# Influence of Bearing Wear on the Stability and Modal Characteristics of a Flexible Rotor Supported on Powder-Lubricated Journal Bearings

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

#### 2.1. Grain Theory

#### 2.2. Application to Bearing Lubrication

- Inertia forces and body forces can be neglected in comparison with viscous forces.
- Pressure is considered constant across the film.
- The terms $\frac{\partial u}{\partial x},\frac{\partial u}{\partial z},\frac{\partial v}{\partial x},\frac{\partial v}{\partial y},\frac{\partial v}{\partial z},\frac{\partial w}{\partial x},\frac{\partial w}{\partial z}$, and their derivatives are negligible in comparison with $\frac{\partial u}{\partial y}$, $\frac{\partial w}{\partial y}$ and their derivatives.
- The term $\frac{\partial \overline{v}}{\partial x}$, $\frac{\partial \overline{v}}{\partial z}$ and their derivatives are negligible in comparison with $\frac{\partial \overline{v}}{\partial y}$.

#### 2.3. Modelling of Bearing Wear

#### 2.4. Modal Analysis of Rotors

## 3. Computational Procedure

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Comparison of present results with published work of Tsai and Jeng [8].

**Figure 8.**Film thickness and pressure profile at the bearing mid-plane $(\overline{z}=0.5)$. (

**a**) $\overline{h}$ vs. $\theta $ at $N=1000$ rpm, (

**b**) $\overline{p}$ vs. $\theta $ at $N=1000$ rpm, (

**c**) $\overline{h}$ vs. $\theta $ at $N=2000$ rpm, (

**d**) $\overline{p}$ vs. $\theta $ at $N=2000$ rpm, (

**e**) $\overline{h}$ vs. $\theta $ at $N=3000$ rpm, (

**f**) $\overline{p}$ vs. $\theta $ at $N=3000$ rpm, (

**g**) $\overline{h}$ vs. $\theta $ at $N=4000$ rpm, (

**h**) $\overline{p}$ vs. $\theta $ at $N=4000$ rpm, (

**i**) $\overline{h}$ vs. $\theta $ at $N=5000$ rpm, and (

**j**) $\overline{p}$ vs. $\theta $ at $N=5000$ rpm.

**Figure 9.**Pressure contours on the unwrapped bearing surface for various speed and worn depth conditions.

**Figure 10.**Variation in the dimensionless stiffness coefficients, (

**a**) ${K}_{XX}$, (

**b**) ${K}_{XY}$, (

**c**) ${K}_{YX}$, and (

**d**) ${K}_{YY}$ with rotor spin speed.

**Figure 11.**Variation in the dimensionless damping coefficients, (

**a**) ${C}_{XX}$, (

**b**) ${C}_{XY}$, (

**c**) ${C}_{YX}$, and (

**d**) ${C}_{YY}$ with rotor spin speed.

**Figure 12.**Campbell diagram showing first four whirl frequencies vs. rotor spin speed for (

**a**) smooth bearing, (

**b**) bearing with ${\overline{d}}_{0}=0.1$, (

**c**) bearing with ${\overline{d}}_{0}=0.2$, and (

**d**) bearing with ${\overline{d}}_{0}=0.3$. (FW—Forward Whirl, BW—Backward Whirl).

**Figure 13.**Mode shapes of first four modes at rotor spin speed of 2500 rpm and of 3200 rpm for smooth bearing (FW—Forward Whirl, BW—Backward Whirl).

**Figure 14.**Modal damping factor for first four modes vs. rotor spin speed for (

**a**) smooth bearing, (

**b**) bearing with ${\overline{d}}_{0}=0.1$, (

**c**) bearing with ${\overline{d}}_{0}=0.2$, and (

**d**) bearing with ${\overline{d}}_{0}=0.3$.

Parameter | Notation | Unit | Value |
---|---|---|---|

Shaft diameter | d_{s} | mm | 34.0 |

Shaft length | L_{s} | mm | 1000 |

Disc mass | m_{d} | kg | 20.4 |

Rotor unbalance | m_{d}u_{d} | kg·m | 0.02 |

Bearing length | L | mm | 34.0 |

Bearing diameter | D | mm | 34.0 |

Radial clearance | c | mm | 0.1 |

Maximum wear depth | ${\overline{d}}_{0}$ | - | 0.1, 0.2, 0.3 |

Grain diameter | d | μm | 1.0 |

t/q | 1.0 | ||

γ/r | 0.0004 | ||

B/U | 4.0 |

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**MDPI and ACS Style**

Rahmani, F.; Makki, E.; Giri, J.
Influence of Bearing Wear on the Stability and Modal Characteristics of a Flexible Rotor Supported on Powder-Lubricated Journal Bearings. *Lubricants* **2023**, *11*, 355.
https://doi.org/10.3390/lubricants11090355

**AMA Style**

Rahmani F, Makki E, Giri J.
Influence of Bearing Wear on the Stability and Modal Characteristics of a Flexible Rotor Supported on Powder-Lubricated Journal Bearings. *Lubricants*. 2023; 11(9):355.
https://doi.org/10.3390/lubricants11090355

**Chicago/Turabian Style**

Rahmani, Faisal, Emad Makki, and Jayant Giri.
2023. "Influence of Bearing Wear on the Stability and Modal Characteristics of a Flexible Rotor Supported on Powder-Lubricated Journal Bearings" *Lubricants* 11, no. 9: 355.
https://doi.org/10.3390/lubricants11090355