Research on the Nonlinear Stiffness Characteristics of Double-Row Angular Contact Ball Bearings under Different Working Conditions
Round 1
Reviewer 1 Report
The paper presents an analytical model for the stiffness of double row angular contact bearings, and analyzed numerically the changes of radial, axial and angular stiffness under different load and speed conditions for the three typical configurations of representative double row angular contact bearings. The analytical model and the calculation results are valuable for the design and application of double row angular contact bearings. a few minor modifications are suggested for revision.
1) For the DR-ACBB with a split inner ring as shown in Fig.2a, how are the radial load, Fy, partitioned between the two rows? Is it assumed to be equally partitioned, or not?
2) How large is the value of the radial clearance for the radially loaded bearings? It is not listed and described in the context.
3) Please show the real contact angles of the left and right row bearings under combined load conditions.
4) There is a typo in the caption of Fig.14, which is wrongly numbered as Fig.12.
Author Response
Manuscript Title: Research on the nonlinear stiffness characteristics of the double row angular contact ball bearings under different working conditions
Manuscript Number: lubricants-2157001
Reviewer 1:
The paper presents an analytical model for the stiffness of double row angular contact bearings, and analyzed numerically the changes of radial, axial and angular stiffness under different load and speed conditions for the three typical configurations of representative double row angular contact bearings. The analytical model and the calculation results are valuable for the design and application of double row angular contact bearings. a few minor modifications are suggested for revision.
- For the DR-ACBB with a split inner ring as shown in Fig.2a, how are the radial load, Fy, partitioned between the two rows? Is it assumed to be equally partitioned, or not?
Response: Thanks for your question. For your question, it can be roughly divided into two cases.
- When the double-row angular contact ball bearing is only subject to radial force except the initial preload, the radial force will be evenly distributed to the two rows of ball bearings.
- When the double-row ball bearing is subject to combined load, the load conditions of the two types of ball bearings will no longer be consistent.
- How large is the value of the radial clearance for the radially loaded bearings? It is not listed and described in the context.
Response: Thanks for your question, the size of the radial clearance for the double-row angular contact ball bearing used in the paper has been added to Table 1(i.e., written in red letter).
- Please show the real contact angles of the left and right row bearings under combined load conditions.
Response: Thanks for your suggestion, the reason why we did not give the results of the contact angles for the double-row angular contact ball bearing are listed as follows:
- First of all, the sizes of contact angles are not the core variables of this manuscript. As the key variable of ACBB, the sizes of the contact angles have an important impact on the its high-speed and load-bearing characteristics. In general, the smaller the initial contact angle of ball bearing, the smaller axial stiffness and the better of high-speed characteristic. Therefore, the contact angle analysis of ball bearings is often used to study the high-speed characteristics of ball bearing under pure axial load (please refer to [9]);
- Secondly, for ball bearings under combined load, the contact angles of balls at different angular positions inside ball bearing are different due to the effect of radial force, so it is difficult to analyze the dynamic characteristics of ball bearings directly through the contact angles. Especially for the research in this paper, it involves the ball-raceway separation phenomena inside ball bearing under the action of a large radial load. At this time, part of the balls inside the bearing are in free state, resulting in the disappearance of the contact angles.
Therefore, to study the change law of contact angle of the double-row ball bearing under different loads, it needs to involve the comparative analysis of the size and change law of the contact angle of DR-ACBB under different working conditions, which deviates slightly from the main research purpose of this paper. If only the contact angle change of double-row angular contact ball bearing bearing under combined load is given separately, the continuity and integrity may be damaged. Therefore, the size and change law of contact angle are not given in the original manuscript. So we hoped the reviewer can understand.
- There is a typo in the caption of Fig.14, which is wrongly numbered as Fig.12.
Response: Sorry for this mistake, the wrong number (i.e., Fig.14) of Fig. 14 has been revised. Thank you for pointing out the above mistake and improving the quality of the manuscript.
Reviewer 2 Report
The paper deals with a model for double row contact ball bearings and how nonlinear stiffness charecteristics can be derived. The paper is well written. I have one major and some minor points.
Major point:
- I understand that "stiffness" defines the correlation between force and displacement, but this somehow irritates me when centrifugal forces are considered. As far as I understand, the centrifugal forces have an impact on the equilibrium position so that it changes the ratio between force and displacement but in terms of a material and kinematics property, the term "stiffness" sounds strange to me in this respect. I understand that in MBS-models the charecteristics of such a bearing is implemented like this, but it is no physical but rather a numerical stiffness for such type of bearings from my point of view. Maybe the authors can clarify this point.
Minor points:
- Fig. 1b: the bottom part of the drawing looks wrong to me (it is identical to Fig 1a but should be mirrored to the top part of Fig.1b.
- The fact that DB and DF is identical for the axial and radial stiffness is obvious to me as it is symmetric in this respect. This should be mentioned in the text.
- In Figure 5 in the top box you write that you define external loads in advance. As you are aiming at calculating stiffnesses this somehow confuses me, as it should be sufficient to define the displacement in advance and calculate the corresponding forces and moments. As there is also no corresponding loop in this Figure, I assume that the external load is not defined in advance or am I wrong?
- In line 169 you state: "In order to shorten the article space, this paper will not repeat derivation process of the ball inside the right-side ACBB.". I would prefer to read such derivation in an appendix.
- line 190: "calculation" -> "calculate"
Author Response
Manuscript Title: Research on the nonlinear stiffness characteristics of the double row angular contact ball bearings under different working conditions
Manuscript Number: lubricants-2157001
Reviewer 2:
The paper deals with a model for double row contact ball bearings and how nonlinear stiffness characteristics can be derived. The paper is well written. I have one major and some minor points.
- I understand that "stiffness" defines the correlation between force and displacement, but this somehow irritates me when centrifugal forces are considered. As far as I understand, the centrifugal forces have an impact on the equilibrium position so that it changes the ratio between force and displacement but in terms of a material and kinematics property, the term "stiffness" sounds strange to me in this respect. I understand that in MBS-models the characteristics of such a bearing is implemented like this, but it is no physical but rather a numerical stiffness for such type of bearings from my point of view. Maybe the authors can clarify this point.
Response: Thank you for putting forward a very interesting question. I will try to give some explanations about this question. I hope you can be satisfied. First of all, for the linear system, the definition of its stiffness is relatively simple, that is, the ratio between load and displacement, and the magnitude is basically constant, which represents the ability of the system to resist deformation. For the object studied in this paper, it is a typical nonlinear system (the load-displacement curve cannot be described by a simple first-order function, which may include quadratic, cubic or even more degree polynomials), and its size cannot be determined by a simple ratio between load and displacement. But for the bearing-rotor system, we need to determine its natural frequency to further dynamic characteristics. At this time, we need to give a concept of equivalent linear stiffness, whose size is defined by the slope of the load-displacement interval at different positions, that is, the Jacobian matrix used in our original text, which has been a consensus in the research of bearing system. Therefore, I think that the stiffness of the nonlinear system represented by ball bearings is more often referred to as its equivalent linear stiffness, which is a little similar to the numerical stiffness you mentioned. Of course, its size changes with the change of load and displacement.
- 1b: the bottom part of the drawing looks wrong to me (it is identical to Fig 1a but should be mirrored to the top part of Fig.1b.
Response: Thank you very much for helping me find the errors in the paper. We have revised them in the revision version.
- The fact that DB and DF is identical for the axial and radial stiffness is obvious to me as it is symmetric in this respect. This should be mentioned in the text.
Response: Thanks for your suggestion. We have made supplementary explanations in the original manuscript.
- In Figure 5 in the top box you write that you define external loads in advance. As you are aiming at calculating stiffnesses this somehow confuses me, as it should be sufficient to define the displacement in advance and calculate the corresponding forces and moments. As there is also no corresponding loop in this Figure, I assume that the external load is not defined in advance or am I wrong?
Response: Thanks for your question, I will try to explain your confusion. In this study, our main goal is to calculate the stiffness of DR-ACBB under different load conditions, so we assume that the external forces act on the bearing is the known variables, and then obtain the ring relative displacements and contact deformations through the iterative operation, so as to obtain the complete stiffness curve with displacement of DR-ACBB. Therefore, we design a two-nest iterative algorithm, and the inner iteration is applicable to calculate the local variables of the balls, and the outer iteration is used to calculate the relative deformation of the rings. Of course, you mentioned another calculation idea that taking the relative displacement of the rings as the known variables. For bearings under the action of pure axial force and radial force, the second iteration idea is indeed feasible and more efficient, because it omits the outer iteration used to calculate the relative displacement of the rings. But a new problem is generated. For ball bearings under the combined load condition, due to the generation of angular displacement and the effect of non-diagonal stiffness coupling terms, we do not know how to obtain the desired stiffness load-curve by effectively adjusting the magnitude of axial displacement and radial displacement. At this time, we can only obtain it by taking the load as a known variable and carrying out double-layer iterative calculation.
- - In line 169 you state: "In order to shorten the article space, this paper will not repeat derivation process of the ball inside the right-side ACBB.". I would prefer to read such derivation in an appendix.
Response: Thanks for your suggestion, the modeling process of single-row bearing on the right-side is the same as that of single-row ball bearing on the left-side (only the variables substitution is used). Without affecting the readers' reading and understanding and considering the simplicity of the article, it is not repeated in the article. I hope you can understand this. Thanks.
- line 190: "calculation" -> "calculate"
Response: Thank you very much for helping me find this error in the paper. We have revised them in the revision version.
Round 2
Reviewer 2 Report
Thank you for your comments. From my point of view, the paper can be published in its present form.
