Taking the underground powerhouse of a pumped storage power station as the engineering background, this study established a 3D finite element model of the main and auxiliary powerhouse and performed the dynamic harmonica calculation for its fluctuating pressure. Based on the power flow theory, the ANSYS Parametric Design Language (APDL) procedure was completed to calculate the power transmission in the powerhouse. The law of dominant path recognition was first proposed to assess the structure’s dominant transmission using a numerical solution on nodes in the model. The conductivity of the closed-cell foam that filled the structure’s joints was examined, as were the dynamic transmission features of the rock around and beneath the powerhouse. The results indicated that, as a structural joint filler, closed-cell foam could actively restrict vibration transmission, and the directions of dynamic transmission were mainly perpendicular to and along the river in the foundation rock. Approximately 20 percent of the foundation rock beneath the auxiliary powerhouse was disturbed by the concrete around the spiral case and induced vibrations in the powerhouse’s lower floors. Vibration in the higher floors was derived from downstream rock, and the dynamic transmission effect had a clear advantage along the horizontal direction.
Underground powerhouses have many advantages compared to other types of powerhouses, including safety and freedom from external disturbances, so they are used widely around the world. Considering the need for management and repairs, auxiliary powerhouses are always arranged underground and close to the main powerhouses, forming a unified structure of both powerhouses. When running turbine machines, the auxiliary powerhouse absorbs the vibration energy from the main powerhouse. This energy may induce local resonance on the auxiliary powerhouse floors and walls, harming the equipment and other aspects of the structure [
Although there are not so many research results for the underground houses, there are many underground hydropower houses in China, so our team is always studying this point. For example, Zhi and Ma [
Using the theory of multidegree of freedom vibration transmission, this study investigated the power transmission in a hydropower house and the surrounding rock with hydraulic fluctuating pressure in a spiral case and a draft tube under normal operation. This study addressed questions related to energy distribution properties and energy transmission between the main and auxiliary powerhouses. The concept of a dominant power threshold value (DPTV) was defined for a concrete structure using the universality of power transmission and its probability distribution, and the law of dominant path recognition based on EFEM was proposed to provide an effective method of recognising structural dynamic transmission paths. The results would be helpful in further studies on dynamic transmission properties from a vibration source and the transmission path and in establishing a theoretical foundation for vibration isolation and dumping in hydropower house. The research in this paper could provide a reference method for recognising dynamic transmission paths in damaged structures.
Power is defined as the work performed by a dynamic load within the
Power can reflect not only the combined features of a force and its structural response but also structural impedance characteristics. Therefore, power flow plays an important role in structural vibration transmissions when assessing power transmission paths in complex constructions. In order to emphasize the process of power transmission, the process of power transmission can be as power flow. So power flow means power transmission in structures.
If a load can be simplified as a harmonic load, its structural response velocity is also presented as a series of harmonic changes. Therefore, the function can be represented by (
Manipulating (
Based on finite element method, the dynamic equation can be expressed as
By differentiating (
Substituting (
The dynamic transmission effect exists generally in a complex 3D model, but a method for identifying a 3D transmission path for underground hydropower station projects is much more difficult than other industrial projects. It is known that there are three types of dynamic loadings in hydropower station, hydraulic loading, mechanics loading, and electromagnetic loading. When they are inspired by high-pressure water in spiral case, hydroturbine, and generator (vibration sources), the hydroenergy changed to electric energy and at the same time the hydropower plant vibrates and the vibration transmits to other parts, such as the auxiliary powerhouse. From the vibration sources to the auxiliary powerhouse there will be many paths for power flow transmission, and it is very important to search the main paths and weaken the vibration of the auxiliary powerhouse by taking effective vibration reducing measure. This section describes the concept of a dominant power threshold value (DPTV) and confirms the law of dominant path recognition.
Using the intermediate value theory, it can be confirmed that
In fact, several vibration transmission paths exist at the same time in most projects. The dominant and absolute transmission paths are not only the objects of dynamic transmission recognition but also the critical paths for structural damping and isolation vibration.
The concept of insertion loss is involved in generalising the conclusion made in this section; therefore, the methods of Dimensionless Power Flow (DPF) and Power Decay Rate (PDR) are used to analyse the decrease in power on every vibration transmission path. The DPF and PDR can be described by (
The 3D finite element model of the main and auxiliary powerhouse, as shown in Figure
Profile View and the FEM of underground hydropower house.
Profile view of underground hydropower house
FEM of underground hydropower house
The 3D finite element model of the main and auxiliary powerhouse, as shown in Figure
As for the load for this research, fluctuating pressure was used by the numerical simulation analysis measured 0.153 MPa at the entrance of the spiral case, assuming the same value on every inner surface of the steel liner. The dominant frequency was measured as approximately 75 Hz (multiplied by the turbine rotation frequency and the number of turbine blades). The dynamic model was calculated by the harmonic response analysis.
The dynamic transmission characteristics were determined for a typical load (a harmonic load) and specific structure (underground hydropower house). In the 3D structure’s interior (unrelated to the shell), the shear wave and longitudinal wave took on the role of conducting energy, while the flexible wave effect was negligible. The rock was assumed to have material isotropy and the joints between the rock and concrete structure were assumed to be elastic connections. The filler in the structural joints between the main and auxiliary powerhouses was a nonlinear material that was in compression but not in tension.
Based on the first assumption, the difference between the positive and negative power flows in the 3D structure was negligible, and the effective power flow value was accepted for its fluctuant theory. Based on the second assumption, three translation DOFs were used to estimate the vibration and transmission characteristics of the concrete and rock. Details about the vibration transmission of the floors (six DOFs) are available in our other paper.
In previous study [
The conductivity of the filler in the system’s structural joints was defined as the ratio of output power in the nodes located on the main powerhouse floor’s boundary to the input power in the nodes located on the auxiliary powerhouse floor’s boundary. The conductivity of different fillers (asphalt wood plate, rubber plate, and closed-cell foam plate) is always less than 10%. A closed-cell foam plate was chosen as the fill material for the structural joints in this study [
Dynamic transmission in structure joint at 1288 m.
Power flow transmission on
Power flow transmission on
The power threshold values of the auxiliary powerhouse at 1288.8 m were calculated using (
DPTV of the auxiliary powerhouse floor at 1288.8 m. Units: W.
Factor |
Perpendicular to the river ( |
Along the river ( |
Vertical direction ( |
Remark | |||
---|---|---|---|---|---|---|---|
|
|
|
|
|
| ||
|
|
1.31 |
|
1.24 |
|
1.33 | |
|
|
1.38 |
|
1.23 |
|
1.42 | |
|
|
|
|
1.38 |
|
| |
|
|
2.23 |
|
|
|
2.19 |
As mentioned above, the surrounding rock played an important role in the vibration transmission through the main and auxiliary powerhouse. Therefore, the vibration transmission of the rocks located at the bottom of and surrounding the powerhouse required further analysis. Based on the FEM in Figure
DPTV of the rock. Units: W.
Height and |
Perpendicular to the river ( |
Along the river ( |
Vertical direction ( |
Remark | |||
---|---|---|---|---|---|---|---|
|
|
|
|
|
| ||
1269.0 m | |||||||
|
0.0004 | 1.472 | 0.0005 | 1.3585 | 0.001 | 1.4234 | The first floor nearby the draft tubes |
|
|
1.6309 | 0.0007 | 1.4798 | 0.0014 | 1.4941 | |
|
0.0012 | 1.8778 |
|
1.5385 |
|
2.0099 | |
1273.0 m | |||||||
|
0.001 | 1.2925 | 0.0009 | 1.3936 | 0.0014 | 1.3935 | The second floor nearby the draft tubes |
|
0.0015 | 1.4358 |
|
1.5735 |
|
1.5255 | |
|
0.0025 | 1.4142 | 0.0027 | 1.5965 | 0.0041 | 1.9193 | |
1275.8 m | |||||||
|
0.0008 | 1.4514 | 0.0007 | 1.37 | 0.0011 | 1.4969 | The third floor nearby the spiral cases |
|
|
1.7067 | 0.001 | 1.393 |
|
1.6826 | |
|
0.0015 | 2.3985 |
|
1.7017 | 0.003 | 2.4476 | |
1282.8 m | |||||||
|
0.0036 | 1.3079 | 0.0038 | 1.2554 | 0.0019 | 1.2966 | The fourth floor nearby the turbine |
|
|
1.5584 | 0.005 | 1.431 |
|
1.5207 | |
|
0.0074 | 1.8851 |
|
1.7212 | 0.0037 | 2.0477 | |
1288.8 m | |||||||
|
0.0047 | 1.2785 | 0.0025 | 1.3129 | 0.0023 | 1.4352 | The fifth floor nearby the generatrix cables |
|
0.0065 | 1.308 | 0.003 | 1.4138 | 0.0037 | 1.3578 | |
|
|
1.6645 |
|
2.2874 | 0.0056 | 1.4485 | |
1295.0 m | |||||||
|
|
1.5799 |
|
1.7096 |
|
1.7021 | The sixth floor nearby the generator sets |
|
0.0112 | 1.6649 | 0.005 | 1.8397 | 0.007 | 1.8457 | |
|
0.0186 | 2.4219 | 0.0148 | 2.6641 | 0.011 | 2.7651 |
The concrete around the spiral case connected the rock downstream of the main powerhouse, located from 1275.8 m to 1282.8 m, and the dynamic transmission was higher here than in other regions. From Figure
DPTV distribution.
It should be noted that the dominant transmission path must exist because of the DPTV during the vibration transmission, and the path distributes in belts corresponding to the DPTV. Taking the section from 1269.0 m to 1288.8 m as an example, the dominant transmission path distribution of the two heights is drawn in Figure
Dominant transmission path at 1269.0 m.
Dominant transmission distribution in direction perpendicular to the river
Dominant transmission distribution in the vertical direction
The dominant transmission path in the direction along the river was similar to that in the perpendicular direction; therefore, the figure is not provided here. This path’s vertical transmission curves are presented in Figure
Therefore, regardless of the direction of transmission, the decay transmission in the auxiliary powerhouse’s foundation rock was clear, and the dominant transmission region could not exceed 20 percent of the auxiliary powerhouse.
Dominant transmission path at 1288.8 m.
Dominant transmission distribution in direction along the river
Dominant transmission distribution in direction perpendicular to the river
Because there were no DPTVs in the vertical direction, the vibration transmission was simply an ordinary transmission; therefore, a figure is not provided here. The path could have run through the main powerhouse to up- or downstream rock, with little vibration power transmitted to the auxiliary powerhouse.
In short, the dominant vibration path ran from the downstream wall of the main powerhouse, along downstream rock, to the floor of the auxiliary powerhouse at 1288.8 m. Above all, the power flow in the direction along the river was superior to that perpendicular to the river, and the vertical vibrations were not transmitted to the auxiliary powerhouse by the surrounding rock.
A unified main and auxiliary powerhouse structure is popular in underground hydropower engineering; therefore, the laws of vibration transmission are beneficial to the structural optimisation of such systems. Due to this study’s limitations, only portions of the power flow and the transmission paths in special positions were analysed. However, the conclusions drawn from the results reflected the general laws of vibration transmission for underground powerhouses. The filler in the structural joints between the main and auxiliary powerhouses had an effect on structural vibration transmission, but the close-cell foam plane actively isolated the vibration on both sides of the structural joints, making it a remarkable structural filling. In our analysis of the auxiliary powerhouse’s foundation rock, the dynamic transmission in the vertical direction and that along the river were more obvious than that in the direction perpendicular to the river. Approximately 20 percent of the auxiliary powerhouse’s foundation rock was disturbed by the vibrations coming from the main powerhouse. The first floor of the auxiliary powerhouse was significantly affected by the foundation rock, while the other floors were not. On higher floors, the power came from downstream rock at the same elevation in the directions along and perpendicular to the river. The columns transmitted only vertical power flow, and the power transmission in these floors clearly decreased with increasing elevation. As for hydropower house damping vibration and isolation vibration, two aspects should be considered: extending the path and cutting off the path. As for foundation rock, the distance from spiral case concrete to auxiliary powerhouse foundation should be extended but, for surrounding rock, the method of cutting off the transmission path can be adopted, such as cutting off the connection of rock and auxiliary powerhouse walls and the columns, as well as the floors.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (Grant no. 51379030).