# A Predictive Analysis Method of Shafting Vibration for the Hydraulic-Turbine Generator Unit

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Principle of Finite Element Modal Analysis

#### 2.2. Model Simplification and Assumptions

- (1)
- It is assumed that the rotating shaft system is a constant system and also a linear elastomer;
- (2)
- The model material is considered to be isotropic and the density distribution is uniform;
- (3)
- It is assumed that the displacement and deformation are small, i.e., small deformation;
- (4)
- The connection between the components of the shafting shall be treated as an integrated connection, and the shafting shall be regarded as a whole after connection.

#### 2.3. Predictive Analysis Steps and Processes

#### 2.3.1. Establishing the Solid Geometric Model of the Shaft System

#### 2.3.2. Establishing a Finite Element Calculation Model

#### 2.3.3. Pre-Processing

#### 2.3.4. Calculating and Solving

#### 2.3.5. Post-Processing

## 3. Case Study

#### 3.1. Basic Parameters of Hydraulic-Turbine Generator Unit

#### 3.2. Shafting Composition and Geometric Dimensions

#### 3.3. Material Properties of Shaft Components

#### 3.4. Shafting’s Solid Geometry Modeling

## 4. Result and Analysis

#### 4.1. Material Defining and Meshing

#### 4.2. Boundary Conditions and Calculation Settings

#### 4.3. The First Ten-Order Mode Shapes

#### 4.4. The First Ten Order Natural Frequency and Critical Speed

## 5. Conclusions

- (1)
- The mode shapes of the shaft system from low order to high order is a process of change from simple to complex, and generally represent a certain change law starting from translation, swing, bending, and torsion, to complex continuous bending and torsion. The higher the mode order, the more complex the bending and torsional deformation of the shafting.
- (2)
- In practical engineering testing, only the elastic mode of the shaft structure is usually considered, while the rigid body mode is ignored. In this paper, the rigid body mode of the shaft system is restored through numerical calculation, which can completely describe the modal characteristics of the shaft system in theory.
- (3)
- Through the analysis of the critical speed, it can be discovered that the shaft system of the studied case could not cause resonance due to the rotational frequency when it operates at the rated speed, but when the Unit has a runaway accident, there would be a possibility of resonance caused by the rotational frequency.
- (4)
- In view of the structural design of the shaft system, some measures for structural optimization design are proposed by means of this predictive analysis method: one is increasing the stiffness of the shaft system, thereby increasing the critical speed, and making the runaway speed lower than the first-order critical speed and maintaining a certain safety margin. The other is improving the mass distribution of the shafting structure to make it more balanced.

## 6. Patents

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Hydro-Turbine | Generator | ||
---|---|---|---|

Type | Francis | Rated power | 20,265 kw |

Rated water head | 181.26 m | Rated voltage | 10,600 V |

Rated flow rate | 12.5 m^{3}/s | Rated current | 1298.6 A |

Rated power | 20,892 kW | Rated frequency | 50 Hz |

Runner diameter | 1430 mm | Rated rotational speed | 600 r/min |

Moment of inertia | ≥110 ton∙m^{2} | Runaway speed | 1064 r/min |

Shafting System | Parts | Dimension | Unit |
---|---|---|---|

Generator rotor | Rotor height | 5342 | mm |

Rotor diameter | 2510 | mm | |

Trust bearing height | 770 | mm | |

No. of Magnetic pole | 10 | Pcs. | |

Turbine shaft | Height | 2805 | mm |

Runner | Runner diameter | 1430 | mm |

Runner height | 625 | mm | |

No. of Blade | 15 | Pcs. |

Main Parts | Material | Elastic Modulus (MPa) | Poisson’s Ratio | Density (kg/m^{3}) |
---|---|---|---|---|

Runner | Stainless Steel (06Cr13Ni4Mo) | 210,000 | 0.3 | 7.90 × 10^{3} |

Shaft | Forged steel (45A) | 209,000 | 0.269 | 7.89 × 10^{3} |

Magnetic yoke | Forged steel (45A) | 209,000 | 0.269 | 7.89 × 10^{3} |

Magnetic pole | Various materials (use Copper instead) | 110,000 | 0.32 | 8.85 × 10^{3} |

Thrust bearing | Casting steel (ZG270-500) | 202,000 | 0.3 | 7.80 × 10^{3} |

Mirror plate | Forged steel (45A) | 209,000 | 0.269 | 7.89 × 10^{3} |

Computational Domain | Physical Characteristics | Mesh | |||
---|---|---|---|---|---|

X-axis | 2.4830 m | Volume | 7.1741 m³ | Components | 19 |

Y-axis | 8.7600 m | Weight | 56,362 kg | Nodes | 59,022 |

Z-axis | 2.4926 m | Scale | 1:1 | Elements | 28,879 |

Rotational Velocity | X (rad/s) | Y(rad/s) | Z (rad/s) |
---|---|---|---|

1 | 0 | 0 | 0 |

2 | 0 | 250 | 0 |

3 | 0 | 500 | 0 |

4 | 0 | 750 | 0 |

5 | 0 | 1000 | 0 |

Mode | Critical Speed | 0 rad/s | 250 rad/s | 500 rad/s | 750 rad/s | 1000 rad/s |
---|---|---|---|---|---|---|

1 | NONE | 0 Hz | 0 Hz | 0 Hz | 0 Hz | 0 Hz |

2 | NONE | 2.2 × 10^{−4} Hz | 2.2 × 10^{−4} Hz | 2.223 × 10^{−4} Hz | 2.2 × 10^{−4} Hz | 2.2 × 10^{−4} Hz |

3 | 9.359 rad/s | 1.5345 Hz | 0.33331 Hz | 0.17165 Hz | 0.1151 Hz | 8.6498 × 10^{−2} Hz |

4 | 13.257 rad/s | 1.8555 Hz | 6.6546 Hz | 9.1421 Hz | 10.485 Hz | 11.458 Hz |

5 | 104.45 rad/s | 18.669 Hz | 13.774 Hz | 11.866 Hz | 10.795 Hz | 10.033 Hz |

6 | 130.21 rad/s | 18.754 Hz | 22.536 Hz | 22.536 Hz | 22.536 Hz | 22.536 Hz |

7 | 169.81 rad/s | 22.536 Hz | 29.146 Hz | 36.915 Hz | 41.121 Hz | 43.9 Hz |

8 | 281.48 rad/s | 50.433 Hz | 45.178 Hz | 42.163 Hz | 39.947 Hz | 38.233 Hz |

9 | 748.44 rad/s | 50.796 Hz | 63.473 Hz | 88.105 Hz | 119.31 Hz | 152.96 Hz |

10 | 428.76 rad/s | 68.237 Hz | 68.24 Hz | 68.238 Hz | 68.244 Hz | 68.251 Hz |

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**MDPI and ACS Style**

Wang, W.; Shang, Y.; Yao, Z. A Predictive Analysis Method of Shafting Vibration for the Hydraulic-Turbine Generator Unit. *Water* **2022**, *14*, 2714.
https://doi.org/10.3390/w14172714

**AMA Style**

Wang W, Shang Y, Yao Z. A Predictive Analysis Method of Shafting Vibration for the Hydraulic-Turbine Generator Unit. *Water*. 2022; 14(17):2714.
https://doi.org/10.3390/w14172714

**Chicago/Turabian Style**

Wang, Wuchang, Yizi Shang, and Zhifeng Yao. 2022. "A Predictive Analysis Method of Shafting Vibration for the Hydraulic-Turbine Generator Unit" *Water* 14, no. 17: 2714.
https://doi.org/10.3390/w14172714