# Research on the Nonlinear Stiffness Characteristics of Double-Row Angular Contact Ball Bearings under Different Working Conditions

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis

- (1)
- Two SR-ACBBs have the same structural parameters, that is, the raceway contact curvature radii r
_{i}and r_{o}, the raceway contact diameters d_{i}and d_{o}, and the ball diameter and number D and Z, respectively; - (2)
- The influence of the lubrication and cage is not considered;
- (3)
- The outer raceway is fixed while the inner ring moves and rotates with the central shaft.

_{c}) and the axial clearance of the split inner ring (${\delta}_{p}$). During the installation process of double-row ball bearings, the axial clearance ${\delta}_{p}$ needs to be eliminated by axial locking force to achieve effective pre-tightening.

#### 2.1. Mechanics Analysis of DR-ACBB

^{L}-X

^{L}Y

^{L}Z

^{L}and O

^{R}-X

^{R}Y

^{R}Z

^{R}. Furthermore, the Y

_{L}and Y

_{R}axes, respectively, pass through the ball centers of the left and right side of the ACBB. On this basis, the mechanical equations of the moving inner raceway for DR-ACBB under a DB arrangement can be given as:

_{m}and D are the pitch diameter and ball diameter, respectively. Furthermore, in this paper, unless otherwise stated, the subscripts i and o, respectively, indicate the parameters used for inner and outer ball raceway action analysis, and the subscript k indicates the parameters used for the k

_{th}ball. The superscripts L and R, respectively, denote the parameters used for modeling the left-side and right-side DR-ACBB.

_{L}and O

_{R}are written as

**F**= {F

_{L}_{xL}, F

_{yL}, F

_{zL}, M

_{yL}, M

_{zL}} and

**F**= {F

_{R}_{xR}, F

_{yR}, F

_{zR}, M

_{yR}, M

_{zR}}, respectively. Then, the expressions of the mechanical equations of two SR-ACBBs are presented in Equations (2)–(5):

**d**and

_{L}**d**) of two SR-ACBBs at the local coordinate center points O

_{R}_{L}and O

_{R}can be calculated by the relative displacement vector of DR-ACBB at the global coordinate center points O (

**d**= {${\delta}_{x}$,${\delta}_{y}$,${\delta}_{z}$,${\theta}_{y}$,${\theta}_{z}$}):

_{L}and O

_{R}can be determined by the matrix transformation of the relative displacement vector of DR-ACBB. Furthermore, the same initial preload displacement (${\delta}_{p}/2$) is also considered in the relative displacement vector calculation of two SR-ACBB.

- i.
- DF arrangement:

- ii.
- DT arrangement:

**N**

_{3}and

**N**

_{4}used in the above expressions are written as:

_{th}ball inside the left-side ACBB to build the local mechanical equations [9]:

_{c}is the ball centrifugal force, and then the ball raceway’s normal contact load Q and tangential friction load T can be further given as [13]:

_{th}ball inside the left-side ACBB for DR-ACBB under a DB arrangement is shown in Figure 4, and the initial positions of the inner-ring’s curvature center and ball center are changed from points and ${O}_{bk}^{L}$ to points ${\dot{O}}_{ik}^{L}$ and ${\dot{O}}_{bk}^{L}$, respectively. Therefore, the ball raceway’s normal deformation is given as:

#### 2.2. Iterative Calculation of the Proposed Model

**d**and stiffness matrix

**K**of DR-ACBB play an important role in the iterative operation. The displacement vector

**d**is continuously modified through the stiffness matrix

**K**until the iteration errors are less than the set threshold value, and the stiffness matrix

**K**needs to be updated continuously by the calculation results of the inner iteration. Furthermore, the parallel calculation scheme can be used in ball mechanical equations solutions to further improve the calculation efficiency.

#### 2.3. Analytical Formulation of the Stiffness Matrix of DR-ACBB

**K**of DR-ACBB are given as the Jacobian matrix of the external load vector

**F**to the relative displacement vector

**d**:

**K**and

_{L}**K**denote the stiffness matrix of two SR-ACBBs. Substituting Equation (25) into Equation (24), one can obtain:

_{R}**F**and displacement vector

_{L}**d**at the local coordinate center points O

_{L}_{L}. Considering that the relationships of the external forces and deformations of SR-ACBB are determined by both the explicit and the implicit equations, the intermediate variables

**x**= {X

_{k}_{1k}, X

_{2k}, A

_{1k}, A

_{2k}} are introduced to divide the stiffness matrix calculation into two steps:

## 3. Numerical Simulation and Discussions

#### 3.1. Analysis of Nonlinear Stiffness Characteristic of the Axially Loaded DR-ACBB

#### 3.2. Analysis of Nonlinear Stiffness Characteristic of the Radially Loaded DR-ACBB

#### 3.3. Analysis of Nonlinear Stiffness Characteristic of the Combined-Loaded DR-ACBB

## 4. Conclusions

- (1)
- DR-ACBBs under DB and DF configurations have the same variation rule in axial and radial stiffness, and DR-ACBBs under DB and DF configurations show the nonlinear spring characteristics of soft first and then hard with the increase in external load;
- (2)
- The SR-ACBB or part of the balls may be unloaded for DR-ACBB under the large load ranges, which further leads to the sudden change in the nonlinear stiffness characteristics of DR-ACBB;
- (3)
- The initial preload has a great influence on the nonlinear stiffness characteristics of DR-ACBB, and it can effectively increase the external load range corresponding to the stiffness attenuation of DR-ACBB;
- (4)
- DR-ACBB under a DB configuration have higher angular stiffness and bending moment resistance than those of DR-ACBB under a DF configuration.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Bercea, I.; Nelias, D.; Cavallaro, G. A unified and simplified treatment of the non-linear equilibrium problem of double-row rolling bearings. Part 1: Rolling bearing model. Proc. Inst. Mech. Eng. Part J.-J. Eng. Tribol.
**2003**, 217, 205–212. [Google Scholar] [CrossRef] - Gunduz, A.; Singh, R. Stiffness matrix formulation for double row angular contact ball bearings: Analytical development and validation. J. Sound Vib.
**2013**, 332, 5898–5916. [Google Scholar] [CrossRef] - Gunduz, A.; Dreyer, J.T.; Singh, R. Effect of bearing preloads on the modal characteristics of a shaft-bearing assembly: Experiments on double row angular contact ball bearings. Mech. Syst. Signal. Pr.
**2012**, 31, 176–195. [Google Scholar] [CrossRef] - Petersen, D.; Howard, C.; Prime, Z. Varying stiffness and load distributions in defective ball bearings: Analytical formulation and application to defect size estimation. J. Sound Vib.
**2015**, 337, 284–300. [Google Scholar] [CrossRef] - Xu, T.; Yang, L.; Wu, W.; Wang, K. Effect of angular misalignment of inner ring on the contact characteristics and stiffness coefficients of duplex angular contact ball bearings. Mech. Mach. Theory.
**2021**, 157, 104178. [Google Scholar] [CrossRef] - Poplawski, J.V.; Maureillo, J.A. Skidding in Lightly Loaded High Speed Ball Thrust Bearings; ASME: New York, NY, USA, 1969. [Google Scholar]
- Harris, T.A. An analytical method to predict skidding in thrust-loaded, angular-contact ball bearings. J. Lubr. Technol.
**1971**, 93, 17–23. [Google Scholar] [CrossRef] - Harris, T.A. Ball motion in thrust-loaded, angular contact bearings with Coulomb friction. J. Lubr. Technol.
**1971**, 93, 32–38. [Google Scholar] [CrossRef] - Harris, T.A. Rolling Bearing Analysis, 4th ed.; John Wiley and Sons, Inc.: New York, NY, USA, 2000. [Google Scholar]
- Ding, C.A.; Zhou, F.Z.; Zhu, J.; Zhang, L. Raceway control assumption and the determination of rolling element attitude angle. Chin. J. Mech. Eng.
**2001**, 37, 58–61. [Google Scholar] [CrossRef] - Tomović, R. Calculation of the boundary values of rolling bearing deflection in relation to the number of active rolling elements. Mech. Mach. Theory.
**2012**, 47, 74–88. [Google Scholar] [CrossRef] - Tomović, R. Calculation of the necessary level of external radial load for inner ring support on q, rolling elements in a radial bearing with internal radial clearance. Int. J. Mech. Sci.
**2012**, 60, 23–33. [Google Scholar] [CrossRef] - Wang, W.Z.; Hu, L.; Zhang, S.G.; Zhao, Z.Q.; Ai, S.Y. Modeling angular contact ball bearing without raceway control hypothesis. Mech. Mach. Theory.
**2014**, 82, 154–172. [Google Scholar] [CrossRef] - Yan, K.; Wang, Y.; Zhu, Y.; Hong, J.; Zhai, Q. Investigation on Heat Dissipation Characteristic of Ball Bearing Cage and Inside Cavity at Ultra High Rotation Speed. Tribol. Int.
**2016**, 93, 470–481. [Google Scholar] [CrossRef] - Yang, Z.; Yu, T.; Zhang, Y.; Sun, Z. Influence of Cage Clearance on the Heating Characteristics of High-Speed Ball Bearings. Tribol. Int.
**2017**, 105, 125–134. [Google Scholar] [CrossRef] - Ren, X.; Zhai, J.; Ren, G. Calculation of radial load distribution on ball and roller bearings with positive, negative and zero clearance. Int. J. Mech. Sci.
**2017**, 131, 1–7. [Google Scholar] - Liu, J.; Shao, Y. Dynamic modeling for rigid rotor bearing systems with a localized defect considering additional deformations at the sharp edges. J. Sound. Vib.
**2017**, 398, 84–102. [Google Scholar] [CrossRef] - Liu, J. A dynamic modelling method of a rotor-roller bearing-housing system with a localized fault including the additional excitation zone. J. Sound. Vib.
**2020**, 469, 115144. [Google Scholar] [CrossRef] - Gargiulo, E.P. A Simple Way to Estimate Bearing Stiffness. Mach. Des.
**1980**, 52, 107–110. [Google Scholar] - Wardle, F.P.; Lacey, S.J.; Poon, S.Y. Dynamic and static characteristics of a wide speed range machine tool spindle. Precis. Eng.
**1983**, 5, 175–183. [Google Scholar] [CrossRef] - Lim, C.T.; Singh, R. Vibration transmission through rolling element bearings, part I: Bearing stiffness formulation. J. Sound Vib.
**1990**, 139, 179–199. [Google Scholar] [CrossRef] - Lim, C.T.; Singh, R. Vibration transmission through rolling element bearings, part II: System studies. J. Sound Vib.
**1990**, 139, 201–225. [Google Scholar] [CrossRef] - Lim, C.T.; Singh, R. Vibration transmission through rolling element bearings, Part III: Geared rotor system studies. J. Sound Vib.
**1991**, 151, 31–54. [Google Scholar] [CrossRef] - Houpert, L. A uniform analytical approach for ball and roller bearings calculations. J. Tribol.
**1997**, 119, 851–858. [Google Scholar] [CrossRef] - Hernot, X.; Sartor, M.; Guillot, J. Calculation of the stiffness matrix of angular contact ball bearings by using the analytical approach. J. Mech. Des.
**2000**, 122, 83–90. [Google Scholar] [CrossRef] - Sheng, X.; Li, B.; Wu, Z.; Li, H. Calculation of ball bearing speed-varying stiffness. Mech. Mach. Theory.
**2014**, 81, 166–180. [Google Scholar] [CrossRef] - Noel, D.; Ritou, M. Complete analytical expression of the stiffness matrix of angular contact ball bearings. J. Tribol.
**2013**, 135, 041101. [Google Scholar] [CrossRef] - Liu, J.; Tang, C.; Wu, H.; Xu, Z.; Wang, L. An analytical calculation method of the load distribution and stiffness of an angular contact ball bearing. Mech. Mach. Theory.
**2019**, 142, 103597. [Google Scholar] [CrossRef] - While, M.F. Rolling element bearing vibration transfer characteristics: Effect of stiffness. J. Appl. Mech.
**1979**, 46, 677–684. [Google Scholar] [CrossRef] - Yang, Z.H.; Li, B.T.; Yu, T.X. Influence of structural parameters and tolerance on stiffness of high-speed ball bearings. Int. J. Precis. Eng. Man.
**2016**, 17, 1493–1501. [Google Scholar] [CrossRef] - Yang, Z.; Chen, H.; Yu, T. Effects of rolling bearing configuration on stiffness of machine tool spindle, P. I. Mech. Eng. C.-J. Mec.
**2018**, 232, 775–785. [Google Scholar] [CrossRef] - Li, J.; Zhu, Y.; Yan, K.; Yan, X.; Liu, Y.; Hong, J. Research on the axial stiffness softening and hardening characteristics of machine tool spindle system. Int. J Adv. Manuf. Tech.
**2018**, 99, 951–963. [Google Scholar] [CrossRef] - Zhang, J.; Fang, B.; Zhu, Y.; Hong, J. A comparative study and stiffness analysis of angular contact ball bearings under different preload mechanisms. Mech. Mach. Theory.
**2017**, 115, 1–17. [Google Scholar] [CrossRef] - Zhang, J.; Fang, B.; Hong, J.; Wan, S.; Zhu, Y. A general model for preload calculation and stiffness analysis for combined angular contact ball bearings. J. Sound. Vib.
**2017**, 411, 435–449. [Google Scholar] [CrossRef] - Fang, B.; Zhang, J.; Yan, K.; Hong, J.; Wang, M. A comprehensive study on the speed-varying stiffness of ball bearing under different load conditions. Mech. Mach. Theory
**2019**, 136, 1–13. [Google Scholar] [CrossRef]

**Figure 1.**Three configurations of DR-ACBB: (

**a**) DB arrangement; (

**b**) DF arrangement; (

**c**) DT arrangement.

**Figure 2.**The coordinate systems and force state of DR-ACBB under DB arrangement: (

**a**) the diagram of DR-ACBB; (

**b**) the diagram of the single-side ACBB.

**Figure 4.**The geometric analysis of local ball of the left-side ball bearing for the double-row ball bearing in DB arrangement.

**Figure 6.**The result curves of the axial displacements and load distributions versus the axial load for DR-ACBB under different configurations (N = 1000 rpm): (

**a**) the axial displacement; (

**b**) the axial load distribution.

**Figure 7.**The axial and radial stiffness variation curves versus the axial external load for DR-ACBB at three different speeds: (

**a**) the axial stiffness; (

**b**) the radial stiffness.

**Figure 8.**The change curves of the axial stiffness of two SR-ACBBs versus the axial external load for DR-ACBB under DB configuration: (

**a**) rotating speed = 1000 rpm; (

**b**) rotating speed = 5000 rpm; (

**c**) rotating speed = 8000 rpm.

**Figure 9.**The comparison results of the angular stiffness versus axial external load of DR-ACBB under DB and DF configurations: (

**a**) DB configuration; (

**b**) DF configuration.

**Figure 10.**The effects of the speed and preload on the nonlinear stiffness variations of DR-ACBB: (

**a**) the actual preload and axial stiffness varying with the speed; (

**b**) the axial stiffness varying with axial load of DR-ACBB with different initial preloads (δ

_{p}/2).

**Figure 11.**The results of the radial stiffness and ball number in loaded area versus radial load of DR-ACBB at static: (

**a**) Y-axis radial stiffness; (

**b**) Z-axis radial stiffness.

**Figure 12.**The results of the radial stiffness versus radial load for DR-ACBB under three different rotating speeds: (

**a**) Y-axis radial stiffness; (

**b**) Z-axis radial stiffness.

**Figure 13.**The comparison results of the angular stiffness versus radial load of DR-ACBB under DB and DF configurations: (

**a–1**,

**a–2**) DB configuration; (

**b–1**,

**b–2**) DF configuration.

**Figure 14.**The effects of the initial preload on the radial stiffness variations of DR-ACBB: (

**a**) Y-axis radial stiffness; (

**b**) Z-axis radial stiffness.

**Figure 15.**The results of the nonlinear stiffness variation for DR-ACBB subjected to constant radial loads and varying axial load (N = 1000 rpm): (

**a**) axial stiffness; (

**b**) radial stiffness.

**Figure 16.**The results of the nonlinear stiffness variation for DR-ACBB subjected to constant axial loads and varying radial load (N = 1000 rpm): (

**a**) axial stiffness; (

**b**) radial stiffness.

**Figure 18.**The radial and angular displacement and stiffness of double-row ball bearing in different configurations: (

**a**) radial displacement; (

**b**) radial stiffness; (

**c**) angular offset; (

**d**) angular stiffness.

Parameters | 320 |
---|---|

Curvature radius (inner-raceway) r_{i} (mm) | 4.54 |

Curvature radius (outer-raceway) r_{o} (mm) | 4.54 |

Contact diameter (inner-raceway) d_{i} (mm) | 61.22 |

Contact diameter (outer-raceway) d_{o} (mm) | 78.78 |

Ball number Z | 12 |

Ball diameter D (mm) | 8.73 |

Pitch diameter dm (mm) | 70 |

Preload displacement ${\delta}_{p}$(μm) | 12 |

Radial clearance (μm) | 100 |

The ball center distance (mm) | 15 |

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**MDPI and ACS Style**

Fang, B.; Zhang, J.; Hong, J.; Yan, K.
Research on the Nonlinear Stiffness Characteristics of Double-Row Angular Contact Ball Bearings under Different Working Conditions. *Lubricants* **2023**, *11*, 44.
https://doi.org/10.3390/lubricants11020044

**AMA Style**

Fang B, Zhang J, Hong J, Yan K.
Research on the Nonlinear Stiffness Characteristics of Double-Row Angular Contact Ball Bearings under Different Working Conditions. *Lubricants*. 2023; 11(2):44.
https://doi.org/10.3390/lubricants11020044

**Chicago/Turabian Style**

Fang, Bin, Jinhua Zhang, Jun Hong, and Ke Yan.
2023. "Research on the Nonlinear Stiffness Characteristics of Double-Row Angular Contact Ball Bearings under Different Working Conditions" *Lubricants* 11, no. 2: 44.
https://doi.org/10.3390/lubricants11020044