# Research on the Power Loss of High-Speed and High-Load Ball Bearing for Cryogenic Turbopump

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Models of Ball Bearing for Cryogenic Turbopump

#### 2.1. Frictional Coefficients of Bearing’s Contact Interfaces

#### 2.2. Dynamic Model of Ball Bearing

#### 2.2.1. Coordinate System

#### 2.2.2. Nonlinear Dynamics Differential Equations of the jth Ball

#### 2.2.3. Nonlinear Dynamics Differential Equations of the Cage

#### 2.2.4. Nonlinear Dynamics Differential Equations of the Inner Ring

#### 2.3. Power Loss Model of the Ball Bearing

- (1)
- The power loss of the ball sliding along the direction of the short axis:

- (2)
- Power loss due to the spinning sliding of the ball:

- (3)
- Power loss due to the elastic hysteresis

- (4)
- Power loss due to the contact between the ball/ ring and the cage:

- (5)
- Power loss due to churning and drag

- (6)
- Total power loss

## 3. Analysis of Power Loss of the Ball Bearings of a Cryogenic Turbopump

#### 3.1. Component of Power Loss

#### 3.2. Influence of Structural Parameters on Power Loss

#### 3.2.1. Influence of Outer Raceway Diameter D_{e} on Power Loss

#### 3.2.2. Influence of Inner Raceway Diameter d_{i} on Power Loss

#### 3.2.3. Influence of Inner Raceway Curvature Radius Coefficient f_{i} on Power Loss

_{i}.

#### 3.2.4. Influence of Outer Raceway Curvature Radius Coefficient f_{o} on Power Loss

#### 3.3. Influence of Working Conditions on Power Loss

#### 3.3.1. Influence of Axial Load F_{a} on Power Loss

#### 3.3.2. Influence of Radial Load F_{r} on Power Loss

#### 3.3.3. Influence of Bearing Speed n_{i} on Power Loss

## 4. Temperature Field Analysis and Test Verification

## 5. Conclusions

- The total of ${H}_{S}$, ${H}_{drag}$, and ${H}_{c}$ represents more than 80% of the power loss of a ball bearing within a cryogenic turbopump, and in particular ${H}_{Si}$ represents the largest percentage (over 45%) throughout. So special attention should be paid to the spin-roll ratio $S{R}_{i}$ of the ball, which can be a key indicator for this type of ball bearing. At the same time, ${H}_{drag}$ and ${H}_{c}$ cannot be ignored when the ball bearing is working at high speed. The structural design of the cage and the flow of cryogenic fluid should be the focus of the next study.
- A relatively small radial clearance and contact angle of a ball bearing within a cryogenic turbopump are suggested.
- An inner raceway curvature radius coefficient ${f}_{\mathrm{i}}$ with a larger value is suggested to reduce the power loss, but this will increase the maximum contact stress ${P}_{i}$ significantly. Therefore, there is a reasonable range of ${f}_{\mathrm{i}}$ to balance power loss and fatigue life. The outer raceway curvature radius coefficient ${f}_{\mathrm{o}}$ has a minor effect on the power loss compared to ${f}_{\mathrm{i}}$, but a larger ${f}_{\mathrm{o}}$ leads to a larger contact stress ${P}_{o}$ that is harmful to the bearing’s fatigue life. Therefore, a relatively small value of ${f}_{\mathrm{o}}$ is suggested. For the ball bearing in this paper, ${f}_{\mathrm{i}}$ = 0.540 and ${f}_{\mathrm{o}}$ = 0.520 are suggested.
- When a ball bearing is working at a larger ratio of ${F}_{a}$ to ${F}_{r}$, the power loss of the ball bearing does not change much. A larger axial force ${F}_{a}$ is the key factor to impact the working states of the ball bearing, which leads to a significant change in the power loss.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Change of frictional coefficient with sliding velocity [35].

**Figure 3.**Ball-disc test specimens. (

**a**) 440C-Ag-coated disk. (

**b**) 440C-PTFE-coated disk. (

**c**) 440C-solid PTFE disk.

**Figure 4.**The comparisons between the computation results and the experimental data. (

**a**) 440C-Ag coating. (

**b**) 440C-PTFE coating. (

**c**) 440C-solid PTFE.

**Figure 16.**Influence of inner raceway curvature radius coefficient ${f}_{\mathrm{i}}$ on power loss.

**Figure 25.**Experimental results for the temperature of the outer ring under 16,000 rpm and 17,000 rpm.

Contact Interface | 440C-Ag Coating | 440C-PTFE Coating | 440C-PTFE | ||||||
---|---|---|---|---|---|---|---|---|---|

Contact Stress | 2.5 GPa | 3.0 GPa | 3.5 GPa | 2.5 GPa | 3 GPa | 3.5 GPa | 12 MPa | 16 MPa | 20 MPa |

S_{m} (m/s) | 3.0 | 3.3 | 3.6 | 2.44 | 2.56 | 2.66 | 2.58 | 2.52 | 2.46 |

K_{m} | 0.138 | 0.128 | 0.121 | 0.130 | 0.126 | 0.119 | 0.125 | 0.127 | 0.137 |

K_{∞} | 0.124 | 0.116 | 0.102 | 0.109 | 0.105 | 0.100 | 0.106 | 0.110 | 0.117 |

Coordinate System Name | Coordinate System Symbol | Coordinate System Definition |
---|---|---|

Inertial coordinate system | ${S}_{O}=\left\{O;X,Y,Z\right\}$ | X-axis coincides with rotating axis of bearing, and YZ-plane parallels to radial plane through bearing center. |

Coordinate system of the jth ball | ${s}_{bj}=\{{o}_{bj};{x}_{bj},{y}_{bj},{z}_{bj}\}$ | o_{bj} coincides with ball’s mass center, y_{bj} axis is along radial direction of bearing, and z_{bj} axis is along circumferential direction of bearing. |

Coordinate system of cage’s | ${s}_{c}=\left\{{o}_{c};{x}_{c},{y}_{c},{z}_{c}\right\}$ | x_{c}-axis coincides with rotating axis of cage, y_{c}z_{c}-plane parallels to radial plane through cage center, o_{c} coincides with geometric center of cage. |

Coordinate system of inner ring | ${s}_{i}=\left\{{o}_{i};{x}_{i},{y}_{i},{z}_{i}\right\}$ | x_{i}-axis is along with rotating axis of inner ring, y_{i}z_{i}-plane parallels with radial plane through inner ring mass center, o_{i} coincides with geometric center of inner ring. |

Coordinate system of the jth cage pocket center | ${s}_{pj}=\left\{{o}_{pj};{x}_{pj},{y}_{pj},{z}_{pj}\right\}$ | o_{pj} coincides with geometric center of cage pocket, y_{pj}-axis is along radial direction of bearing, and z_{pj}-axis is along circumferential direction of bearing. |

Item | Value |
---|---|

Bearing outside diameter (mm) | 218 |

Bearing bore diameter (mm) | 118 |

Bearing width (mm) | 40 |

Ball diameter (mm) | 26.988 |

Material of inner ring, outer ring, ball | 440C |

Material of cage | PTFE |

Material of raceway coating | Ag |

No. | Type | Variable |
---|---|---|

1 | Inlet liquid nitrogen supply speed, pressure, temperature, area | Q_{in}, P_{in}, T_{in}, S_{in} |

2 | Outlet pressure, temperature | P_{out}, T_{out} |

3 | cage revolution speed, ball rotation speed | nc, nr |

4 | Inner surface temperature, outer surface temperature | T_{ic}, T_{oc} |

Medium | Density kg/m ^{3} | Specific Heat J/(kg·K) | Thermal Conductivity W/(m·K) | Viscosity kg/m-s | Moles kg/kmol |
---|---|---|---|---|---|

LN2 | 808.4 | 1040 | 0.026 | 0.0001 | 28.01 |

PTFE | 2160 | 960 | 0.25 | - | - |

440C | 7750 | 481 | 29.3 | - | - |

n_{i}(rpm) | n_{c}(rpm) | n_{r}(rpm) | α (°) | Ball-Outer Raceway (W) | Ball-Inner Raceway (W) | Ball-Cage (W) | Ball-Liquid (W) | Cage-Liquid (W) |
---|---|---|---|---|---|---|---|---|

16,000 | 6805 | 43,177 | 17.6 | 190 | 1638 | 30 | 443 | 238 |

17,000 | 7239 | 45,833 | 18.0 | 211 | 1729 | 32 | 462 | 271 |

T_{in}(K) | T_{ic}(K) | T_{oc}(K) | T_{out}(K) | P_{in}(MPa) | P_{out}(MPa) | Q_{in}(kg/s) |
---|---|---|---|---|---|---|

80 | 98 | 98 | 88 | 3.6 | 3.45 | 14 |

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**MDPI and ACS Style**

Zhang, W.; Zhang, C.; Miao, X.; Li, L.; Deng, S.
Research on the Power Loss of High-Speed and High-Load Ball Bearing for Cryogenic Turbopump. *Machines* **2022**, *10*, 1080.
https://doi.org/10.3390/machines10111080

**AMA Style**

Zhang W, Zhang C, Miao X, Li L, Deng S.
Research on the Power Loss of High-Speed and High-Load Ball Bearing for Cryogenic Turbopump. *Machines*. 2022; 10(11):1080.
https://doi.org/10.3390/machines10111080

**Chicago/Turabian Style**

Zhang, Wenhu, Chaojie Zhang, Xusheng Miao, Liang Li, and Sier Deng.
2022. "Research on the Power Loss of High-Speed and High-Load Ball Bearing for Cryogenic Turbopump" *Machines* 10, no. 11: 1080.
https://doi.org/10.3390/machines10111080