quite simple and
the dealing problems of study is still located just in exercises.
However there is some originality of the unique results which readers
of the journal would be interested in.  The authors use the procedure
"Extended Galerkin Method (EGM)" in the title and the body of the
manuscript. However, in the reviewer's opinion, it is not extension of
Galerkin method, but just an application of Galerkin method.
At least the authors should reconsider more suitable title.  The reviewer
recommends "Application of Galerkin Method for Asymptotic Solutions of Nonlinear Vibration Equations".

Author Response

Dear Reviewer,

Happy New Year!

Thank you very much for your reading and comments.  Please find our reply in the file.  I choose not to change the name as you have suggested with my explanations.  Please forgive to my insistence.

Regards,

Ji Wang

Author Response File: Author Response.doc

Round 2

Reviewer 1 Report

Although some of the proposals have not been taken into account, I consider accepting the manuscript in its present form.