A Perturbation Approach for Lateral Excited Vibrations of a Beam-like Viscoelastic Microstructure Using the Nonlocal Theory

Round 1

Reviewer 1 Report

The proposed paper presents a perturbation approach for lateral excited vibration of a beam-like viscoelastic microstructure based on the nonlocal elasticity theory. The perturbation method is used to solve the equation of motion for microstructure and author performed  some parametric study between the variables. It showed good performance on comprehension. If properly revised, the study may be possible interests for readers of applied sciences.

Major revision

Most of the results and discussions are graphs analyzed after dimensionless substitution of values in formulas, and physical understanding is not easy.  (E.G., Line 234-238, Author claimed that internal characteristic scale could be ignored in comparison with the external characteristic scale. However, the lack of detailed explanations is insufficient to explain whether the claims are supported.)

minor comments

1. Figure 1 looks poor resolution especially for the characters. Author can refine and replace it with good quality.
2. It is natural that there are many equations depending on the aspect of the paper in which the formulas are developed, but from the reader's point of view, the length of the formulas is too long or there are many unorganized parts, so there is a problem of readability. Equations such as 26, 27 needs to be re-written and simplify after minimizing duplicated parts.
3. There is a misprint. Line 332 5. .Conclusion.
4. Captions for the figures are not well aligned in the paragraphs.

Author Response

The authors thank the reviewer very much for the constructive comments on the manuscript. The manuscript has been revised in accordance with the reviewer’s comments. The reply to comments and details of revision are listed as follows.

The proposed paper presents a perturbation approach for lateral excited vibration of a beam-like viscoelastic microstructure based on the nonlocal elasticity theory. The perturbation method is used to solve the equation of motion for microstructure and author performed some parametric study between the variables. It showed good performance on comprehension. If properly revised, the study may be possible interests for readers of applied sciences.

Major revision:

1. Query: Most of the results and discussions are graphs analyzed after dimensionless substitution of values in formulas, and physical understanding is not easy.  (E.G., Line 234-238, Author claimed that internal characteristic scale could be ignored in comparison with the external characteristic scale. However, the lack of detailed explanations is insufficient to explain whether the claims are supported.).

Reply: Thanks for the constructive suggestion. Some analyses are presented to explain the claims. In addition to the dimensionless numerical results, there are some physical results with the dimensional variables, e.g. Figs. 5 and 6, we try to discuss them in detail physically. Moreover, although most of the numerical results in this paper are shown in dimensionless form which cannot be directly compared with the classical excited vibration results, many conclusions reported herein are consistent with the classical excited vibration from the qualitative trend. Hence the present numerical results can be verified qualitatively.

Minor comments:

2. Query: Figure 1 looks poor resolution especially for the characters. Author can refine and replace it with good quality.

Reply: Yes, Fig. 1 has been refined in the revised manuscript.

3. Query: It is natural that there are many equations depending on the aspect of the paper in which the formulas are developed, but from the reader's point of view, the length of the formulas is too long or there are many unorganized parts, so there is a problem of readability. Equations such as 26, 27 needs to be re-written and simplify after minimizing duplicated parts.

Reply: Thanks for such a good suggestion. These equations are re-written in the revised manuscript.

4. Query: There is a misprint. Line 332 5. .Conclusion.

Reply: Yes, it is one duplicate point is deleted.

5. Query: Captions for the figures are not well aligned in the paragraphs.

Reply: Yes, it is adjusted accordingly.

Author Response File: Author Response.doc

Reviewer 2 Report

Major revision of the article:

 

1. Major revision of the concepts
2. “Tuning parameter”-line 20 of the article and line 213 - is detuning parameter as defined in “Nonlinear Oscillations- Nayfeh A.H, ,Mook D.T.- John Wiley & Sons Publishing House, New York, USA, 1979, ISBN-0-471-03555-6” (p. 8, p. 163, p. :175,178,179,185,188,189,199,230, 291, 293, 308).
3. The detuning parameter is a notation for the mathematical solution of a nonlinear vibration with cubic nonlinearity that involves only the method of multiple scale.
4. “cosine excitation”- line 10 of the article-  is the harmonic excitation as define in “ Mechanical Vibrations- Rao S.S.- Prentice Hall publishing House, 5^th Edition, NJ, USA, 2011- p. 271, p. 278, p. 281.
5. At line 115 “The force equilibrium diagram of an element” is – the dynamic equilibrium chart for a beam element.
6. Equations (15)-(23) did not respect the use of multiple scale method for the primary resonance (excitation)- see “Nonlinear Oscillations- Nayfeh A.H, ,Mook D.T.- John Wiley & Sons Publishing House, New York, USA, 1979, ISBN-0-471-03555-6”-pp. 196-198.
7. Because of 5, the results and conclusions may be corrupted.
8. The solution given by equation (27) must be put in a form such as :, in order to remark the cubic nonlinearity.
9. Major revision of the equations and relations ( the authors didn’t use Math Type in the Word template as demands Type of Manuscript template, the had used Equation Editor that is not allowing the modifications):
10. All the mathematical notations in the equations must be Palatino Lino Type straight 10(as in the Type of Manuscript template).
11. The equations are not centered as in the Type of Manuscript template and numbering align right.
12. The relations and the mathematical notations in the text are not straight without italic as in the Type of Manuscript template straight without italic as in the Type of Manuscript template.

 

1. Major revision of the graphical material:
2. All the figures and figures caption are not presented as demanded in the Type of Manuscript template, align left indent (figures and caption).
3. All the notations inside the figures are not straight without italic as in the Type of Manuscript template (Palatino Lino type 10).
4. The figures are not clearly presented.

Author Response

The authors thank the reviewer very much for the constructive comments on the manuscript. The manuscript has been revised in accordance with the reviewer’s comments. The reply to comments and details of revision are listed as follows.

Major revision of the concepts:

1. Query: “Tuning parameter”-line 20 of the article and line 213 - is detuning parameter as defined in “Nonlinear Oscillations- Nayfeh A.H, ,Mook D.T.- John Wiley & Sons Publishing House, New York, USA, 1979, ISBN-0-471-03555-6” (p. 8, p. 163, p. :175,178,179,185,188,189,199,230, 291, 293, 308).

Reply: Thanks very much for pointing out the terminology mistakes. They are replaced by ‘detuning parameter’ in the revised version.

2. Query: The detuning parameter is a notation for the mathematical solution of a nonlinear vibration with cubic nonlinearity that involves only the method of multiple scale.

Reply: Thanks very much for pointing out such a rule. We mention it in the revised manuscript.

3. Query: “cosine excitation”- line 10 of the article-  is the harmonic excitation as define in “ Mechanical Vibrations- Rao S.S.- Prentice Hall publishing House, 5^thEdition, NJ, USA, 2011- p. 271, p. 278, p. 281.

Reply: Thank you very much. We revise it in the revised manuscript with a citation.

4. Query: At line 115 “The force equilibrium diagram of an element” is – the dynamic equilibrium chart for a beam element.

Reply: Yes it is modified.

5. Query: Equations (15)-(23) did not respect the use of multiple scale method for the primary resonance (excitation)- see “Nonlinear Oscillations- Nayfeh A.H, ,Mook D.T.- John Wiley & Sons Publishing House, New York, USA, 1979, ISBN-0-471-03555-6”-pp. 196-198.

Reply: Thank you very much. We rewrite the harmonic excitation frequency to order the damping, nonlinearity and excitation so that they appear at the same time in the perturbation scheme.

6. Query: Because of 5, the results and conclusions may be corrupted.

Reply: We check the results and related conclusions carefully. Although most of the numerical results in this paper are shown in dimensionless form which cannot be directly compared with the classical excited vibration results, many conclusions reported herein are consistent with the classical excited vibration from the qualitative trend. Hence the present numerical results can be verified qualitatively. The conclusions are consistent with the classical counterparts due to the existence of external harmonic excitation and internal viscoelasticity. In addition, some phenomena unique to the excited vibration of microbeams are found in the numerical results due to the existence of nonlocality.

7. Query: The solution given by equation (27) must be put in a form such as :, in order to remark the cubic nonlinearity.

Reply: Yes it is modified.

8. Query: Major revision of the equations and relations ( the authors didn’t use Math Type in the Word template as demands Type of Manuscript template, they had used Equation Editor that is not allowing the modifications):

Reply: Thanks for the careful reviews. The revised version is based on the manuscript template and the equations are entered using Math Type.

9. Query: All the mathematical notations in the equations must be Palatino Lino Type straight 10(as in the Type of Manuscript template).

Reply: Yes, it has been done using the Manuscript template.

10. Query: The equations are not centered as in the Type of Manuscript template and numbering align right.

Reply: As required, they are adjusted in the revised manuscript.

11. Query: The relations and the mathematical notations in the text are not straight without italic as in the Type of Manuscript template straight without italic as in the Type of Manuscript template.

Reply: Yes, the relations and the mathematical notations in the text are changed to be straight.

Major revision of the graphical material:

12. Query: All the figures and figures caption are not presented as demanded in the Type of Manuscript template, align left indent (figures and caption).

Reply: Yes, it is finished.

13. Query: All the notations inside the figures are not straight without italic as in the Type of Manuscript template (Palatino Lino type 10).

Reply: As required, all the notations inside the figures are changed to be straight.

14. Query: The figures are not clearly presented.

Reply: Yes, the presentations are refined in the revised version.

Author Response File: Author Response.doc

Round 2

Reviewer 1 Report

Thanks for considering the suggestion of reviewer`s positively. Indent and capital letter after equations need to be corrected. 

Author Response

Query: Thanks for considering the suggestion of reviewer`s positively. Indent and capital letter after equations need to be corrected. 

Reply: Thanks for the careful review. They are corrected.

Author Response File: Author Response.doc

Reviewer 2 Report

Concerning the revised form of the Manuscript: applsci-1500575-v2 entitled “ A Perturbation Approach for Lateral Excited Vibrations of a Beam-like Viscoelastic Microstructure using the Nonlocal Theory”.

 

1. The authors revised the article with respect to comments and recommendations 1-14, but there is still one mistake inside the figure 3 line 247 “detuning parameter” instead of “tuning parameter”.
2. The development of future work is not yet emphasized in paragraph “4. Results and Discussion”

Author Response

Query: The authors revised the article with respect to comments and recommendations 1-14, but there is still one mistake inside the figure 3 line 247 “detuning parameter” instead of “tuning parameter”.

Reply: Thanks for the careful review. It is revised.

Query: The development of future work is not yet emphasized in paragraph “4. Results and Discussion”.

Reply: Thanks for the good advice. We emphasize it in “4. Results and Discussion”.

 

Author Response File: Author Response.doc