Analysis of Dynamic Response of a Two Degrees of Freedom (2-DOF) Ball Bearing Nonlinear Model

Round 1

Reviewer 1 Report

Dear Authors

Congratulations to the Authors of an interesting study on the dynamics of deep groove ball bearings. The analysis of the dimensionless mathematical model of the dynamics of the single-row deep groove ball bearing, included in the second part of the manuscript, deserves a special mention. As for the first part of the study, probably due to the desire to simplify the model, several parameters have been omitted that have a greater or lesser impact on the dynamics, i.e. also on the values of internal forces (friction) and bearing durability. However, as the authors wrote, the simplification of the model was justified by (a successful according to the reviewer) attempt to avoid calculation errors caused by the multiplication of very small values of deformations and damping with very large values of stiffness in the dynamics equations. At the same time, the very troublesome determination of the contact stiffness of the ball-inner ring was avoided. The assumption of a rigid outer ring together with the housing simplifies the model a bit, as they are in practice less stiff compared to the stiffness of the inner ring with the shaft journal. Not to mention the case of the movable outer ring of the bearing. It is a pity that no additional external force thrust (axial) was applied to the bearing, since it is a deep groove (Conrad). Since the eccentricity of the applied load was taken into account, it was also possible not to ignore the gyroscopic effect. Because at high speeds of rotation of the balls in the bearings, forces are generated due to the change of direction of their spin axes.

In the opinion of the reviewer, one cannot have any critical remarks to the method of analyzing the issue and to the results obtained. They are particularly interesting and at the same time informative for practitioners, e.g. it is important to avoid eccentricities or misalignments of shafts. And this is due, among others, to the fact that the factories were skillfully selected, which have a large impact on the dynamics of the bearing. The method of analysis is also largely original. It is difficult to find a study where such valuable results were obtained on the basis of a relatively simple model. Perhaps only apart from paper number 1 (attached below), where only slightly less important results were obtained on the basis of an even more simple model. This paper could be included in the literature list as well as the next paper (number 2) where the contact stiffness is explained very thoroughly on the basis of Hertzian theory, also used in the reviewed manuscript.

• K. Kankar, Satish C. Sharma, S.P. Harsha: Vibration based performance prediction of ball bearings caused by localized defects. Nonlinear Dyn (2012) 69:847–875

• Liu J, Xu Y, Wang L, Xu Z, Tang C. Influence of the local defect distribution on vibration characteristics of ball bearings. Eksploatacja i Niezawodnosc – Maintenance and Reliability 2019; 21 (3): 485–492, http://dx.doi.org/10.17531/ein.2019.3.15

Best regards, Reviewer.

Comments for author File: Comments.pdf

Author Response

Dear Reviewer,

 

thank you for the substantive review of our manuscript. In the attached PDF file, please find our response to the review. 

Author Response File: Author Response.pdf

Reviewer 2 Report

This manuscript discusses the analysis of a dimensionless mathematical model for the single-row ball bearing by the recurrence-based methods. The text appears interesting, but there are some minor issues that should be addressed.

1. There are some grammatical errors that should be addressed throughout the manuscript.

2. Please provide references for the equations listed in the text.

3. The authors discuss chaotic motion in the section on orbit plots and phase portraits. The authors should give a brief discussion on chaotic behavior in the introduction section. The following references should help with this:

J. Appl. Mech.-Trans. ASME 2020, 87, 10; Chaos, Solitons & Fractals 2018, 116, 166-175; Int. J. Non-Linear Mech. 2013, 50, 1-10

Author Response

Dear Reviewer,

 

thank you for the substantive review of our manuscript. In the attached PDF file, please find our response to the review. 

Author Response File: Author Response.pdf

Reviewer 3 Report

The reviewer feels that the paper is interesting, and it is within the scope of the Journal, but one additional revision should be done. The written English needs a careful revision. Please see the following comments:

Specific comments:

line 61, improve the statement: “… direction by which one of the rings can move to the other.”, but also the statement between lines 73-76.

In the paragraph comprising lines 106 and 112 a reference to Figure 1 it would help the reader. In line 125, please, remove the ":" after where. The usual way of saying this is: where ... is the angular position of the first ball, i is the ...etc. This kind of written appears through all the paper, please, perform a careful revision of this issue. In line 126, authors write about the rotational velocity of the cage, please represent this velocity on Figure 1. Authors, should explain the meaning of the statement presented at line 135.

authors should give information about the notation used within Figure 2. the angular velocity presented in this figure is the shaft velocity? if it is the case, the first circle should have the variable m_s too.

what is the meaning of letter n in equations 7 and 8? please verify if these two equations are well written? the notation of contact forces seems to be incoherent. A more careful explanation should be given in order to obtain equation(10).

Sections related with the recurrence analysis are very interesting and show how a different tool can be used to study this kind of problems. 

Author Response

Dear Reviewer,

 

thank you for the substantive review of our manuscript. In the attached PDF file, please find our response to the review. 

Author Response File: Author Response.pdf

Reviewer 4 Report

This paper proposes a 2DOF nonlinear ball bearing model for comprehensive dynamic analysis. Remarkably, the recurrence plots are introduced into the numerical investigation which is quite new in such research field. Interesting results are presented, and validations are provided. I recommend it for publication subject to revisions. The following are a few suggestions and comments that may be helpful. 1. First, is it possible to conduct an experimental study regarding the proposed model and theory? With such demonstrations, the significance will be further promoted. 2. It is encouraging that the authors considered raceway waviness on the model. Sinusoidal variations are used. However, it is an ideal assumption and inaccurate. There could be more reasonable descriptions and depends only on statistics which are more reliable for engineering products. Such as Gaussian and non-Gaussian surfaces. 3. Based on the work in [27], a further consideration of the factors, as stated in the paper, i.e., the variable eccentricity of the shaft, shape errors caused by manufacturing imperfections and so on can all be categorized by parametric uncertainties, whether in probability measure or interval model. They are reflected on the bearing stiffness or damping and then affects the dynamic responses of rotor-bearing systems (Journal of Sound and Vibration, 466, 2020, 115047). It is advised to consider that aspect. 4. Please include more on the recurrence analysis in bearing or rotor systems, including what it can mainly achieve and how relevant discussion could help the dynamic response analysis.

Author Response

Dear Reviewer,

 

thank you for the substantive review of our manuscript. In the attached PDF file, please find our response to the review. 

Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

It can be accepted.