# Study on Vibration Transmission among Units in Underground Powerhouse of a Hydropower Station

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Field Structural Vibration Test

#### 2.1. Field Tests Overview

^{3}, and the adjusted storage capacity is 4.91 billion m

^{3}. The installed capacity of the power station is 3600 MW, the annual utilization hour is 4616 h, and the annual power generation is 166.20 billion kWh. All the units are lined up in the main powerhouse from #1 to #6, and the rated capacity of single unit is 600 MW. Total length of the main powerhouse is 204.52 m, the excavation height is 68.80 m, and the width of the main powerhouse along river is 25.90 m. The main powerhouse is shown as Figure 2.

#### 2.2. Preliminary Tests Results

#### 2.3. Vibration Transmission Rules of Tests

#### 2.3.1. Vibration Intensity

#### 2.3.2. Signal Component

## 3. Study of Vibration Transmission Mechanism

#### 3.1. Simplication of Powerhouse Structure

_{1}and m

_{4}. The bedrocks below m

_{1}and m

_{4}are represented by two homogeneous elastic blocks m

_{2}and m

_{3}, respectively. The bottom and the lateral sides of bedrocks m

_{2}and m

_{3}are restrained by normal constraints. It is easy to know m

_{1}= m

_{4}, let m

_{1}= m

_{4}= m; similarly, m

_{2}= m

_{3}= M.

#### 3.2. Establishment of Vibration Models

_{2}and m

_{3}. They are connected to the left and right boundary by springs; Deformation of the bedrock between m

_{2}and m

_{3}is represented by the stretching and compression of a spring to simulate the interaction of axial force. Load F(t) is applied on the lumped mass m

_{1}. x

_{1}, x

_{2}, x

_{3}and x

_{4}denote displacements of m

_{1}, m

_{2}, m

_{3}, and m

_{4}, respectively. Considering dynamic load only, equations of motion for the four lumped masses are listed in Equation (2).

_{12}refers to the shear force between m

_{1}and m

_{2}; Q

_{34}refers to the shear force between m

_{3}and m

_{4}. K

_{2x}refers to the compression stiffness between m

_{2}and left boundary; K

_{3x}refers to the compression stiffness between m

_{3}and right boundary; K

_{23}refers to the compression stiffness between m

_{2}and m

_{3}. For homogeneous elastic structure, the compression stiffness can be calculated according to K = EA/l, then the compression stiffness of horizontal vibration model is obtained as K

_{2x}= K

_{3x}= 2Eh/l, K

_{23}= Eh/l. l refers to the length of a single unit, h refers to the depth of bedrock considered, and E refers to the elastic modulus of bedrock. Let K

_{x}= Eh/l, then K

_{2x}= K

_{3x}= 2K

_{x}, K

_{23}= K

_{x}. According to the kinematic relationship between units and surrounding rocks, x

_{1}= x

_{2}and x

_{3}= x

_{4}can be drawn. After simplification of Equation (2) according to the above formula, Equation (3) is derived.

_{1}and x

_{4}should also be in the simple harmonics form. Let x

_{4}= Psin(ωt), substitute it into the Equation (3). Equation (4) can be derived.

_{1}to x

_{4}, to describe the influence on m

_{4}caused by vibration of m

_{1}in X direction.

_{2}and m

_{3}. They are connected to the bottom boundary and above masses m

_{1}and m

_{4}by springs. Interaction between bedrocks and units are represented by shear force Q

_{23}. Load F(t) is applied on m

_{1}. Considering dynamic loads only, equations of motion for the four lumped masses are listed in Equation (5).

_{12}refers to the compression stiffness between m

_{1}and m

_{2}; K

_{2z}refers to the compression stiffness between m

_{2}and the bottom boundary; K

_{34}refers to the compression stiffness between m

_{3}and m

_{4}; K

_{3z}refers to the compression stiffness between m

_{3}and the bottom boundary. According to K = EA/l, then the compression stiffness of vertical vibration model is obtained as K

_{12}= K

_{2z}= K

_{34}= K

_{3z}= 2El/h. Let K

_{z}= El/h, then K

_{12}= K

_{2z}= K

_{34}= K

_{3z}= 2K

_{z}. In addition, shear force should be calculated as Q = K′GA(∂z/∂x) based on mechanics of materials. For this model, Q

_{23}= K′Gh(z

_{2}− z

_{3})/l = G

_{z}(z

_{2}− z

_{3}). Let G

_{z}= K′Gh/l, then Q

_{23}= G

_{z}(z

_{2}− z

_{3}). K′ refers to the section shape coefficient and G refers to the shear modulus of bedrock. According to actual condition, the units and the bedrocks are always in contact. The relationship between z

_{1}and z

_{2}can be derived as z

_{1}= 2z

_{2}, as well as z

_{4}= 2z

_{3}. After simplification of Equation (5), Equation (6) is derived.

_{1}to z

_{4}, to describe the influence on m

_{4}that is caused by the vibration of m

_{1}in Z direction.

#### 3.3. Rules of Vibration Transmission Ratios

_{x}, vertical compression stiffness of bedrock K

_{z}, shear stiffness G

_{z}, and frequency of vibration source load ω. Vibration transmission ratios among units are determined by these parameters in the simplified model.

_{x}= K

_{z}= E is derived. For the rectangular section, the section shape coefficient ${K}^{\prime}$ is 1.2, then G

_{z}= 1.2G = 1.2E/[2(1 + μ)]. After the simplification, Equation (4) and Equation (7) are simplified as two expressions of elastic modulus E, Poisson ratio μ, mass m, and M, and frequency ω. As shown in Equations (8) and (9).

^{6}to 10

^{7}kg for large hydroelectric unit. According to the previous research and load characteristics of the powerhouse, low frequency tail fluctuation and rotation of hydraulic generator are the main vibration sources of powerhouse structural vibration. Their frequencies are within 0–5 Hz, especially in the case of severe vibration. Substituting above data into Equations (8) and (9), it can be found that both (m + M)ω

^{2}and (2m + M)ω

^{2}are 1 to 2 orders of magnitude smaller than 3K

_{x}and (2K

_{z}+ G

_{z}) for low frequency loads. Consequently, the transmission ratio of horizontal vibration in Equation (8) can be approximated, as Equation (10).

_{x}= 0.25 for horizontal vibration, and q

_{z}= 0.16 for vertical vibration. It can be conclude that, for the vibration transmission among units, the vibration transmission ratio of lateral-river vibration is significantly larger than that of longitude-river vibration and vertical vibration. This is in coincidence with the results obtained from the field tests in Figure 10.

## 4. Numerical Simulation

#### 4.1. Establishment of Finite Element Model

#### 4.2. Results of Numerical Simulation

#### 4.2.1. Transmission Rules among Units of Vibration in Three Directions

_{1}refers to the frequency of low frequency tail fluctuation with a value between 0.167 and 0.6 times rotational frequency, according to the previous research results and experience [9,31]; it is set as 1 Hz based on the frequency spectrum analysis of field tests in this paper. ω

_{2}refers to the rotational frequency, set as 2.4 Hz. a and b represent the proportion of two vibration sources, and set as 0.8 and 0.2, respectively, according to the analysis of field tests. Time history of load is shown in Figure 19.

#### 4.2.2. Transmission Rules under Different Vibration Sources

_{1}and ω

_{2}are set as 1 Hz and 2.4 Hz, which are the typical frequency of low frequency tail fluctuation and rotation of hydraulic generator. After calculation, the vibration displacements of the nodes corresponding to the location of sensors are extracted. The RMS of vibration displacements of unit #1 are taken as reference values. Results of normalized displacements are shown in Table 3.

## 5. Discussion

## 6. Conclusions

- (a)
- Vibration transmission ratio of lateral-river vibration is significantly larger than those of longitude-river vibration and vertical vibration. The transmission ratio between adjacent units of lateral-river vibration is about 15–25%, while those of longitude-river vibration and vertical vibration are about 10–15%.
- (b)
- Low frequency tail fluctuation and the rotation of hydraulic generator are the main vibration sources of powerhouse structural vibration. Vibration transmission ratios of the vibration caused by the two sources are basically equal.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A | Amplitude of load |

E | Elastic modulus of bedrock |

F(t) | Load applied on the unit |

G | Shear modulus of bedrock |

h | Depth of bedrock considered |

K′ | Section shape coefficient |

K_{2x} | Compression stiffness between m_{2} and left boundary |

K_{3x} | Compression stiffness between m_{3} and right boundary |

K_{23} | Compression stiffness between m_{2} and m_{3} |

K_{12} | Compression stiffness between m_{1} and m_{2} |

K_{2z} | Compression stiffness between m_{2} and the bottom boundary |

K_{34} | Compression stiffness between m_{3} and m_{4} |

K_{3z} | Compression stiffness between m_{3} and the bottom boundary |

l | Length of a single unit |

m | Mass of the unit |

m_{1} | Lumped Mass of the unit #1 |

m_{2} | Lumped Mass of the bedrock under the unit #1 |

m_{3} | Lumped Mass of the bedrock under the unit #2 |

m_{4} | Lumped Mass of the unit #2 |

M | Mass of the bedrock |

Q_{12} | Shear force between m_{1} and m_{2} |

Q_{34} | Shear force between m_{3} and m_{4} |

x_{1}, x_{2}, x_{3}, x_{4} | Vibration displacement of m_{1}, m_{2}, m_{3}, m_{4} in X direction |

$\ddot{{x}_{1}}$, $\ddot{{x}_{2}}$, $\ddot{{x}_{3}}$, $\ddot{{x}_{4}}$ | Vibration acceleration of m_{1}, m_{2}, m_{3}, m_{4} in X direction |

z_{1}, z_{2}, z_{3}, z_{4} | Vibration displacement of m_{1}, m_{2}, m_{3}, m_{4} in Z direction |

$\ddot{{z}_{1}}$, $\ddot{{z}_{2}},\text{}\ddot{{z}_{3}},\text{}\ddot{{z}_{4}}$ | Vibration acceleration of m_{1}, m_{2}, m_{3}, m_{4} in Z direction |

β | Power amplification factor |

θ | Frequency of the modal frequency |

μ | Poisson ratio |

ω | Frequency of vibration source load |

FE | Finite element |

MRA | Multi-Resolution Analysis |

PSD | Power spectral density |

RMS | Root mean square |

## References

- Xu, X.P.; Han, Q.K.; Chu, F.L. Review of electromagnetic vibration in electrical machines. Energies
**2018**, 11, 1779. [Google Scholar] [CrossRef] - Mollasalehi, E.; Wood, D.; Sun, Q. Indicative fault diagnosis of wind turbine generator bearings using tower sound and vibration. Energies
**2017**, 10, 1853. [Google Scholar] [CrossRef] - Cachafeiroa, H.; Arevaloa, L.F.; Vinuesaa, R.; Goikoetxeab, J.; Barrigab, J. Impact of solar selective coating ageing on energy cost. Energy Procedia
**2015**, 69, 299–309. [Google Scholar] [CrossRef] - Liu, X.; Luo, Y.Y.; Wang, Z.W. A review on fatigue damage mechanism in hydro turbines. Renew. Sustain. Energy Rev.
**2016**, 54, 1–14. [Google Scholar] [CrossRef] - Shen, K.; Zhang, Z.Q.; Liang, Z. Hydraulic Vibration Calculation of Yantan Hydropower House. Water Resour. Power
**2003**, 1, 73–75. [Google Scholar] [CrossRef] - Kurzin, V.B.; Seleznev, V.S. Mechanism of emergence of intense vibrations of turbines on the Sayano-Shushensk hydro power plant. J. Appl. Mech. Tech. Phys.
**2010**, 4, 590–597. [Google Scholar] [CrossRef] - Yang, J.D.; Zhao, K.; Li, L.; Wu, P. Analysis on the causes of units 7 and 9 accidents at Sayano-Shushenskaya hydropower station. J. Hydroelectr. Eng.
**2011**, 4, 226–234. (In Chinese) [Google Scholar] - Dorji, U.; Ghomashchi, R. Hydro turbine failure mechanisms: An overview. Eng. Fail. Anal.
**2014**, 44, 136–147. [Google Scholar] [CrossRef] - Mohanta, R.K.; Chelliah, T.R.; Allamsetty, S.; Akula, A.; Ghosh, R. Sources of vibration and their treatment in hydro power stations—A review. Eng. Sci. Technol. Int. J.
**2017**, 20, 637–648. [Google Scholar] [CrossRef] - Zhi, B.P.; Ma, Z.Y. Disturbance analysis of hydropower station vertical vibration dynamic characteristics: The effect of dual disturbances. Struct. Eng. Mech.
**2015**, 2, 297–309. [Google Scholar] [CrossRef] - Ma, Z.Y.; Dong, Y.X. Dynamic response of hydroelectric set by hydraulic lateral force on turbine runner. J. Hydroelectr. Eng.
**1990**, 2, 31–39. (In Chinese) [Google Scholar] - Song, Z.Q. Research on Coupling Vibration Characteristics of Generator Set and Hydropower House. Ph.D. Thesis, Dalian University of Technology, Dalian, China, 2009. (In Chinese). [Google Scholar]
- Zhi, B.P. Study of Vibration Transmission Path about Hydropower Station Units and Powerhouse with Complex Disturbance. Ph.D. Thesis, Dalian University of Technology, Dalian, China, 2014. (In Chinese). [Google Scholar]
- Zhou, J.Z.; Peng, X.L.; Li, R.H.; Xu, Y.H.; Liu, H.; Chen, D.Y. Experimental and finite element analysis to investigate the vibration of Oblique-Stud stator frame in a large hydropower generator unit. Energies
**2017**, 10, 2175. [Google Scholar] [CrossRef] - Zhang, C.H.; Zhang, Y.L. Nonlinear dynamic analysis of the Three Gorge Project powerhouse excited by pressure fluctuation. J. Zhejiang Univ. Sci. A
**2009**, 9, 1231–1240. [Google Scholar] [CrossRef] - Wang, X.; Li, T.C.; Zhao, L.H. Vibration analysis of large bulb tubular pump house under pressure pulsations. Water Sci. Eng.
**2009**, 1, 86–94. [Google Scholar] [CrossRef] - Lian, J.J.; Qin, L.; He, C.L. Structure vibration of hydropower house based on prototype observation. J. Tianjin Univ.
**2006**, 2, 176–180. (In Chinese) [Google Scholar] - Lian, J.J.; Qin, L.; Wang, R.X.; Hu, Z.G.; Wang, H.J. Study on the dynamic characteristics of the power house structure of two-row placed units. J. Hydroelectr. Eng.
**2004**, 2, 55–60. (In Chinese) [Google Scholar] - Lian, J.J.; Zhang, Y.; Liu, F.; Yu, X.H. Vibration source characteristics of a roof overflow hydropower station. J. Vib. Shock
**2013**, 18, 8–14. [Google Scholar] [CrossRef] - He, L.J.; Lian, J.J.; Ma, B. Intelligent damage identification method for large structures based on strain modal parameters. J. Vib. Control
**2014**, 12, 1783–1795. [Google Scholar] [CrossRef] - He, L.J. Study on Coupled Vibration Characteristics and Response Prediction of Underground Powerhouse. Ph.D. Thesis, Tianjin University, Tianjin, China, 2010. (In Chinese). [Google Scholar]
- Zhang, Y. Vibration Characteristics of the Overflow Powerhouse with Bulb Tubular Unit. Ph.D. Thesis, Tianjin University, Tianjin, China, 2012. (In Chinese). [Google Scholar]
- Mao, L.D.; Wang, H.J. Research on load feedback of structure vibration of underground house of hydropower station. J. Water Resour. Water Eng.
**2014**, 3, 79–82. [Google Scholar] [CrossRef] - Wang, H.J.; Bai, B.; Li, K. Research on vibration propagation regular of adjacent unit-blocks for hydropower house. J. Water Resour. Water Eng.
**2016**, 1, 141–146. [Google Scholar] [CrossRef] - Wei, Y.B.; Chen, J.; Ma, Z.Y. Vibrational travel and behavior of pumped storage power station underground powerhouse. J. Water Resour. Arch. Eng.
**2017**, 4, 101–106. [Google Scholar] [CrossRef] - Ameen, M.S.A.; Ibrahim, Z.; Othman, F.; Al-Ansari, N.; Yaseen, Z.M. Minimizing the principle stresses of powerhoused Rock-Fill dams using control turbine running units: Application of Finite Element Method. Water-SUI
**2018**, 10, 1138. [Google Scholar] [CrossRef] - Gupta, S.; Stanus, Y.; Lombaert, G.; Degrande, G. Influence of tunnel and soil parameters on vibrations from underground railways. J. Sound Vib.
**2009**, 327, 70–91. [Google Scholar] [CrossRef] - Chen, M.; Lu, W.B.; Yi, C.P. Blasting vibration criterion for a rock-anchored beam in an underground powerhouse. Tunn. Undergr. Space Technol.
**2007**, 22, 69–79. [Google Scholar] [CrossRef] - Xia, X.; Li, H.B.; Li, J.C.; Liu, B.; Yu, C. A case study on rock damage prediction and control method for underground tunnels subjected to adjacent excavation blasting. Tunn. Undergr. Space Technol.
**2013**, 35, 1–7. [Google Scholar] [CrossRef] - Kuo, K.A.; Hunt, H.E.M.; Hussein, M.F.M. The effect of a twin tunnel on the propagation of ground-borne vibration from an underground railway. J. Sound Vib.
**2011**, 330, 6203–6222. [Google Scholar] [CrossRef] - Dörfler, P.; Sick, M.; Coutu, A. Flow-Induced Pulsation and Vibration in Hydroelectric Machinery, 1st ed.; Springer: London, UK, 2013; pp. 31–60. ISBN 978-7-5684-0070-1. [Google Scholar]
- Samanta, A.; Vinuesa, R.; Lashgari, I.; Schlatter, P.; Brandt, L. Enhanced secondary motion of the turbulent flow through a porous square duct. J. Fluid Mech.
**2015**, 784, 681–693. [Google Scholar] [CrossRef] - Vinuesa, R.; Bartrons, E.; Chiu, D.; Dressler, K.M.; Rüedi, J.D.; Suzuki, Y.; Nagib, H.M. New insight into flow development and two dimensionality of turbulent channel flows. Exp. Fluids
**2017**, 55, 1759. [Google Scholar] [CrossRef] - Vinuesa, R.; Schlatter, P.; Nagib, H.M. Role of data uncertainties in identifying the logarithmic region of turbulent boundary layers. Exp. Fluids
**2017**, 55, 1751. [Google Scholar] [CrossRef] - Wang, H.J. Research on Composite Structural Analysis and Dynamic Identification of Hydropower House. Ph.D. Thesis, Tianjin University, Tianjin, China, 2005. (In Chinese). [Google Scholar]

**Figure 4.**Vibration displacement sensors used in field tests: (

**a**) Close-up; and (

**b**) Field installation photo.

**Figure 7.**Variation of root mean square (RMS) of displacement with unit #1, unit #2 and unit #3 in operation respectively: (

**a**) Unit #1 in operation; (

**b**) Unit #2 in operation; and, (

**c**) Unit #3 in operation.

**Figure 8.**Time histories of vibration displacements in three directions: (

**a**) X direction; (

**b**) Y direction; and, (

**c**) Z direction.

**Figure 9.**RMS of vibration displacements with different units in three directions: (

**a**) X direction; (

**b**) Y direction; (

**c**) Z direction.

**Figure 11.**Spectrums of vibration displacements in three directions: (

**a**) X direction; (

**b**) Y direction; and, (

**c**) Z direction.

**Figure 12.**Energy proportions of vibrations in three directions: (

**a**) X direction; (

**b**) Y direction; and (

**c**) Z direction.

**Figure 17.**Diagram of simplified models of six units: (

**a**) Horizontal vibration; and, (

**b**) Vertical vibration.

Signal Decomposed | a7 | d7 | d6 | d5 | d4 | d3 | d2 | d1 |

Frequency Range (Hz) | 0–1.56 | 1.56–3.13 | 3.13–6.25 | 6.25–12.5 | 12.5–25 | 25–50 | 50–100 | 100–200 |

Direction | #1 | #2 | #3 | #4 | #5 | #6 |
---|---|---|---|---|---|---|

X | 100 | 22.55 | 12.69 | 8.89 | 5.33 | 1.62 |

Y | 100 | 12.63 | 3.02 | 1.93 | 1.31 | 1.08 |

Z | 100 | 10.11 | 2.89 | 1.32 | 1.13 | 0.99 |

Frequency (Hz) | #1 | #2 | #3 | #4 | #5 | #6 |
---|---|---|---|---|---|---|

1 | 100 | 22.55 | 12.69 | 8.89 | 5.33 | 1.62 |

2.4 | 100 | 22.57 | 12.69 | 8.90 | 5.33 | 1.61 |

Direction | Simplified Model | Field Test | Numerical Simulation |
---|---|---|---|

X | 25 | 17.69 | 22.55 |

Y | 16 | 10.69 | 12.63 |

Z | 16 | 10.74 | 10.11 |

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**MDPI and ACS Style**

Lian, J.; Wang, H.; Wang, H.
Study on Vibration Transmission among Units in Underground Powerhouse of a Hydropower Station. *Energies* **2018**, *11*, 3015.
https://doi.org/10.3390/en11113015

**AMA Style**

Lian J, Wang H, Wang H.
Study on Vibration Transmission among Units in Underground Powerhouse of a Hydropower Station. *Energies*. 2018; 11(11):3015.
https://doi.org/10.3390/en11113015

**Chicago/Turabian Style**

Lian, Jijian, Hongzhen Wang, and Haijun Wang.
2018. "Study on Vibration Transmission among Units in Underground Powerhouse of a Hydropower Station" *Energies* 11, no. 11: 3015.
https://doi.org/10.3390/en11113015