# Research on the Optimal Design Approach of the Surface Texture for Journal Bearings

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## Abstract

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## 1. Introduction

## 2. Model Description

#### 2.1. Geometric Model

#### 2.2. Lubrication Equations

#### 2.3. Film Thickness Equation

#### 2.4. Asperity Contact Model

#### 2.5. Lubricant Properties

#### 2.6. Equation of Dynamics

#### 2.7. Performance Parameters

## 3. Texture Design and Optimization Approach

#### 3.1. Optimization Method

#### 3.2. Design Variables

#### 3.3. Optimization Objectives

#### 3.4. Constraints

#### 3.5. Optimization Process

- Initialization of algorithm parameters: population size is 50, the maximum number of iterations is 100, the archive is 40, and boundary constraints should be initialized.
- Transient simulation and evaluation of untextured journal bearing system:
- (a)
- Input the simulation conditions and set the time step to 0.5 crank angle per step. For the journal bearing system during an engine cycle, 0 to 720 degrees is one calculation cycle.
- (b)
- Assume the initial position of the journal center.
- (c)
- Calculate the hydrodynamic pressure using Equations (1)–(4) while updating the viscosity and density according to Equations (7)–(9). If the hydrodynamic pressure converges, the asperity contact pressure is obtained using Equations (5) and (6). If it does not converge, then recalculate.
- (d)
- Determine whether the load is balanced or not; if not, re-adjust the journal center position according to Equations (10) and (11) until it is balanced.
- (e)
- The time steps should be advanced one by one and the whole process repeated until the corresponding calculations are completed.
- (f)
- Evaluate the performance indicators of the lubrication system.

- Determine the journal center position: the performance at a certain moment in the transient state process is optimized, and the journal center position corresponding to that moment is set as the position during steady-state optimization.
- Determine the objective function and optimization variables, establish the steady-state optimization model based on Equations (24) and (25), and at the same time, initialize the wolf pack (objective functions) according to Equations (12)–(15).
- Judge whether the current objective solution satisfies the constraints according to Equation (26). If satisfied, execute the next step; otherwise, return to step (iv) to search again.
- Calculate the objective function value according to the iterative Equations (17)–(23), and update the non-dominated solution set and external population archive.
- Determine whether the constraints and maximum number of iterations are satisfied. If they are satisfied, stop the optimization; otherwise, return to step (vi) to continue the optimization.
- Output the obtained non-dominated objective solution and optimal texture size.
- Reapply the obtained optimal texture to the transient simulation process to improve the tribological performance.

## 4. Results and Discussion

#### 4.1. Model Validation

#### 4.2. Application Example and Discussion

#### 4.2.1. Simulation Results of Untextured Journal Bearing System

#### 4.2.2. Performance and Texture Optimization Results at 2000 rpm

#### 4.2.3. Performance and Texture Optimization Results at 4000 rpm

#### 4.2.4. Transient State Simulation Results at 2000 rpm

#### 4.2.5. Transient State Simulation Results at 4000 rpm

## 5. Conclusions

- Different texture schemes will lead to different friction reduction effects. The optimal texture scheme at a specific journal center position may not be optimal at other positions, and a comprehensive evaluation in transient-state simulation is required.
- The variations in texture dimension parameters and operating conditions can lead to different tribological performances. Therefore, during the design phase of surface texture, it is recommended to conduct a comprehensive analysis of various parameters.
- At low speed (2000 rpm), the journal center is often in a high eccentricity ratio state. However, at high speed (4000 rpm), the journal center is often in a moderate eccentricity ratio state. The high-speed state helps to avoid the generation of asperity contact friction force.
- Adaptive scale texture exhibits strong adaptability and achieves remarkable friction reduction benefits at both 2000 rpm and 4000 rpm speeds.
- A reasonable surface texture is advantageous in increasing the minimum oil film thickness and reducing the probability of asperity contact during the mixed lubrication phase.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**,

**b**) Schematic of a textured journal bearing system; (

**c**) overview of the lubrication regimes.

**Figure 2.**(

**a**) Unfolded schematic of journal bearings with square textures; (

**b**) depth of the texture; (

**c**) schematic of square texture unit.

**Figure 5.**Validation cases: (

**a**) comparison of hydrodynamic pressure results under steady-state conditions and (

**b**) comparison of journal center evolution results under transient-state conditions.

**Figure 6.**The variations in applied load at different speeds: (

**a**) the x component, (

**b**) the y component.

**Figure 7.**Results of untextured journal bearing system at 2000 $\mathrm{r}\mathrm{p}\mathrm{m}$: (

**a**) journal center position; (

**b**) eccentricity ratio; (

**c**) film thickness ratio. Results of untextured journal bearing system at 4000 $\mathrm{r}\mathrm{p}\mathrm{m}$: (

**d**) journal center position; (

**e**) eccentricity ratio; (

**f**) film thickness ratio.

**Figure 8.**Non-dominated solutions under different cases: (

**a**) Case 1 ($\lambda =9$); (

**b**) Case 2 ($\lambda =6$); (

**c**) Case 3 ($\lambda =3$). Texture profiles under different cases: (

**d**) Case 1 ($\lambda =9$); (

**e**) Case 2 ($\lambda =6$); (

**f**) Case 3 ($\lambda =3$).

**Figure 9.**Non-dominated solutions under different cases: (

**a**) Case 4 ($\lambda =15$); (

**b**) Case 5 ($\lambda =30$); (

**c**) Case 6 ($\lambda =5$). Texture profiles under different cases: (

**d**) Case 4 ($\lambda =15$); (

**e**) Case 5 ($\lambda =30$); (

**f**) Case 6 ($\lambda =5$).

**Figure 10.**Comparison of simulation results for untextured and textured journal bearing systems at the speed of 2000 $\mathrm{r}\mathrm{p}\mathrm{m}$: (

**a**) minimum oil film thickness; (

**b**) viscous friction and asperity contact friction; (

**c**) total friction; (

**d**) energy loss.

**Figure 11.**Comparison of simulation results for untextured and textured journal bearing systems at the speed of 4000 $\mathrm{r}\mathrm{p}\mathrm{m}$: (

**a**) minimum oil film thickness; (

**b**) viscous friction and asperity contact friction; (

**c**) total friction; (

**d**) energy loss.

Parameters | Value |
---|---|

$\mathrm{Journal}\mathrm{elastic}\mathrm{modulus}/{E}_{1}$ | $200\mathrm{G}\mathrm{P}\mathrm{a}$ |

$\mathrm{Bearing}\mathrm{elastic}\mathrm{modulus}/{E}_{2}$ | $65\mathrm{G}\mathrm{P}\mathrm{a}$ |

$\mathrm{Journal}\mathrm{Poisson}\prime \mathrm{s}\mathrm{ratio}/{\mu}_{1}$ | 0.3 |

$\mathrm{Bearing}\mathrm{Poisson}\prime \mathrm{s}\mathrm{ratio}/{\mu}_{2}$ | 0.3 |

$\mathrm{Surface}\mathrm{roughness}\mathrm{of}\mathrm{journal}/{\sigma}_{1}$ | $0.36\mathsf{\mu}\mathrm{m}$ |

$\mathrm{Surface}\mathrm{roughness}\mathrm{of}\mathrm{bearing}/{\sigma}_{2}$ | $0.407\mathsf{\mu}\mathrm{m}$ |

$\eta $ | $0.04394{\mathsf{\mu}\mathrm{m}}^{-2}$ |

$\beta $ | $5.9874\mathsf{\mu}\mathrm{m}$ |

$\eta \beta \sigma $ | 0.1430 |

Parameters | Value |
---|---|

$\mathrm{Bearing}\mathrm{circumference}/L$ | $170\mathrm{m}\mathrm{m}$ |

$\mathrm{Bearing}\mathrm{width}/B$ | $16\mathrm{m}\mathrm{m}$ |

$\mathrm{Bearing}\mathrm{radial}\mathrm{clearance}/C$ | $40\mathsf{\mu}\mathrm{m}$ |

$\mathrm{Journal}\mathrm{mass}/{M}_{jour}$ | $3.2\mathrm{k}\mathrm{g}$ |

$\mathrm{Lubricant}\mathrm{temperature}/T$ | $80\xb0\mathrm{C}$ |

$\mathrm{Rotational}\mathrm{speed}\mathrm{of}\mathrm{the}\mathrm{journal}/U$ | $2000\mathrm{r}\mathrm{p}\mathrm{m}$$,4000\mathrm{r}\mathrm{p}\mathrm{m}$ |

**Table 3.**Load-carrying capacity and friction results of the untextured journal bearing system at different journal center positions.

Speed | Type | $\mathbf{Journal}\mathbf{Center}\mathbf{Position}({\mathit{X}}_{\mathit{c}},{\mathit{Y}}_{\mathit{c}}$) | $\mathit{L}\mathit{C}\mathit{C}$ (N) | $\mathit{F}\mathit{r}\mathit{i}\mathit{c}\mathit{t}\mathit{i}\mathit{o}\mathit{n}$ (N) |
---|---|---|---|---|

Case $1(\lambda =9$) | (0.87287, 0.0898) | 1713.440 | 10.937 | |

$2000\mathrm{r}\mathrm{p}\mathrm{m}$ | Case $2(\lambda =6$) | (0.91307, 0.10068) | 3338.533 | 13.369 |

Case $3(\lambda =3$) | (0.72001, 0.63188) | 9298.238 | 23.432 | |

Case $4(\lambda =15$) | (0.74153, −0.27986) | 1377.177 | 16.666 | |

$4000\mathrm{r}\mathrm{p}\mathrm{m}$ | Case $5(\lambda =30$) | (0.39056, −0.44479) | 371.486 | 12.484 |

Case $6(\lambda =5$) | (0.76242, 0.53588) | 8851.231 | 31.765 |

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## Share and Cite

**MDPI and ACS Style**

Gu, C.; Cui, Y.; Zhang, D.
Research on the Optimal Design Approach of the Surface Texture for Journal Bearings. *Lubricants* **2024**, *12*, 111.
https://doi.org/10.3390/lubricants12040111

**AMA Style**

Gu C, Cui Y, Zhang D.
Research on the Optimal Design Approach of the Surface Texture for Journal Bearings. *Lubricants*. 2024; 12(4):111.
https://doi.org/10.3390/lubricants12040111

**Chicago/Turabian Style**

Gu, Chunxing, Yumin Cui, and Di Zhang.
2024. "Research on the Optimal Design Approach of the Surface Texture for Journal Bearings" *Lubricants* 12, no. 4: 111.
https://doi.org/10.3390/lubricants12040111