Shock and Vibration

Volume 2016 (2016), Article ID 5039578, 9 pages

http://dx.doi.org/10.1155/2016/5039578

## Study of the Vibration Transmission and Path Recognition of an Underground Powerhouse Using Energy Finite Element Method

^{1}School of Civil and Hydraulic Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China^{2}College of Water Resource, Shenyang Agricultural University, Shenyang, Liaoning 110866, China

Received 19 August 2015; Revised 7 October 2015; Accepted 15 October 2015

Academic Editor: Marcello Vanali

Copyright © 2016 Wei Xu and Zhen-yue Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2]. Up to now, a number of studies have been made on the vibration of powerhouse structure under pressure fluctuation, but most of them only examined stress and displacement of the main powerhouses [3, 4]; there were less related researches on auxiliary powerhouse and on the path of vibration transmission. The harmonic calculation in dynamic analysis is the primary method used to analyse complex structures, but the transmission function achieved by a structural frequency response cannot reflect all of the information on a vibratory transmission path. Therefore, harmonic calculations cannot directly identify vibration transmission paths and, as a result, there are no reasonable measures in place for the dangers caused by structural vibrations. Recently, researchers [5, 6] studied vibration power flow and its application on the passenger car for identification of vibration transmission path; the results showed that the method could reduce the structure-borne noise level about 5 dBA. Vander et al. [7] focused on vibration transmissions with single excitation and multipoint impact in a car. De Klerk and Rixen [8] proposed a component transfer path analysis procedure based on frequency response functions for test bench dynamics. Le Bot and Bou Chakra [9] presented an experiment to measure the dependence of friction noise versus the nominal contact area. They found that the vibration energy was proportional to the contact area sometimes, but on the other hand the vibration energy was constant. Renno and Mace [10] calculated the reflection and transmitting coefficients of the joints by coupling finite element with a wave and finite element, and they considered the wave travelling in the structure. Wang et al. [11] investigated the natural frequencies and mode shapes of structures with mixed random and interval parameters by using a hybrid stochastic and interval approach; the examples showed that the method could be also applicable to solve pure random and pure interval problems. Besides, power flow method has also been widely applied in other aspects, for examples, beam structures [12], flexible manipulator [13], powertrain [14], complex frame structure [15], compress system [16], and electric vehicle [17].

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 [18, 19] have performed a contrastive analysis on a numerical vibratory model of an underground powerhouse, while the work of Xu et al. [20] is focusing on using power flow theory and the energy finite element method (EFEM) to analyse the response of main powerhouse’s vibrations by fluctuating pressure in a spiral case.

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.

#### 2. Basic Theory

##### 2.1. Power Flow Theory and Power Flow FEM

Power is defined as the work performed by a dynamic load within the th node interval, as described bywhere and are the force and velocity, respectively.

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 (2) when calculating the power flow in a cycle as follows:where is the effective value of power flow in one cycle, is the harmonic load (a complex vector), is velocity of a point in the structure (also a complex vector), is the angular frequency of the simple harmonic vibration, and means the phase angle of the velocity and load.

Manipulating (2) with complex operation when and are expressed by complex vector, can be showed as expression in where the symbol “” represents a conjugate vector and the structural power flow transmission can be obtained by a theoretical derivation or experiment. Here if the excited load is the force, then is power flow in time domain, while the load on the structure is defined as a force spectral density, and then can be expressed by in frequent domain, where means frequency and is the power flow spectral density.

Based on finite element method, the dynamic equation can be expressed aswhere is stiffness matrix, is damping matrix, is mass matrix, is the vector composed by nodes’ displacement, is a vector composed by nodes’ force, and and are frequency and imaginary symbol, respectively.

By differentiating (4) with respect to the time, node velocity can be expressed by (5) in frequent domain as follows: where is the node velocity and other symbols are shown as mentioned above.

Substituting (5) into (3) yields (6), and power flow on node can be obtained as follows:where and are the real part and the imaginary part of the variables, respectively.

##### 2.2. Transmission Path Recognition

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.

*(1) Dominant Power Threshold Value*. Based on the theory of a significance test in mathematical statistics, (7) is the discriminate of the main transmission path for structural vibrations: where represents the power flow at a node of the finite element model, the meaning of symbol is the probability, is a constant (), denotes the significant factor, and represents the probability.

Using the intermediate value theory, it can be confirmed that must exist within the transmission region to satisfy (7), and is the power flow value at a specific node, so can be defined as the DPTV in this equation.

*(2) Law of Dominant Path Recognition*. The power flow transmission path has spatiality in a spatial structure, and the intensive level of power distribution, magnitude of the peak value, and radiation effect of the power transition all have an effect on the transmission path recognition. A spatial structure can, therefore, be approximated using a 3D finite element model, and its nodes’ power flow is calculated based on the power flow finite element method. When = 0.2~0.9, DPTV is calculated eight times separately, and eight transmission domains are confirmed. The transmission intensity can then be obtained usingwhere is the average power flow (W), represents the power flow value at the th node of the finite element model, represents the volume of the th node with unit m^{3}, and stands for the total number of nodes in the transmission domain.

*The Law of Dominant Path Recognition* is as follows: , where is the transmission dominance ratio of the adjacent significance factor , and is the power flow discrepancy in the adjacent significance factor . If , the transmission region is defined as the ordinary transmission path; if , the transmission region is defined as the domain transmission path; and if , the transmission region is defined as the absolute transmission path.

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.

##### 2.3. Method of Dimensionless Power Flow and Power Decay Rate

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 (9) and (10), respectively, as follows: where is the DPF, is a reference power flow value used instead of the DPTV (W/m^{3}), is the significance factor of the dominant transmission, = 0.2~0.9, is the PDR (db/m), and are the DPFs between any two points, and is the corresponding distance (m). The higher the value of becomes, the more the energy has been absorbed, while a lower value indicates lower absorption and vibrations transmitted far from the powerhouse.

#### 3. Finite Element Model

##### 3.1. Finite Element Model and Load

The 3D finite element model of the main and auxiliary powerhouse, as shown in Figure 1, is established based on Hohhot Pump Storage Station, which has the main powerhouse, erection bay, and auxiliary powerhouse, and four hydroelectric generating sets are located on the main powerhouse and 22 meters distance at a line. Five floors from the lowest to the highest in the main powerhouse are draft tube floor, spiral case floor, turbine floor, generatrix floor, and generator floor, respectively. The auxiliary powerhouse nearby the main powerhouse has seven floors, including libraries, storerooms, and offices. The auxiliary powerhouse and the part of the main powerhouse with one hydroelectric generating set are chosen for researching on power transmission between the structures. For the main powerhouse, the floors, the walls, the generator supports, wind covers, spiral case, and draft tube are the main structures, and the thickness of the walls is 1.0 meter, the spiral case is made of steel, the diameter of the inlet pipeline is 2.0 meters, and the concrete thickness outside of the spiral case is from 0.8 meter to 2.0 meters. The draft tube belongs to elbow style, and the concrete thickness outside of the draft tube is 1.35 meters. The auxiliary powerhouse is separated from the main powerhouse by settlement joint; it has seven floors as shown in Figure 1. The walls of the main powerhouse and the concrete of draft tube are connected with rocks around the powerhouse, while peripheral columns and the lowest floor of the auxiliary powerhouse are also connected with rocks nearby the structures.