A Method to Obtain Frequency Response Functions of Operating Mechanical Systems Based on Experimental Modal Analysis and Operational Modal Analysis

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The study demonstrates a certain degree of novelty and requires minor revisions before it can be published in the journal. The specific comments are as follows:

 

(1) The abstract should highlight the experimental results, which are currently missing. Additionally, a significant portion of the abstract is dedicated to background information, which is unnecessary.

(2) Is the main procedure of operational FRF estimation proposed by the authors themselves? If it is referenced from other articles, the relevant references should be included.

(3) Many formulas in the manuscript lack references.

(4) The conclusion section needs to highlight the limitations of the study.

Author Response

Thank you very much for taking the time to review this manuscript. The comments are very helpful for revising and improving our paper, as well as the important guiding significance to other researches. We have studied the comments carefully and made corrections which we hope meet with approval. The main corrections are highlighted in blue color in the manuscript, and the responds to the reviewer’s comments are as follows.

 

 

Comments 1: The abstract should highlight the experimental results, which are currently missing. Additionally, a significant portion of the abstract is dedicated to background information, which is unnecessary.

 

Response 1: Thank you for pointing this out and we agree with this comment. We have updated the abstract, with less background, more contents on the method and the experiments results.

 

 

Comments 2: Is the main procedure of operational FRF estimation proposed by the authors themselves? If it is referenced from other articles, the relevant references should be included.

 

Response 2: To the best of our knowledge, the main procedure in the paper has not been proposed by others, and we have not published any papers using this procedure.

 

 

Comments 3: Many formulas in the manuscript lack references.

 

Response 3: Some formulas were not referenced in the context or were considered not so important, so we did not label them originally. We have revised this mistake. Thank you for your guidance.

 

 

Comments 3: The conclusion section needs to highlight the limitations of the study.

 

Response 3: Agree. Previously, we only pointed out that the method is not very applicable to high damping structures, which was not sufficient. In the revised paper, we have supplemented two more limitations: (1) applying this method to OFRFs in high-frequency zone may also be challenging due to difficulties in identifying modal parameters and numerical issues in the principal square root method; (2) changes in position or pose of sub-structures are not considered in the operating state, making the proposed method unsuitable for gesture-dependent OFRFs of robots. This revision can be seen in the last paragraph of the main article.

 

 

Thank you again for your guidance.

Reviewer 2 Report

Comments and Suggestions for Authors

The description of the boundary conditions illustrated by triangles in Fig. 2 is unclear. The exact position of the supports should be defined. The coordinates of points B,C,D are missing in Table 1. How many beam elements are used between the corner points of the plate and point P, respectively? A comparison between the natural frequencies of the original and targeted structure should be provided. Are the spring coefficients aligned with the global coordinate system? The discrete spring and damper elements are orientated in the global z-direction. The spring stiffnesses in the x and y directions should therefore be identical. You have used different length units for the spring (meter) and damper (millimeter). Is that correct?

What is the meaning of a "static" mode shape?

All equations should be given a consecutive number for referencing!

The generalized eigenvalue problem for undamped vibrations is given by

K X = M X \Lambda

where X denotes the real mode shapes and \Lambda is a diagonal matrix, that contains the squared natural frequencies.

Gateux instead of Gateus See https://en.wikipedia.org/wiki/Gateaux_derivative

In Table 1 use plate instead of plane.

The FRFs are given in [dB]. So which reference value is used here?

Which (commercial) finite element program was used for the simulation example?

Comments on the Quality of English Language

The paper should be revised to ensure the reproducibility of the results presented.  Some formulations are unclear and do not correspond to common vocabulary in the literature.

Author Response

Thank you very much for taking the time to review this manuscript. The comments are very helpful for revising and improving our paper, as well as the important guiding significance to other researches. We have studied the comments carefully and made corrections which we hope meet with approval. The main corrections are highlighted in red color in the manuscript, and the responds to the reviewer’s comments are as follows.

 

 

Comments 1: (1) The description of the boundary conditions illustrated by triangles in Fig. 2 is unclear. The exact position of the supports should be defined. The coordinates of points B,C,D are missing in Table 1.

(2) How many beam elements are used between the corner points of the plate and point P, respectively?

(3) A comparison between the natural frequencies of the original and targeted structure should be provided.

(4) Are the spring coefficients aligned with the global coordinate system? The discrete spring and damper elements are orientated in the global z-direction. The spring stiffnesses in the x and y directions should therefore be identical.

(5) You have used different length units for the spring (meter) and damper (millimeter). Is that correct?

 

Response 1: Thank you for pointing this out and it was indeed our mistake. We have revised the paper accordingly. Here are the details:

• We have added Appendix B to provide the coordinates of the critical nodes in the simulation example, ensuring that other researchers can reproduce the results accurately. Related text revision can be seen in line 304, highlighted in red.
• We have added the information in Table 1 that we use only one beam element per beam in our model. This addition is highlighted in red.
• The changes in natural frequencies are not so crucial to the proposed method, so we did not list them previously. However, they do reflect the variations in dynamic properties between the original and targeted structure. Therefore, we have added Appendix C to show the comparisons. Three comparisons involving all cases are provided. Related text can be found in lines 323-324 and 367-368 in Section 4.1, and lines 432-434 in Section 4.2.
• These are really our mistakes. Initially, we defined the springs in an asymmetric way to add some complexity into the model, so the stiffness and damping along x and y direction are not the same. But we failed to clearly specify these directions. The added description can be seen in lines 298-300.
• We have to appreciate the reviewer for this comment. The issues with stiffness values and units were significant. Errors in values and units make the stiffness in original paper decrease by many orders of magnitude. Fortunately, the reviewer noted the stiffness unit, which led us to identify and correct this error. Related revision can be seen in Table 1 and line 303.

 

Given the modifications to parameters in the FEM model, we want to demonstrate that these were indeed editing errors, and that there is no data fraud in the subsequent figures. We decide to provide the FEM models as a supplementary file. These models were completed in 2019, prior to our receiving the reviewer’s report.

We built the model with Hypermesh and the solver was Nastran. The stiffness values can be examined in the PBUSH card, while the directions of the springs are detailed in the CBUSH cards.

Additionally, the coordinate system used in FEM model differs from that in the paper, so the coordinates of nodes in the model are not identical to those presented in Appendix B. A transformation in the coordinate systems is necessary if the reviewer wishes to verify those coordinates.

We can provide more files if the reviewer has some further problems in validation.

 

 

Comments 2: What is the meaning of a "static" mode shape?

 

Response 2: Thank you for this comment. We took it granted that the readers could under-stand our meaning. We actually mean static mode shapes by the mode shapes identified in EMA. We have added the explanation in the introduction, highlighted in red in lines 82-83. Additionally, in lines 331-332 of Section 4, we clarify that “static mode shapes” in the figure legends refers to the method that utilizes static mode shapes instead of operational mode shapes for FRF synthesis

 

 

Comments 3: All equations should be given a consecutive number for referencing!

 

Response 3: Thank you for your guidance. Some formulas were not referenced in the context or were considered not very important, so we did not label them originally. We have now corrected this mistake.

 

 

Comments 4: The generalized eigenvalue problem for undamped vibrations is given by

K X = M X \Lambda

where X denotes the real mode shapes and \Lambda is a diagonal matrix, that contains the squared natural frequencies.

 

Response 4: There is only a difference of a negative sign between the equation provided by the reviewer and the Eq. (5) in our paper. Mathematically, the generalized eigenvalue problem is defined as

                                                   AV=BVD

where V and D denote matrices composed of eigenvectors and eigenvalues, respectively. But in mechanics, since the ordinary differential equation is

                                                  Mx''+Kx=F

the corresponding eigenvalue problem is

                                                  KX=-MX\Lambda

Frankly speaking, we wonder whether the reviewer wants us to revise the Eq. (5) in the paper, or simply to number the equation in the original paper by that sentence.

 

 

Comments 5: Gateux instead of Gateus See https://en.wikipedia.org/wiki/Gateaux_derivative

 

Response 5: Thank you for pointing this out. We have made the revision in line 238.

 

 

Comments 6: In Table 1 use plate instead of plane.

 

Response 6: Thank you for pointing this out. We have made the revision.

 

 

Comments 7: The FRFs are given in [dB]. So which reference value is used here?

 

Response 7: The unit of force is N, and units of acceleration in Section 4.1 and 4.2 are mm/s² and g, respectively. The reference for dB calculation is 1. Related revision can be seen in lines 328-330 and 437-439.

 

 

Comments 8: Which (commercial) finite element program was used for the simulation example?

 

Response 8: We use Nastran for simulation. Related information has been added in lines 307-308. In addition, we use SOL 109 in the impulse excitation simulation, which solves the transient response in a direct way. Lines 350-351 describe this information. Line SOL 112 is used in the random simulation, performing mode analysis first to uncouple the differential equations, followed by the calculation of responses. Modes beyond 6000Hz are excluded, as mentioned in line 373.

 

 

Thank you again for your comments.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

Why aren't the names of the authors of the article numbered with the number 1 and 2. I recommend adding the exact address of the authors.

Why is the logo of the magazine Machines wrong on the first page.

Why is formula (4) numbered on page 4 and also formula (5) on page 5 and there are no numbered formulas in line 151, 160, 162 and 164. I recommend adding the labeling of all formulas (2), (3) (4) and ( 5). According to the template of the given magazine, each sentence should be marked with a number.

The quality and resolution and description of the Figure in this article is good and high quality.

Figure 9 shows a color graph where there are blue areas, red areas and yellow areas. I have an error in the description of what value the red, yellow, green and blue area answers.

The authors showed the results of a number of experiments that they did as part of the article.

The attached document shows the value of Quotes Excluded 10%. The authors of the article drew from almost 60 different sources. The value of one resource does not exceed 1%. So the article is original and shows the current research of the authors.

Although the article did not really interest me, it does not show an attractive topic, but it examines the given issue and after minor adjustments it can be published in the given magazine.

Author Response

Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted in cyan color in the re-submitted files.

 

 

Comments 1: Why aren't the names of the authors of the article numbered with the number 1 and 2. I recommend adding the exact address of the authors.

 

Response 1: The two authors share the same address, therefore, we think there is no need to number their names. Additionally, we designate only the second author as the corresponding author, whose name is labeled and the e-mail is presented.

 

 

Comments 2: Why is the logo of the magazine Machines wrong on the first page?

 

Response 2: We used the LaTeX template provided by the journal for paper editing. Unlike editing with Word, the layout and style of this paper are determined entirely by LaTeX code and are not under our control. It may be the editor’s responsibility to present the magazine’s logo. We have ensured that the word “Machines” is spelled correctly.

 

 

Comments 3: Why is formula (4) numbered on page 4 and also formula (5) on page 5 and there are no numbered formulas in line 151, 160, 162 and 164. I recommend adding the labeling of all formulas (2), (3) (4) and ( 5). According to the template of the given magazine, each sentence should be marked with a number.

 

Response 3: Thank you for your guidance. Some formulas were not referenced in the context or were considered not so important, so we did not number them initially. Now we have labeled all the equations.

 

 

Comments 4: Figure 9 shows a color graph where there are blue areas, red areas and yellow areas. I have an error in the description of what value the red, yellow, green and blue area answers.

 

Response 4: Thank you for pointing this out. Fig. 9(b) is a contour map, showing the mode shape at 110 Hz obtained from the FEM program Nastran. It is really our mistake that we didn’t explain the color filling scheme. We have added a color bar beside the contour map, which describes how the colors change from 0 to maximum modal displacement. The related text modifications can be seen in lines 406-408, highlighted in cyan.

 

 

Thank you again for your significant comments.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

static mode shape should be replaced by experimental mode shape

 

Author Response

Comment: static mode shape should be replaced by experimental mode shape

Response: Thank you for your advice. We agree that the word "static" could cause some misunderstanding. However, since operational mode shapes are also identified through experiments, using the term "experimental" may also introduce ambiguity. Instead, we opted the term "idle", which has been used in the following papers.

1. Yi, T.Y.; Nikravesh, P.E. A method to identify vibration characteristics of modified structures for flexible vehicle dynamics. In Proceedings of the Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 2002, pp.55-63. https://doi.org/10.1243/0954407021528904.
2. Deng, K.; Gao, D.; Zhao, C.; Lu, Y. Prediction of in-process frequency response function and chatter stability considering pose and feedrate in robotic milling. Robotics and Computer-Integrated Manufacturing 2023, 82, 102548. https://doi.org/10.1016/j.rcim.2023.102548.

"idle" is used to describe "static condition" in the two papers, and conventional FRF is denoted as H_idle in the first paper. So we think this term is appropriate to replace "static".

The first instance of the term change can be seen in lines 82-83, highlighted in red. The following "static mode shapes" have also been changed, although they are not highlighted. Legends in figures have been revised as well.