# Thermal Tribo-Dynamic Behaviors of Water-Lubricated Bearings during Start-Up with Journal Shape Error

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematic Model

#### 2.1. Transient Lubrication Gap

_{J}and O

_{B}are the geometric center of the WLB and the transient center position of the journal, respectively. e is expressed as the eccentric distance between the geometric center and transient center position. During startup, the transient lubrication gap at the contact interface is a superposition of nonlinear quantities such as transient elastic deformation, thermal deformation, and rotor trajectory. The governing equation is described as [31]:

_{1}and r

_{2}, respectively. Eventually, ${\delta}_{SE}\left(\theta ,t\right)$ is presented by

#### 2.2. Dynamic Equations

_{J}and W are the rotor mass and static load, respectively; F represents the dynamic forces (including transient hydrodynamic pressure force, contact force, and mixed friction force); the subscripts κ and ζ indicate the direction of the coordinate system; the subscripts h and c denote hydrodynamic and contact, respectively; the subscript fric represents friction; and t and r are the start-up time and rotor unbalance.

#### 2.3. Transient Hydrodynamic Model

#### 2.3.1. Transient Reynolds Equation

_{B}and h are the bearing radius and water film thickness, respectively; z and θ are the circumferential angle (θ∈(0, 2π)) and coordinate along the axial direction of the bearing, respectively; ρ and η denote the density and viscosity of the lubricant, respectively; P

_{h}and σ are the hydrodynamic pressure and composite surface roughness, respectively; ϕ

_{s}, ϕ

_{c}represent the shear and contact factors, respectively; and ϕ

_{θ}and ϕ

_{z}indicate the flow factors of the WLB in the circumferential and axial directions, respectively. Each flow coefficient is related to the ratio of the water film thickness to the combined surface roughness and can be determined by the following equation.

#### 2.3.2. Hydrodynamic Forces

_{h}of the water film in the solution domain at each instant of time. And the transient hydrodynamic forces can be obtained by integrating the hydrodynamic pressure, which can be expressed by

#### 2.4. 3D Thermal Model

_{B},T

_{W},T

_{J}) is the temperature; the subscripts B and J stand for bearings and journals, respectively; the W subscripts represent the lubricating medium water; and the heat source Φ in the mixed frictional contact zone of the bearing contains the heat of viscous shear dissipation Φ

_{W}of the water film shear and the heat of frictional contact Φ

_{C}of the asperity friction. For more information on the thermal model, see [16,34].

#### 2.5. Transient Contact Model

#### 2.5.1. Transient Asperity Contact

^{*}and ${\omega}_{c}^{*}$ are normalized by the composite surface roughness. I

_{c}denotes the integral operation. The calculation equation of I

_{c}and ${\omega}_{c}^{*}$ can be found in [31].

#### 2.5.2. Contact Forces

#### 2.6. Transient Deformation

#### 2.7. Transient Friction Forces

#### 2.8. Boundary Conditions

#### 2.8.1. Thermal Boundary

- (a)
- Internal heat exchange boundary conditions (BC1-BC2);
- (b)
- External heat exchange boundary conditions (BC3-BC8);
- (c)
- Cavitation boundary conditions.

_{rup}and h, where h

_{rup}and h represent the rupture film thickness and the standard film thickness, respectively.

#### 2.8.2. Cavitation Boundary

## 3. Numerical Schemes

#### 3.1. Numerical Scheme of the Reynolds Equation

_{j,k}and Δz

_{j,k}denote the mesh size of the control volume (j, k), respectively. Equation (24) can be simplified in terms of the interface node coefficients, K, for the control volume in each direction.

#### 3.2. Numerical Scheme of the Dynamic Equation

## 4. Results and Discussion

#### 4.1. Verification of Present Model

#### 4.2. Setting of Simulation Parameters

#### 4.3. Effect of Different Journal Shape Error Amplitude

#### 4.4. Effect of Different Journal Shape Error Waviness Orders

#### 4.5. Effect of Different Starting Speeds on Bearing Performance

## 5. Conclusions

- (1)
- The greater the amplitude of the journal shape error, the more pronounced the temperature increase during start-up and the greater the fluctuation in lubrication performance. The location of the maximum temperature during the bearing’s start-up state remains unaffected by changes in error amplitude. Temperature changes are not significant when the amplitude of the shape error is less than 9‰ of the bearing radius clearance.
- (2)
- The temperature effect reduces the displacement in the vertical direction during the start-up state of the WLB, which is more significant at higher waviness orders of journal shape error.
- (3)
- The higher the starting speed of the bearing, the easier it is to reach the EHL state, but the temperature rise becomes faster and the shaft track becomes larger. And when the speed is lower, although the temperature is lower, the friction and contact force are greater, so one must be careful when choosing the starting speed.
- (4)
- The neglect of thermal effects leads to an underestimation of the hydrodynamic effect during the startup of WLBs with journal shape errors, resulting in errors in the prediction of the friction dynamics behavior.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Verification of radial bearing transient circumferential temperature distribution during start-up state.

**Figure 7.**Effect of journal shape error amplitude on transient contact force and transient hydrodynamic force of water-lubricated bearing: (

**a**) $\Delta r=0.02\u20130.09$; (

**b**) $\Delta r=0.002\u20130.009$.

**Figure 8.**Comparison of dynamic characteristics of bearings with and without consideration of thermal effects: (

**a**) contact force; (

**b**) deformation; (

**c**) journal trajectory.

**Figure 9.**Effect of journal shape error amplitude on transient maximum temperature of the WLB: (

**a**) $\Delta r=0.02\u20130.09$; (

**b**) $\Delta r=0.002\u20130.009$.

**Figure 10.**The temperature distribution for different journal shape error amplitudes (within 2 s startup time).

**Figure 11.**Influence of journal shape error order on transient contact force and transient hydrodynamic force of water-lubricated bearing ($\Delta r=0.033$): (

**a**) even-order; (

**b**) odd-order.

**Figure 12.**Temperature effects on hydrodynamic pressure for shape errors with different waviness orders.

**Figure 13.**The influence of shape error waviness order on the axis locus of WLBs: (

**a**,

**c**) even order; (

**b**,

**d**) odd order; (

**a**,

**b**) in the ζ direction; (

**c**,

**d**) in the κ direction.

**Figure 14.**Influence of journal shape error order on transient maximum temperature of water-lubricated bearing: (

**a**) even order; (

**b**) odd order.

**Figure 15.**Influence of journal shape error order on transient contact pressure distribution of WLBs: (

**a**) even order; (

**b**) odd order.

**Figure 16.**Effect of different rotational speeds on transient contact forces of WLBs at start-up: (

**a**) low speed, (

**b**) high speed.

**Figure 17.**Comparison of dynamic characteristics of bearings with and without consideration of thermal effects: (

**a**) deformation; (

**b**) journal trajectory.

**Figure 18.**Effect of different rotational speeds on the instantaneous maximum temperature of water-lubricated bearing during start-up: (

**a**) low speed, (

**b**) high speed.

**Figure 19.**Effects of different rotational speeds on starting journal trajectories of water-lubricated bearings: (

**a**) low speed in horizontal direction, (

**b**) low speed in vertical direction, (

**c**) high speed in horizontal direction; (

**d**) high speed in vertical direction.

**Table 1.**Kucinschi’s experimental parameters [41].

Parameter | Value | Parameter | Value |
---|---|---|---|

Bearing radius | 50 mm | Bearing width | 80 mm |

Bearing specific heat capacity | 380 J/(kg·K) | Journal specific heat capacity | 490 J/(kg·K) |

Radius clearance | 0.123 mm | Lubricant specific heatcapacity | 2000 J/(kg·K) |

Journal thermal expansivity | 12 μm/(m·K) | Bearing thermal expansivity | 17 μm/(m·K) |

Bearing Poisson ratio | 0.3 | Bearing elastic modulus | 120 GPa |

Journal elastic modulus | 210 GPa | lubricating temperature | 30 °C |

Journal Poisson ratio | 0.33 | Lubricant viscosity (30 °C) | 0.05 Pa·s |

Bearing thermal conductivity | 65 W/m·K | Journal thermal conductivity | 50 W/m·K |

Bearing density | 8940 kg/m^{3} | Journal density | 7700 kg/m^{3} |

Lubricant thermal conductivity | 0.13 W/m·K | Lubricant density | 870 kg/m^{3} |

Start-up time | 7 s |

Parameter | Value | Parameter | Value |
---|---|---|---|

Inner radius/R_{B} | 22.5 mm | Water specific heat capacity/C_{PW} | 4200 J/(kg·K) |

Outer radius/R_{O} | 24 mm | Journal density ρ_{J} | 7800 kg/m^{3} |

Bearing width/L | 20 mm | Journal elastic modulus/E_{J} | 210 GPa |

Radius clearance/C | 0.06 mm | Journal Poisson ratio/ν_{J} | 0.3 |

Bearing elastic modulus/E_{B} | 3.89 GPa | Journal thermal conductivity/k_{J} | 50 W/(m·K) |

Bearing Poisson ratio/ν_{B} | 0.4 | Journal specific heat capacity/C_{PJ} | 460 J/(kg·K) |

Bearing density/ρ_{B} | 1300 kg/m^{3} | Journal thermal expansivity | 11.9 μm/(m·K) |

Bearing thermal conductivity/k_{B} | 11 W/(m·K) | Convection heat transfercoefficient/h_{h} | 80 W/(m·K) |

Bearing specific heat capacity/C_{PJ} | 1005 J/(kg·K) | Inlet temperature/T_{inlet} | 20 °C |

Bearing thermal expansivity | 50 μm/(m·K) | Water thermal conductivity/k_{W} | 0.599 W/(m·K) |

Water density/ρ_{W} | 1000 kg/m^{3} | Journal surface roughness/σ_{J} | 0.2 μm |

Water viscosity/η_{W} | 0.001 Pa·s | Bearing surface roughness/σ_{B} | 1.6 μm |

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**MDPI and ACS Style**

Chen, S.; Cai, J.; Zhang, J.; Liu, Z.
Thermal Tribo-Dynamic Behaviors of Water-Lubricated Bearings during Start-Up with Journal Shape Error. *Lubricants* **2024**, *12*, 106.
https://doi.org/10.3390/lubricants12040106

**AMA Style**

Chen S, Cai J, Zhang J, Liu Z.
Thermal Tribo-Dynamic Behaviors of Water-Lubricated Bearings during Start-Up with Journal Shape Error. *Lubricants*. 2024; 12(4):106.
https://doi.org/10.3390/lubricants12040106

**Chicago/Turabian Style**

Chen, Shouan, Jianlin Cai, Junfu Zhang, and Zaixin Liu.
2024. "Thermal Tribo-Dynamic Behaviors of Water-Lubricated Bearings during Start-Up with Journal Shape Error" *Lubricants* 12, no. 4: 106.
https://doi.org/10.3390/lubricants12040106