A Study on the Static and Dynamic Characteristics of the Spindle System of a Spiral Bevel Gear Grinding Machine

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper studied the static and dynamic characteristics of the spindle system of spiral bevel gear grinding machine. Despite it is an interesting topic with a specific engineering background, some major problems should be emphasized. My detailed comments are presented as follows:

(1) Based on the research theme of this paper, it is necessary to define the main evaluation indexes of the dynamic and static characteristics before detailed discussions. In addition, it should also be explained clearly how these indexes influence the performance of the spindle system.

(2) I noticed that the authors have carried out the analysis concerning static stiffness. What is the situation of the dynamic stiffness? It seems that dynamic stiffness is the key factor affecting the operating performance.

(3) In Section 2, the theoretical analysis is implemented based on many assumptions and approximations. All your assumptions should be listed in one area and justified if they are practical or not, e.g., superposition principle, small deformation, linear treatment, etc.

(4) Some of your equations are not clear. It is not only about filling papers with maths whereas you need to make everything clear. The expressions of Ti, Ci, and C should be further explained. In addition, is T defined as a scalar or a variable in Eq. (20)? The readability of Figs. 5 and 7 should be improved.

(5) More detailed descriptions concerning Figs. 8 and 9 are required. By the way, can the relevant conclusions be drawn directly just based on engineering experience? The highlights and new findings of your research should be further clarified and discussed.

(6) The research status in the Introduction should be further improved. Comparisons among different methods and the novelty of this paper should be discussed in detail.

Comments on the Quality of English Language

Minor editing of English language is required.

Author Response

Answers to reviewers:

Reviewer #1:

We thank the reviewer for his/her constructive criticisms that have helped us to improve our manuscript. The point-by-point response to the comments is given below.

Comment 1: 

Based on the research theme of this paper, it is necessary to define the main evaluation indexes of the dynamic and static characteristics before detailed discussions. In addition, it should also be explained clearly how these indexes influence the performance of the spindle system. 

Reply:

Thanks for the reviewer’s valuable comment. The main evaluation indexes of static performance is static stiffness, which reflects the spindle system's ability to resist static external loads. The main evaluation indexes of dynamic performance include frequency response (dynamic stiffness), damping characteristics, and interference resistance. Frequency response refers to the ability of the spindle components to produce corresponding frequency changes when subjected to external forces of different frequencies. Damping characteristics describe the ability of the spindle components to resist energy loss when exposed to vibration shocks. Interference resistance is the ability of the spindle system to maintain stable operation when subjected to external disturbances. In this paper, we use the static stiffness, natural frequency, mode shapes, and critical rotational speed of the spindle system as evaluation indicators for the static and dynamic characteristics of the dual-rotor system.

The above explanation has been added to revised manuscript.

  

Comment 2: 

I noticed that the authors have carried out the analysis concerning static stiffness. What is the situation of the dynamic stiffness? It seems that dynamic stiffness is the key factor affecting the operating performance.

Reply:

Thanks for the reviewer’s valuable comment and the dynamic stiffness has been provided as shown in the following Figure 1 and 2:

      

Figure 1. Frequency response function of spindle      Figure 2. Frequency response function of eccentric shaft

 

Comment 3: 

In Section 2, the theoretical analysis is implemented based on many assumptions and approximations. All your assumptions should be listed in one area and justified if they are practical or not, e.g., superposition principle, small deformation, linear treatment, etc.

Reply:

Thanks for the reviewer’s valuable comment. The principles of superposition and small deformation assumptions in this paper are derived from Hertzian contact theory. This theory is used to analyze the contact deformation and contact stress between two elastic bodies. The assumptions of this theory include: (1) the deformations of the contact bodies are small; (2) the stress-strain relationship is linear and satisfies the principle of superposition. The theory has been widely used to analyze the stress and strain relationships in various types of bearings, and its validity has been established. This paper applies Hertzian contact theory to analyze the contact deformation of angular contact ball bearings, laying the foundation for further establishment of the static and dynamic model.

The above explanation has been added to revised manuscript.

 

Comment 4: 

Some of your equations are not clear. It is not only about filling papers with maths whereas you need to make everything clear. The expressions of Ti, Ci, and C should be further explained. In addition, is T defined as a scalar or a variable in Eq. (20)? The readability of Figs. 5 and 7 should be improved.

Reply:

Thanks for the reviewer’s valuable comment. In Figure 5, the spindle shaft and eccentric shaft are connected to each other through bearings to form a coupling unit. Therefore, C_i represents the coupling matrix of the i-th unit, and T_i is the transfer matrix of the i-th overall unit. Here, the overall unit refers to considering the spindel shaft and eccentric shaft as a whole. In Eq. (20), T is a matrix.

Using the transfer matrix method to establish the dynamic model of the spindle and eccentric shaft system requires simplifying the actual structures such as the rotor and turntable into a concentrated mass model. Therefore, Figure 5 is a simplified model of the spindle and eccentric shaft system (Figure 3). The simplification principle is: for the turntable, coupling, shaft diameter changes, and the position where the bearing is located, they should be used as nodes to divide the shaft section; The turntable is simplified as a rigid disc without thickness, and the ordinary shaft segment is simplified as a concentrated mass without rigidity and an elastic beam segment without considering mass; For bearings, they are simplified as a combination of a tension spring and a torsion spring. The concentrated mass of each axis segment is distributed to the left and right nodes according to the principle of invariant center of mass, and a natural axis segment can be simplified as an elastic beam and two concentrated masses. Therefore, a main axis can be simplified as n-1 elastic beams and n concentrated masses or rigid disks.

Figure 7 shows the force relationship of the coupled concentrated mass unit, which is formed by connecting the spindle shaft eccentric shaft in the spindle shaft eccentric shaft system in Figure 5 through bearings. Among them, subscript i represents the i-th mass unit, superscript S represents the main axis, E represents the eccentric axis, L represents the left side, and R represents the right side.

The above explanation has been added to revised manuscript.

 

Comment 5: 

 More detailed descriptions concerning Figs. 8 and 9 are required. By the way, can the relevant conclusions be drawn directly just based on engineering experience? The highlights and new findings of your research should be further clarified and discussed.

Reply:

Thanks for the reviewer’s good comment. Static stiffness can be derived from engineering experience, but the first-order critical speed may not be. The focus of this paper is to establish the static-dynamic model of the double-rotor spindle system for the bevel gear grinding machine, analyze the impact of certain structural parameters on the static and dynamic performance of the system, and provide reference for the design optimization of the bevel gear grinding machine. New findings are further discussed in the revised manuscript.

 

Comment 6: 

The research status in the Introduction should be further improved. Comparisons among different methods and the novelty of this paper should be discussed in detail.

Reply:

We accept the reviewer’s suggestion and more comparisons among different methods and the novelty of this paper have been added to the Introduction part in revised manuscript.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper deals with the enhancement of static and dynamic performance of the grinding wheel spindle system. A static mechanics model of the spindle system was established based on Castigliano's theorem. Based on the method of overall transfer matrix a dynamic model of the main spindle-eccentric shaft dual-rotor system was established, taking into account the effects of shear deformation and gyroscopic moments. On this basis, the impact of the spindle span, front and rear overhang of the eccentric shaft, and bearing stiffness on the static stiffness and first-order critical speed of the system was analyzed. Finally, static stiffness experiments, modal tests, and finite element simulation models were conducted to verify the static and dynamic models.

 

The paper contributes to the knowledge base.

I do not suspect the authors of plagiarism, fraud or other ethical concerns. The English language is appropriate and understandable.

 

Specific comments on the paper are given below.

 

Comments:

1.      The novelty of the post is not highlighted enough.

2.      Equations not derived by the authors of the paper should be referred to the literature from which they were taken.

3.      Subchapter 3.2: It is not evident what is meant by “the two-shaft and three-shaft assemblies”.

4.      Line 89, Figure 3: It is not evident what red lines mean.

5.      Line 108, the first Equation 6: Write total bending deformation “Y” in “YA” with a capital letter (the term on the left side of the equation).

6.      Line 215: Order of “30 mm” and “70 mm” should be changed, i.e. “being 70 mm and 30 mm, respectively.”

 

Formal comments:

1.      Write the space “ ” or do not write space “ ” between reference to the literature and previous word, i.e. write “method [5, 6]” or “method[5, 6]”.

2.      Unify the writing omega “Ω” in italics or roman font.

3.      Line 68, Figure 1: Use the same font type to describe individual parts of the spiral bevel gear grinding machine.

4.      Line 76, Figure 2: Enlarge the frame with the inscription “Eccentric shaft”.

5.      Lines 133 and 134: Unify writing “k” in “kt” with small or capital letters.

6.      Line 141: Unify writing “U” in “UZ” and “UP” in the same font type.

7.      Line 154: Write transfer coupling matrix “C” in italics.

8.      Line 162, equation 20: Write all “T” in italics.

9.      Line 198: Write the space “ ” between “(a)” and “Eccentric”.

10.  Line 230: Write “Figure 10.” instead of “Fig. 10” and write it in bold letters.

11.  Line 232: Write “Figure 11.” instead of “Fig. 11” and write it in bold letters.

12.  Line 268, Figure 13: Use the same font type to describe individual parts of the field diagram of the modal test and write “exciting” with the small letter “e” and “sensor” with the small letter “s”.

13.  Line 279, Figure 13: Write “pressure” with the small letter “p” and “force” with the small letter “f”.

14.  Line 289, Figures 15 and 16: Write in the caption of the horizontal axis “position” with one “s”.

15.  Line 374: Write “2019” instead of “2019 5”.

16.  Line 375: Does literature “22” really exist?

Comments on the Quality of English Language

I have no comments on the quality of English language,

Author Response

Answers to reviewers:

We thank the reviewer for his/her constructive criticisms that have helped us to improve our manuscript. The point-by-point response to the comments is given below. In addition, other comments in the attached reviewed manuscript have been studied and the corresponding errors have been corrected in the revised manuscript.

Comment 1: 

The novelty of the post is not highlighted enough.

Reply:

We accept the reviewer’s suggestion and the novelty of this paper have been added to the Introduction part in revised manuscript.

 

Comment 2: 

Equations not derived by the authors of the paper should be referred to the literature from which they were taken.

Reply:

The suggestion has been taken and equations have be referred to the laws or literature from which they were taken.

 

Comment 3: 

Subchapter 3.2: It is not evident what is meant by “the two-shaft and three-shaft assemblies”.

Reply:

Thanks for the reviewer’s good comment. To verify the accuracy of the dynamic model calculation results, multiple modal testing experiments were designed, including modal testing of a single spindle, two-shaft modal testing of the spindle-eccentric shaft combination, and three-shaft modal testing of the spindle-eccentric shaft-sleeve combination, as shown in Figure 13 in the revised manuscript.

 

Figure 13. Field diagram of modal test

 

Comment 4: 

Line 89, Figure 3: It is not evident what red lines mean.

Reply:

Thanks for the reviewer’s good comment. The red solid line has been changed to the red dashed line (as shown in Figure 3), which is the line of action of the bearing, corresponding to the bearing center of the angular contact bearing.

 

Figure 3. Schematic diagram of simplified position of force point of triple angular contact ball bearing

 

Comment 5: 

Line 108, the first Equation 6: Write total bending deformation “Y” in “YA” with a capital letter (the term on the left side of the equation).

Reply:

We accept the reviewer’s suggestion. The total bending deformation “Y” in “YA” has been written with a capital letter.

 

Comment 6: 

Line 215: Order of “30 mm” and “70 mm” should be changed, i.e. “being 70 mm and 30 mm, respectively.”

Reply:

We accept the reviewer’s suggestion. The order of “30 mm” and “70 mm” has be changed to “being 70 mm and 30 mm, respectively.”.

 

Comment 7: 

Write the space “ ” or do not write space “ ” between reference to the literature and previous word, i.e. write “method [5, 6]” or “method[5, 6]”. 

Reply:

Thanks for the reviewer’s valuable comment. the space “ ” between reference to the literature and previous word has been written in the revised manuscript.

 

Comment 8: 

Unify the writing omega “Ω” in italics or roman font. 

Reply:

The suggestion has been taken and the writing omega “Ω” has been unified in italics in the revised manuscript.

 

Comment 9: 

Line 68, Figure 1: Use the same font type to describe individual parts of the spiral bevel gear grinding machine. 

Reply:

We accept the reviewer’s suggestion. The individual parts of the spiral bevel gear grinding machine has been described using the same font type in Figure 1 in the revised manuscript.

 

Comment 10: 

Line 76, Figure 2: Enlarge the frame with the inscription “Eccentric shaft”.

Reply:

Thanks for the reviewer’s valuable suggestion. The frame with the inscription “Eccentric shaft” has been enlarged in Figure 2 in the revised manuscript. 

 

Comment 11: 

Lines 133 and 134: Unify writing “k” in “kt” with small or capital letters. 

Reply:

We accept the reviewer’s suggestion and the writing “k” in “kt” has been unified with capital letters in the revised manuscript.

 

Comment 12: 

Line 141: Unify writing “U” in “UZ” and “UP” in the same font type. 

Reply:

We accept the reviewer’s suggestion. The writing “U” in “UZ” and “UP” has been unified in the same font type in the revised manuscript.

 

Comment 13: 

Line 154: Write transfer coupling matrix “C” in italics.

Reply:

We accept the reviewer’s suggestion and the  transfer coupling matrix “C” has been written in italic in the revised manuscript.

 

Comment 14: 

Line 162, equation 20: Write all “T” in italics.

Reply:

We accept the reviewer’s suggestion and all “T” in equation 20 have been written in italics in the revised manuscript.

 

Comment 15: 

Line 198: Write the space “ ” between “(a)” and “Eccentric”.

Reply:

We accept the reviewer’s suggestion and the space “ ” between “(a)” and “Eccentric” has been written in the revised manuscript.

 

Comment 16: 

Line 230: Write “Figure 10.” instead of “Fig. 10” and write it in bold letters.

Reply:

We accept the reviewer’s suggestion and  “Fig. 10” has been changed to “Figure 10.” in the revised manuscript.

 

Comment 17: 

Line 232: Write “Figure 11.” instead of “Fig. 11” and write it in bold letters.

Reply:

We accept the reviewer’s suggestion and “Fig. 11” has been changed to “Figure 11.” in the revised manuscript.

 

Comment 18: 

Line 268, Figure 13: Use the same font type to describe individual parts of the field diagram of the modal test and write “exciting” with the small letter “e” and “sensor” with the small letter “s”.

Reply:

We accept the reviewer’s suggestion and the individual parts of the field diagram of the modal test have been described in the same font type and the “exciting” has been written with the small letter “e” and, “sensor” has been written with the small letter “s” in Figure 13 in the revised manuscript.

 

Comment 19: 

Line 279, Figure 14: Write “pressure” with the small letter “p” and “force” with the small letter “f”.

Reply:

We accept the reviewer’s suggestion and “pressure” has been written with the small letter “p” and “force” has been written with the small letter “f” in Figure 14 in the revised manuscript.

 

Comment 20: 

Line 289, Figures 15 and 16: Write in the caption of the horizontal axis “position” with one “s”.

Reply:

We accept the reviewer’s suggestion and the writing “possition” has been changed to “position” in Figure 15 and 16 in the revised manuscript.

 

Comment 21: 

Line 374: Write “2019” instead of “2019 5”. 

Reply:

We accept the reviewer’s suggestion and the writing “2019 5” has been changed to “2019” in the revised manuscript.

 

Comment 22: 

Line 375: Does literature “22” really exist?

Reply:

Thanks for the reviewer’s comment. Literature “22” is a literature written in Chinese and we can search it from google scholar.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors All my comments were accepted.