# Effect of Unbalanced Magnetic Pull of Generator Rotor on the Dynamic Characteristics of a Pump—Turbine Rotor System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Finite Element Calculation Model

#### 2.1. Mathematical Model of Rotor Dynamics

_{k}is the frequency of the excitation force.

#### 2.2. Generator Air Gap Unbalance Magnetic Pull Model

_{0}is the average length of the air gap when the rotor is not eccentric, α is the angle at the eccentric position, γ is the angle of rotation of the generator rotor, and e is the radial displacement of the generator rotor axis (i.e., rotor eccentricity distance).

#### 2.3. D Model and Mesh Model of the Rotor-Bearing EM System

^{−1}and the flyaway speed is 632.1 r·min

^{−1}.

^{6}. The grid-independent verification results are shown in Figure 4.

## 3. Analysis of Unbalanced Magnetic Pull on the Rotor of a Pumped Storage Unit Generator

#### 3.1. Effect of Rotor Eccentricity on Unbalanced Magnetic Pull

^{4}N, when e = 47 mm, the increase in the unbalanced magnetic pull of the rotor from 1200 A to 1400 A is 8.77 × 10

^{5}N, and the ratio of the increase in unbalanced magnetic pull under two rotor eccentric distances reaches 68.51, so the increase in the unbalanced magnetic pull of the rotor due to the increase in unbalanced rotor pull, which in turn is due to the excitation current, gradually increases with increasing eccentricity.

^{6}, and when e = 47 mm, the increase in the unbalanced magnetic pull stiffness coefficient of the rotor from 1200 A to 1400 A is 6.35 × 10

^{7}. The increase in the unbalanced magnetic pull stiffness coefficient of the two rotor eccentric distances reaches 14.54.

#### 3.2. Influence of Bearing Stiffness on Critical Speed

^{9}N/m and 2 × 10

^{10}N·m/r, respectively, and the critical speeds of the shaft system were calculated when the upper, lower, and water guide bearing stiffnesses were fixed at 2.0 × 10

^{9}N/m, 2.0 × 10

^{9}N/m, and 1.5 × 10

^{9}N/m, respectively, and then the critical speeds of the shaft system were then calculated when the individual bearing stiffness is increased or decreased by 25%, 50%, and 75%, respectively, to obtain the effect of the change in bearing stiffness on the critical speed of the unit’s shaft system.

^{9}N/m, the critical speed of the rotor system remains basically unchanged at all stages. As the stiffness coefficient of the lower guide bearing increases from 5.0 × 10

^{8}N/m to 3.5 × 10

^{9}N/m, the critical speed of the rotor system increases significantly in the third, fourth, seventh, eighth, tenth, and eleventh steps, and the critical speeds of the third and fourth steps are most affected by the change in the stiffness coefficient of the lower guide bearing. When the lower guide bearing stiffness coefficient is greater than 2.0 × 10

^{9}N/m, the increase in critical speed of each stage of the rotor system remains within 2%, and the effect of increasing the bearing stiffness coefficient on the increase in critical speed of each stage of the rotor system is not obvious. The increase in the water-guided bearing stiffness factor from 3.75 × 10

^{8}N/m to 2.625 × 10

^{9}N/m increases the third-, fourth-, fifth-, sixth-, tenth-, and eleventh-order critical speeds of the rotor system, with the third- and fourth-order critical speeds increasing significantly, while the second-, seventh-, and eighth-order critical speeds are basically unaffected by the change in the water-guided bearing stiffness factor. When the water-guided bearing stiffness coefficient is greater than 1.5 × 10

^{9}N/m, the increase in the critical speed of the rotor system at each stage is less than 1%, so the change in the water-guided bearing stiffness coefficient no longer significantly changes the critical speed of the rotor system.

#### 3.3. Effect of Unbalanced Magnetic Pull on the Dynamic Characteristics of Rotor Systems

^{7}N/m (rotor eccentricity e of 5 mm), −2.157 × 10

^{7}N/m (rotor eccentricity e of 15 mm), and −2 × 10

^{8}N/m (rotor eccentricity e of 46.5 mm), and then the critical rotor system speeds were calculated considering and not considering magnetic tension, respectively. The transverse stiffness coefficients of the upper, lower, and water-guided bearings were taken as 2.0 × 10

^{9}N/m, 2.0 × 10

^{9}N/m, and 1.5 × 10

^{9}N/m, respectively; the vertical stiffness coefficient of the thrust bearing was taken as 2.5 × 10

^{9}N/m; and the torsional stiffness coefficient was taken as 2.0 × 10

^{9}N·m/r.

^{−1}(i.e., the rotational frequency is 8.334 Hz), and the flyaway speed is 632.1 r·min

^{−1}(i.e., the rotational frequency is 10.535 Hz). The low-order mode inherent frequencies of different vibration types of the rotor system are analyzed, and the unbalanced magnetic pull reduces the axial first-order mode’s inherent frequency by 23.86% and has less influence on the axial second-order mode inherent frequency. The unbalanced magnetic tension reduces the axial first-order mode inherent frequency by 23.86%, while it has less effect on the axial second-order mode inherent frequency by only 1.25%, and the rotor eccentricity has no effect on the axial first-order mode inherent frequency. As the unbalanced magnetic pull acts on the rotor surface of the generator, it has an effect on the inherent frequency of the rotor transverse oscillation in a certain order of vibration. The unbalanced magnetic tension has a large impact on the rotor system’s transverse first-order modal vibration inherent frequency; the inherent frequency amplitude drop reaches 34.65%, and the first-order inherent frequency is 7.82 Hz, which is lower than the unit rotation frequency. The unit may produce a strong vibration phenomenon in the process of increasing the load. The third-order inherent frequency drop is 6%–10%, and the second-order inherent frequency is almost unaffected. The unbalanced magnetic pull has less influence on the rotor system’s torsional inherent frequency, the first three orders of the modal inherent frequency drop are within 2%, and the torsional inherent frequency is much larger than the unit flyaway rotation frequency, so the torsional inherent frequency will not affect the dynamic characteristics of the rotor system.

^{8}N/m) was selected for analysis. Figure 7 shows the low-order modal vibration pattern of the rotor system; the transparent area is the undeformed model of the rotor system, and the colored area is the deformed model of the rotor system. The axial first-order modal vibration model shows the overall rigid up and down movement of the rotor system, while the second-order modal vibration model shows the axial expansion and contraction movement of the generator rotor and turbine runner, and the displacement amplitude of the runner is larger. The transverse first-order mode is mainly characterized by the transverse oscillation of the runner, with the largest displacement values occurring in the runner structure and the generator rotor being almost stationary. The second- and third-order modes are characterized by the reverse oscillation of the runner and generator rotor, with the largest second-order-mode displacement values occurring in the runner structure and the largest third-order-mode displacement values occurring in the generator rotor structure. The torsional first-, second-, and third-order modal oscillations are manifested by the torsion of the pump turbine runner, the torsion of the generator rotor, and the torsion of the coupling below the main shaft guide bearing, with the oscillations becoming larger or smaller in the radial direction.

## 4. Conclusions

- (1)
- The unbalanced magnetic pull increases non-linearly with the increase in excitation current and rotor eccentricity. During stable operation, the generator rotor vibration oscillation is small and the unbalance magnetic pull increases linearly. During the transition process, excessive rotor vibration may cause the rotor eccentricity to increase, and the unbalance magnetic pull increases with an obvious non-linear trend.
- (2)
- Changes in the stiffness coefficients of the upper guide bearing, lower guide bearing, and water guide bearing all have a significant effect on the critical speed of the rotor system at each stage. Changes in the stiffness coefficients of the three guide bearings have the greatest effect on the critical speed of the rotor system at the third and fourth stages, and the minimum bearing stiffness coefficients for stable operation of the different guide bearings of the rotor system are obtained.
- (3)
- The unbalanced magnetic tension has an impact on the intrinsic frequency of transverse oscillation in the first-order mode vibration pattern, with the intrinsic frequency amplitude dropping by 34.65%, which is lower than the rotational frequency of the unit, and strong vibration may occur during the unit load increase. The axial mode vibration pattern is characterized by up and down movements of different parts of the rotor system, the transverse mode vibration pattern is characterized by transverse oscillations of different parts, and the torsional mode is characterized by the radial enlargement or reduction in the generator rotor, runner, and coupling parts of the rotor system.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Feng, F.; Chu, F.; Zhang, Z.; Li, C. Analysis and Calculation of Dynamic Characteristics of the Shaft System of a Hydro-Turbine Generator Unit. J. Vib. Meas. Diagn.
**1999**, 3–9+68. [Google Scholar] [CrossRef] - Zhai, L.; Yao, Z.; Huang, Q.; Yan, Z.; Wang, Z. 3D Dynamic Characteristics of Rotor System of Pump Storage Generator Set. Trans. Chin. Soc. Agric. Mach.
**2014**, 45, 107–111+137. [Google Scholar] - Wang, T.; Wang, F. Critical Speeds and Vibration Modes of Shafting for High-Speed Permanent-Magnet Machines. Eng. Mech.
**2012**, 29, 264–269. [Google Scholar] - Wang, Z.; Yu, J.; Fang, Y.; Wen, X.; Cao, J.; Shi, Q. The characteristic analysis of rotor dynamics of large hydraulic generating uint. J. Hydroelectr. Eng.
**2005**, 4, 62–66. [Google Scholar] - Tenhunen, T.P.; Holopainen, A.A. Spatial linearity of an unbalanced magnetic pull in induction motors during eccentric rotor motions. COMPEL Int. J. Comput. Math. Electr. Electron. Eng.
**2003**, 22, 862–876. [Google Scholar] [CrossRef] [Green Version] - Holopainen, T.P.; Tenhunen, A.; Lantto, E.; Arkkio, A. Unbalanced magnetic pull induced by arbitrary eccentric motion of cage rotor in transient operation. Part 1: Analytical model. Electr. Eng.
**2005**, 88, 13–24. [Google Scholar] [CrossRef] - Holopainen, T.P.; Tenhunen, A.; Lantto, E.; Arkkio, A. Unbalanced magnetic pull induced by arbitrary eccentric motion of cage rotor in transient operation. Part 2: Verification and numerical parameter estimation. Electr. Eng.
**2005**, 88, 25–34. [Google Scholar] [CrossRef] - Guo, D.; Chu, F.; Chen, D. The Unbalanced Magnetic Pull and Its Effects On Vibration In a Three-Phase Generator With Eccentric Rotor. J. Sound Vib.
**2002**, 254, 297–312. [Google Scholar] [CrossRef] - Subbiah, R.; Rieger, N.F. On the Transient Analysis of Rotor-Bearing Systems. J. Vib. Acoust. Stress Reliab. Des.
**1988**, 110, 515–520. [Google Scholar] [CrossRef] - Bhat RBSubbiah RSankar, T.S. Dynamic Behavior of a Simple Rotor with Dissimilar Hydrodynamic Bearings by Modal Analysis. J. Vib. Acoust. Stress Reliab. Des.
**1985**, 107, 267–269. [Google Scholar] [CrossRef] - Subbiah RBhat RBSankar, T.S. Response of Rotors Subjected to Random Support Excitations. J. Vib. Acoust. Stress Reliab. Des.
**1985**, 107, 453–459. [Google Scholar] [CrossRef] - Wang, T.; Wang, F.; Bai, H.; Cui, H. Stiffness and Critical Speed Calculation of Magnetic Bearing-rotor System Based on FEA. In Proceedings of the 11th International Conference on Electrical Machines and Systems, Wuhan, China, 17–20 October 2008; Volume 2, pp. 75–78. [Google Scholar]
- Wei, F.; Zhang, K.; Liu, Z.; Liu, B.; Zhang, H.; Mei, J.; Wu, L. Modeling and analysis of unbalanced magnetic pull in synchronous motorized spindle considering magneto-thermal coupling. Int. J. Appl. Electromagn. Mech.
**2022**, 70, 303–321. [Google Scholar] - Ma, C.; Zhao, D.; Liao, W.; Sun, W.; Guo, P.; Xu, Y. Effect of the unbalanced magnetic pull force and oil-film force on nonlinearity of a dual-rotor bearing system. Proc. Inst. Mech. Eng. Part K: J. Multi-Body Dyn.
**2022**, 236, 623–638. [Google Scholar] [CrossRef] - Huang, L.; Shen, G.; Hu, N.; Chen, L.; Yang, Y. Coupled Electromagnetic-Dynamic Modeling and Bearing Fault Characteristics of Induction Motors considering Unbalanced Magnetic Pull. Entropy
**2022**, 24, 1386. [Google Scholar] [CrossRef] - Zhang, L.; Ma, Z.; Song, B. Characteristics Analysis of Hydroelectric Generating Set under Unbalanced Magnetic Pull and Sealing Force of Runner. Water Resour. Power
**2010**, 28, 117–120. [Google Scholar] - Song, Z.; Ma, Z. Nonlinear vibration analysis of an eccentric rotor with unbalance magnetic pull. J. Vib. Shock
**2010**, 29, 169–173+250. [Google Scholar] [CrossRef] - Perers, R.; Lundin, U.; Leijon, M. Saturation Effects on Unbalanced Magnetic Pull in a Hydroelectric Generator With an Eccentric Rotor. IEEE Trans. Magn.
**2007**, 43, 3884–3890. [Google Scholar] [CrossRef] - Perers, R.; Lundin, U.; Leijon, M. Development of synchronous generators for Swedish hydropower: A review. Renew. Sustain. Energy Rev.
**2005**, 11, 1008–1017. [Google Scholar] [CrossRef] - Wallin, M.; Lundin, U. Dynamic Unbalanced Pull from Field Winding Turn Short Circuits in Hydropower Generators. Electr. Power Compon. Syst.
**2013**, 41, 1672–1685. [Google Scholar] [CrossRef] - Wallin, M.; Bladh, J.; Lundin, U. Damper Winding Influence on Unbalanced Magnetic Pull in Salient Pole Generators With Rotor Eccentricity. IEEE Trans. Magn.
**2013**, 49, 5158–5165. [Google Scholar] [CrossRef] - Wallin, M.; Ranlof, M.; Lundin, U. Reduction of Unbalanced Magnetic Pull in Synchronous Machines due to Parallel Circuits. IEEE Trans. Magn.
**2011**, 47, 4827–4833. [Google Scholar] [CrossRef] - Nave, O.; Sharma, M. Singular Perturbed Vector Field (SPVF) Applied to Complex ODE System with Hidden Hierarchy Application to Turbocharger Engine Model. Int. J. Nonlinear Sci. Numer. Simul.
**2020**, 21, 99–113. [Google Scholar] [CrossRef] - Pang, J.; Liu, H.; Liu, X.; Yang, H.; Peng, Y.; Zeng, Y.; Yu, Z. Study on sediment erosion of high head Francis turbine runner in Minjiang River basin. Renew. Energy
**2022**, 192, 849–858. [Google Scholar] [CrossRef] - Shi, D. Calculation of UMP and Analysis of Rotor Dynamic Characteristics of Large Induction Motor; Harbin University of Science and Technology: Harbin, China, 2016. [Google Scholar]

**Figure 6.**Effect of different stiffness factors of the three guide bearings on the critical speed of the rotor system.

Parameters | Values | Parameters | Values |
---|---|---|---|

Generator stator radius R_{0}/m | 2.45 | Air gap fundamental magnetic momentum coefficient k_{i} | 7 |

Rotor radius of generators R/m | 2.403 | Air magnetic permeability μ_{0} | 4π × 10^{−7} |

Rotor length of generators L/m | 3.665 | Excitation current I_{j}/A | 1200–1400 |

Average length of air gap δ_{0}/m | 0.047 | Eccentric distance of generator rotor e/mm | 0–47 |

Components | Materials | Density/(kg/m^{3}) | Modulus of Elasticity/Pa | Poisson’s Ratio |
---|---|---|---|---|

Coils | Copper | 8900 | 1.15 × 10^{9} | 0.33 |

Magnetic yoke poles | Magnetic yoke materials | 7830 | 2.06 × 10^{9} | 0.3 |

Other | steel | 7850 | 2.10 × 10^{11} | 0.3 |

Modal | Number of Steps | No Consideration of Unbalanced Magnetic Pull | Consideration of Unbalanced Magnetic Pull (e = 5 mm) | Consideration of Unbalanced Magnetic Pull (e = 15 mm) | Consideration of Unbalanced Magnetic Pull (e = 46.5 mm) | |||
---|---|---|---|---|---|---|---|---|

Values (Hz) | Values (Hz) | Deviation/% | Values (Hz) | Deviation/% | Values (Hz) | Deviation/% | ||

Axial | 1 | 15.778 | 12.017 | −23.86 | 12.017 | −23.86 | 12.017 | −23.86 |

2 | 89.321 | 90.437 | 1.25 | 90.437 | 1.25 | 90.437 | 1.25 | |

Horizontal | 1 | 11.966 | 7.82 | −34.65 | 7.82 | −34.65 | 7.82 | −34.65 |

2 | 16.136 | 16.079 | −0.35 | 16.064 | −0.45 | 15.661 | −2.94 | |

3 | 17.477 | 16.369 | −6.34 | 16.351 | −6.44 | 15.886 | −9.10 | |

Turning | 1 | 24.443 | 24.595 | 0.62 | 24.595 | 0.62 | 24.595 | 0.62 |

2 | 56.331 | 56.657 | 0.58 | 56.657 | 0.58 | 56.657 | 0.58 | |

3 | 133.04 | 134.770 | 1.3 | 134.77 | 1.3 | 134.77 | 1.3 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wu, W.; Pang, J.; Liu, X.; Zhao, W.; Lu, Z.; Yan, D.; Zhou, L.; Wang, Z.
Effect of Unbalanced Magnetic Pull of Generator Rotor on the Dynamic Characteristics of a Pump—Turbine Rotor System. *Water* **2023**, *15*, 1120.
https://doi.org/10.3390/w15061120

**AMA Style**

Wu W, Pang J, Liu X, Zhao W, Lu Z, Yan D, Zhou L, Wang Z.
Effect of Unbalanced Magnetic Pull of Generator Rotor on the Dynamic Characteristics of a Pump—Turbine Rotor System. *Water*. 2023; 15(6):1120.
https://doi.org/10.3390/w15061120

**Chicago/Turabian Style**

Wu, Weidong, Jiayang Pang, Xuyang Liu, Weiqiang Zhao, Zhiwei Lu, Dandan Yan, Lingjiu Zhou, and Zhengwei Wang.
2023. "Effect of Unbalanced Magnetic Pull of Generator Rotor on the Dynamic Characteristics of a Pump—Turbine Rotor System" *Water* 15, no. 6: 1120.
https://doi.org/10.3390/w15061120