Path Transmissibility Analysis Considering Two Types of Correlations in Hydropower Stations

Abstract

A new vibration model is built by introducing the head-cover vibration transfer path based on a previous analysis of the vertical vibration model for hydropower station units and powerhouses. This research focuses on disturbance- and parameter-related transfer paths in a practical situation. In a complex situation, the application of the stochastic perturbation method is expanded using an algebra synthesis method the Hadamard product, and theoretical analyses, and numerical simulations of transfer paths in the new vibration model are carried out through the expanded perturbation method. The path transfer force, the path transmissibility, and the path disturbance ranges in the frequency domain are provided. The results indicate that the methods proposed in this study can efficiently reduce the disturbance range and can accurately analyze the transfer paths of hydraulic-source vertical vibration in hydropower stations.

1. Introduction

Hydraulic vibration is the main vibration source in hydropower station units and powerhouses. The vertical vibration in units is usually caused by hydraulic pressure fluctuations or other loads on the water turbine’s flow passage components. Field and model tests have shown that there are three main vibration transfer paths running from the water turbine to the powerhouse [1]: (1) runner-shaft-bearing-fixed components (machine frame, head-cover-powerhouse;) (2) flow pressure-spiral case-powerhouse; and (3) runner-runner negative pressure region-head cover-powerhouse. Previous studies concerning the vertical vibration produced by a hydraulic source mainly focus on path (1), while the effects of paths (2) and (3) are usually ignored [2]. However, as the scale and capacity of hydropower stations increase, the flow passage area of the head-cover system continually increases, with corresponding increases in the vibration of the head-cover system. Therefore, the influence of the head-cover system becomes more important in the hydraulic vibration transfer path, and ignoring path (3) will produce a larger error. Therefore, it is imperative that the contribution of path (3) should be analyzed; more specifically, the contribution of the vibration of the head-cover system of the hydropower station vertical vibration transfer should be analyzed. However, due to the presence of randomness, it is difficult to clearly and accurately describe the contribution of this transfer path to the structural vibration.

Theoretical analyses of the vibration transfer path can be classified as a stochastic structural system problem. At present, the Monte-Carlo numerical simulation method (MCSM) [3, 4] and the perturbation method [5–7] are the most popular analysis methods. MCSM is used less frequently because of the large amount of computation required when dealing with a large-scale structure. Conversely, the perturbation method is applied by many researchers in various fields. Collins and Thompson [8] initially employed the perturbation method to analyze stochastic dynamical systematic characteristics in 1969; the perturbation method was later employed in a static analysis by Hisada and Nakagiri [9] and in a dynamic analysis by Liu et al. [10]. Kronecker algebra was introduced to the expansion of the perturbation method by Vetter [11]. After several decades of development, relevant studies on perturbation theory were quite abundant. The primary methods included the L-P method [12], the multiple scale method [13, 14], the average method [15], the KBM (Krylov-Bogoliubov-Mitropolsky) method [15, 16] and the singular perturbation method [17, 18]. Recent developments include the homotopy perturbation method [19]. In the analysis of transfer paths combined with the perturbation method, Zhang et al. built a theoretical model for vibration transfer path analysis by implementing the perturbation method with Kronecker algebra in the static analysis [20], the dynamic analysis [21] and the reliability analysis [22, 23]. This model has been used for the analysis and design of mechanical components. Computations of the transfer path considering multi-vibration sources have been carried out by Zhao and Zhang [24] based on Zhang’s research. These results have been used for isolation vibration analysis. The above research studies focus on additive disturbance analysis. Using multiplicative disturbance analysis, Gao et al. separately analyzed the dynamical characteristics of a truss structure by using the interval factor method [25], the random factor method [26], and nonstationary random excitation [27]. Ma et al. conducted a dynamical characteristic analysis of a linear [28] and nonlinear [29] truss structure using fuzzy variables. In 2010, a two-factor method was proposed by Ma et al. [30], which considers two multiplicative disturbances. However, because of the limitations of these methods, they have not been applied to transfer path analysis with respect to a multiplicative disturbance. Furthermore, previous studies have not jointly analyzed these two types of disturbances. In practice, the two types of disturbances always exist together, and there are some correlations between these two disturbances. There are some correlations among the parameters of the structure as well. Thus, the previous studies were not based on an accurate analysis because they only considered a single disturbance, neglected the correlations between disturbances, or neglected the correlations among parameters. Therefore, it is necessary to jointly analyze the problem of the vibration transfer path of time-invariable parameters for the two types of disturbances and their related parameters. Only a few studies have reported on the two types of correlations in perturbation theory. Previous studies on correlations ignored the two types of correlation or converted relevant variables into irrelevant variables. These analyses did not consider the effects of correlations and thus were not rigorous. At present, only a few researchers have attempted to explore this aspect. Some examples include studies on molecular chemistry [31], nuclear physics [32], and vocal vibration [33, 34]. Of these studies, only the study on vocal vibration considered the correlation parameters, but it did not involve the perturbation method. At present, there are few available studies on structural analysis. Pirrotta [35] conducted a perturbation analysis of delta-correlated processes, but he did not study the correlation between parameters; Ambrogio [36] described the correlations by considering the correlation coefficient between additive and multiplicative disturbances for one parameter, but he also did not refer to the correlation between parameters. Husain et al. [37] solved the problem of parameter correlation, but they only considered computational results and did not extend their results to improve the perturbation method. Khodaparast et al. [38] corrected the structure variation model, but they ignored the correlation between modified parameters and measurement values. This study focuses on the application of perturbation theory with two types of correlation from the perspective of structure analysis.

Hydropower station units and powerhouses are large-scale structures. Most of the parameters’ disturbances should be less than 10% of their mean values in such a large structure. Therefore, problems regarding hydropower stations can be solved by the perturbation method. First, the head-cover system is introduced, which has the same vibration source as a path (1); the elastic foundation constraint is selected; and a new vibration model is built on the basis of the previous vertical vibration model. Furthermore, a method for solving the path transfer force is proposed using the general method of dynamic analysis. Second, for the test signal, correlations between the two types of disturbances and correlations between the parameters are considered based on the single disturbance vibration path analysis. By using the coefficient of variation algebra synthesis method [39], Kronecker algebra [11], and the Hadamard product [40], gradient-sorting estimations of the transfer paths in the frequency domain are carried out, and methods for determining the transfer force and transmissibility and their disturbances are proposed. Finally, the method described in this paper is verified using the model of a large hydropower station.

2. Analysis Model with the Introduction of a Head-Cover System

For a hydroelectric generating unit, regardless of whether it has a suspension or umbrella structure, the weight of its rotating parts is successively transferred to the reinforced concrete machine foundation through the thrust bearing, frame (suspension units containing the stator frame), and sole screw. The head-cover system is always fixed on the base ring strengthening plate, and the head-cover system and the strengthening plate are considered to be one part. Taking a vertical vibration characteristic analysis of the umbrella unit as an example, the model contains a shaft system, thrust bearing, and lower bracket (see Figure 1). The heave shaft can be simplified as a massless elastic continuous beam, and then its mass can be regarded as a mass attached to three nodes, 𝑚₁, 𝑚₂, and 𝑚₃.  𝑚₁ can be defined as the mass of the excitation rotor and the shaft, the half shafting mass, which is measured from the heavy shaft top to the rotor frame, and another mass added on top of the heavy shaft;  𝑚₂ can be defined as the mass of the central body of the rotor frame, the half mass of the whole gate arm, and the half mass of the whole shaft; and 𝑚₃ can be defined as the mass of the water, turbine runner, the additional mass of water and the half shafting mass, which is measured from the rotor frame to the hydraulic turbine. The rotor gate arm can be simplified as a massless elastic continuous rod, and then its mass can be assigned to the runner margin and the central body of the rotor frame. 𝑘₄ can be defined as the sum of the vertical stiffness of the whole gate arm, and 𝑚₄ can be defined as the lumped mass of the runner margin. The outer end of the lower bracket is fixed to the concrete foundation. The lower bracket gate arm can then be simplified as a gravity-free beam if the coupling effect of the foundation is ignored. 𝑘₅₂ can be defined as the vertical stiffness of the lower bracket gate arm. 𝑚₅ can be defined as the lumped mass of one end of the lower bracket, which is close to the heavy shaft, and the half mass of the gate arm; 𝑚₅ can be connected to 𝑚₂ by the thrust bearing, which is simplified by an equivalent stiffness 𝑘₅₁.

Figure 1 

Simplified model of the coupling system between the umbrella unit and the powerhouse.

With the introduction of the head-cover system, the vibration transfer path is as follows: first, the hydraulic vertical vibration is transferred to the head cover by the heavy shaft seal and guide bearing in the water turbine runner chamber; next, the vibration is transferred from the head cover to the spiral case base ring strengthening plate, which is connected to the outer end of the head cover; and finally, the vibration is transferred to the machine foundation by the wrapped concrete outside the spiral case. Ignoring the coupling effect, the control parts and other additional parts on the head cover can be regarded as the attached mass of the head-cover system. As the lumped mass is close to the heavy shaft, 𝑚₆ can be defined as the mass of the central body and the half mass of the whole head-cover system. The head-cover system can be simplified as a gravity-free beam, and 𝑘₆₂ can be defined as the vertical stiffness. 𝑚₆ can be connected to 𝑚₃ by the sealing spring, which lies between the head-cover structure and the water turbine runner. The vertical stiffness of the connection can be simplified by an equivalent stiffness 𝑘₆₁.

𝑘₅₁ is the series stiffness of the elastic oil tank stiffness (thrust bearing support system) and the oil film stiffness. 𝑘₆₁ is the series stiffness of the sealing structure stiffness and the clearance water stiffness. These two parameters exhibit a linear relationship because the unit’s axial water thrust varies with the unit’s conditions. However, 𝑘₅₁ and 𝑘₆₁ are simplified as a single stochastic variable in this study.

In the process of examining the unit’s vertical vibration, the machine foundation pier can be regarded as an elastic foundation. The structure can be treated as a single node, 𝑚₇, which ranges from the machine foundation pier to the concrete floor in the turbine layer. The structure can be treated as a single node, 𝑚₈, which ranges from the turbine layer to the foundation. The stiffness of this structure is replaced by the equivalent stiffness, which is based on the strengthening plate, spiral structure, and concrete structure. Thus, the hydropower station base is treated by dividing it into two nodes at the connection of the head-cover system and the hydropower station base, where 𝑘₆₂ is only connected to 𝑚₈.

For the entire model, the form and meaning of the dampness matrix are similar to those of the stiffness matrix. Assuming that the system is linear, the differential equation of the vibration is found from the Lagrange equation: 𝐌𝐮̈+𝐂𝐮̇+𝐊𝐮=𝐅(𝑡). (1)

By merging these dynamical balance equations for the shaft, the rotor, the lower bracket, the head-cover system, and the machine foundation pier, an equation with 8 degrees of freedom and 26 parameters can be obtained, and the total stiffness matrix can be written as𝑘𝐊=¹−𝑘₁000000−𝑘₁𝑘₁+𝑘₃+𝑘₄+𝑘₅₁−𝑘₃−𝑘₄−𝑘₅₁0000−𝑘₃𝑘₃+𝑘₆₁00−𝑘₆₁000−𝑘₄0𝑘₄00000−𝑘₅₁00𝑘₅₁+𝑘₅₂0−𝑘₅₂000−𝑘₆₁00𝑘₆₁+𝑘₆₂0−𝑘₆₂0000−𝑘₅₂0𝑘₅₂+𝑘₇−𝑘₇00000−𝑘₆₂−𝑘₇𝑘₆₂+𝑘₇+𝑘₈. (2)

The total mass matrix is obtained from the lumped mass: 𝐌=diag𝑚₁,𝑚₂,𝑚₃,𝑚₄,𝑚₅,𝑚₆,𝑚₇,𝑚₈. (3)

Generally, the characteristics of the vertical vibration source in the hydropower station are unique position, simple contact surface with structure, and few ingredients. Conversely, the characteristics of the lateral (radial) vibration are multipositions, strong nonlinear contact surface, and complex excitation. For simplifying the model and the calculation, the total structure of the unit’s powerhouse is considered as suffering vertical harmonic excitation. The vertical harmonic excitation is located on the water turbine runner, and the response of the total structure is in a steady state. In addition, the initial phase remains constant throughout the whole process. Setting the steady-state response as 𝑢_𝑖(𝑡)=𝑈_𝑖𝑒^{𝑖(𝜔𝑡+𝜑𝑖)} and 𝑈_𝑖=𝑈_𝑖𝑒^𝑖𝜑𝑖 yields𝑢_𝑖(𝑡)=𝑈_𝑖𝑒^𝑖𝜔𝑡,𝑢̇_𝑖(𝑡)=𝑖𝜔𝑈_𝑖𝑒^𝑖𝜔𝑡,𝑢̈_𝑖(𝑡)=−𝜔²𝑈_𝑖𝑒^𝑖𝜔𝑡,𝐔=𝑢₁,𝑢₂,𝑢₃,𝑢₄,𝑢₅,𝑢₆,𝑢₇,𝑢₈^𝑇,𝐅(𝑡)=0,0,𝐹₀𝑒^𝑖𝜔𝑡,0,0,0,0,0^𝑇. (4a)(4b)(4c)

Substituting (4a)–(4c) into (1) gives the dynamical balance equation of the hydropower station’s vertical vibration with the head-cover system in the frequency domain: −𝜔²𝐌+𝑖𝜔𝐂+𝐊𝐔=𝐅(𝑡). (5)

The response vector 𝐔 at each node of the structure was obtained by numerical calculations. 𝐹_zhou and 𝐹_ding denote the forces that are transferred to the machine foundation pier from the water turbine by the shaft system and the head-cover system, respectively, and can be written as𝐹_zhou=𝑘_zhou𝑢₃−𝑢₇+𝑐_zhou𝑢̇₃−𝑢̇₇=𝑘_zhou+𝑖𝜔𝑐_zhou𝑈₃−𝑈₇𝑒^𝑖𝜔𝑡,𝐹_ding=𝑘_ding𝑢₃−𝑢₇+𝑐_ding𝑢̇₃−𝑢̇₇=𝑘_ding+𝑖𝜔𝑐_ding𝑈₃−𝑈₇𝑒^𝑖𝜔𝑡. (6a)(6b)

In (6a)–(6b), 𝑘_zhou and 𝑐_zhou denote the path stiffness and the path dampness of the vibration path through the shaft system, respectively. Similarly, 𝑘_ding and 𝑐_ding denote the path stiffness and the path dampness through the head-cover system, respectively. When calculating the path stiffness and dampness of the shaft system, 𝑚₁ and 𝑚₄ are treated as dynamic vibration absorbers with dampness.

3. Vibration Path Sorting Considering Two Types of Correlations

In the analysis of a hydropower station, parameters such as the stiffness, mass, and dampness are attributed to the multiplicative disturbances arising from the material properties, manufacturing technology, and other factors. These parameters, obtained by measurement, are attributed to the additive disturbances due to the testing noise and environmental noise. Furthermore, because the testing method and environment are the same for every parameter, the two disturbances of each parameter are interrelated. These parameters are also interrelated by means of continuous structure discretization.

Given the factors mentioned above, the mass, stiffness, and dampness are described as a random vector 𝐚 with 𝑛 random variables, where every random variable involves two types of disturbances. When the variable disturbance is lower than 15% of the mean value, the stochastic variable can be expressed as 𝑎_𝑖=𝑎¹_𝑖𝑎^𝑑_𝑖+𝑎²_𝑖. (7)

In (7), 𝑎_𝑖 denotes the 𝑖th element in the random vector 𝐚. 𝑎¹_𝑖 and 𝑎²_𝑖 denote the multiplicative and additive disturbance of the random variable 𝑎_𝑖, respectively. Assuming that their mean values are 1 and 0, respectively, 𝑎^𝑑_𝑖 denotes the deterministic component of 𝑎_𝑖, which represents the mean value after multiple samplings.

According to the coefficient of variation algebra synthesis method [40], calculating the mathematical expectation and variance of (7) yields 𝐸_𝑎𝑖=𝐸𝑎¹_𝑖𝑎^𝑑_𝑖+𝑎²_𝑖=𝐸𝑎¹_𝑖𝑎^𝑑_𝑖+𝐸𝑎²_𝑖=𝑎^𝑑_𝑖,𝜎²_𝑎𝑖=Var𝑎_𝑖=𝐸𝑎_𝑖−𝐸_𝑎𝑖²=𝐸𝑎¹_𝑖−𝑎_𝑖𝑎^𝑑_𝑖+𝑎²_𝑖²=𝑎^𝑑_𝑖²𝜎²_𝑎1𝑖+𝜎²_𝑎2𝑖+2𝑎^𝑑_𝑖Cov𝑎¹_𝑖,𝑎²_𝑖, (8) where Cov𝑎¹_𝑖,𝑎²_𝑖=𝜌_{𝑎1𝑖,𝑎2𝑖}𝜎_𝑎¹_𝑖𝜎_𝑎²_𝑖, (9) where 𝜌_𝑎1,𝑎2 is the correlation coefficient of the multiplicative and additive disturbances contained in the parameter 𝑎. If these parameters are interrelated, then Cov𝑎_𝑖,𝑎_𝑗=𝜌_{𝑎𝑖,𝑎𝑗}𝜎_𝑎𝑖𝜎_𝑎𝑗, (10) where 𝜌_{𝑎𝑖,𝑎𝑗} is the correlation coefficient of the random variables 𝑎_𝑖 and 𝑎_𝑗. Generally, if the random variables follow a normal distribution, their linear transformations and multiplication will also follow a normal distribution. For example, if the vector 𝐚 follows a normal distribution, then the function 𝐹_𝑖(𝐚) will follow a normal distribution. Using the Taylor expansion, expanding the transfer force 𝐹_𝑖 at the mean value 𝐹^𝑑_𝑖 yields 𝐹_𝑖=𝐹^𝑑_𝑖+𝜕𝐹^𝑑_𝑖𝜕𝐚^𝑇𝐚−𝐚^𝑑+𝑂𝐚^𝑝=𝐹^𝑑_𝑖+𝜕𝐹^𝑑_𝑖𝜕𝐚^𝑇𝐚^𝑝+𝑂𝐚^𝑝, (11) where 𝐚^𝑝=𝐚−𝐚^𝑑=𝐚¹−1𝐚^𝑑+𝐚². (12)

Neglecting components of the second order and above, the disturbance of the transfer force is 𝐹^𝑝_𝑖=^𝑚_𝑘=1𝜕𝐹^𝑑_𝑖𝜕𝑎_𝑘𝑎^𝑝_𝑘. (13)

In (13), 𝜕𝐹^𝑑_𝑖/𝜕𝑎_𝑘 is the partial derivative of 𝐹_𝑖 with respect to the random variable 𝑎_𝑘, which is also the first-order sensitivity of 𝑎_𝑘. The covariance of 𝐹^𝑝_𝑖 and 𝐹^𝑝_𝑗 is Cov𝐹_𝑖,𝐹_𝑗=𝐸𝐹^𝑝_𝑖𝐹^𝑝_𝑗=𝐸^𝑚_𝑘=1𝜕𝐹^𝑑_𝑖𝜕𝑎_𝑘𝑎^𝑝_𝑘^𝑚_𝑙=1𝜕𝐹^𝑑_𝑗𝜕𝑎_𝑙𝑎^𝑝_𝑙. (14)

That is, Cov𝐹_𝑖,𝐹_𝑗=^𝑚𝑚_𝑘=1𝑙=1𝜕𝐹^𝑑_𝑖𝜕𝑎_𝑘𝜕𝐹^𝑑_𝑗𝜕𝑎_𝑙𝐸𝑎^𝑝_𝑘𝑎^𝑝_𝑙=^𝑚𝑚_𝑘=1𝑙=1𝜕𝐹^𝑑_𝑖𝜕𝑎_𝑘𝜕𝐹^𝑑_𝑗𝜕𝑎_𝑙Cov𝑎_𝑘,𝑎_𝑙. (15)

Equation (16) shows that Cov(𝐹_𝑖,𝐹_𝑗) of the transfer force 𝐹_𝑖 and 𝐹_𝑗 can be expressed by Cov(𝑎_𝑘,𝑎_𝑙) of the random structure parameter: Cov𝐹_𝑖,𝐹_𝑗=^𝑚𝑚_𝑖=1𝑗=1𝜕𝐹^𝑑_𝑖𝜕𝑎_𝑘𝜕𝐹^𝑑_𝑗𝜕𝑎_𝑙𝜌_{𝑎𝑘,𝑎𝑙}𝜎_𝑎𝑘𝜎_𝑎𝑙. (16)

If 𝑖=𝑗, 𝜎²_𝐹𝑖=Cov𝐹_𝑖,𝐹_𝑖=^𝑚𝑚_𝑘=1𝑙=1𝜕𝐹^𝑑_𝑖𝜕𝑎_𝑘𝜕𝐹^𝑑_𝑖𝜕𝑎_𝑙𝜌_{𝑎𝑘,𝑎𝑙}𝜎_𝑎𝑘𝜎_𝑎𝑙. (17)

According to Kronecker algebra [11], the corresponding stochastic analysis theory and Hadamard product [41] yield 𝜎²_𝐹𝑖=𝜕𝐹^𝑑_𝑖𝜕𝐚^𝑇[2]⇀𝝆∘𝜎_𝐚^[2], (18) where the subscript [2] denotes the Kronecker power; that is, 𝑎^[2]=𝑎⊗𝑎; if 𝑎 is of order 𝑛×1, then 𝑎^[2] is of order 𝑛²×1. ^⇀𝝆 is the stacking vector of the matrix of correlation coefficients, which is also of order 𝑛²×1. The symbol ∘ denotes the Hadamard product. Therefore, 𝜎²_𝐹𝑖 is of order 1×1.

The sensitivity matrix |𝜕𝐹^𝑑_𝑖(𝑎)/𝜕𝐚^𝑇| of the transfer force of each parameter is 𝜕𝐹^𝑑_𝑖𝜕𝐚^𝑇=𝜕𝐹^𝑑_𝑖𝜕𝑎₁⋯𝜕𝐹^𝑑_𝑖𝜕𝑎_𝑛. (19)

Substituting (19) into (18), the variance of the transfer force 𝐹_𝑖 for each path can be calculated. Equation (11) only involves the first-order Taylor expansion. A higher-order Taylor expansion will improve the accuracy, but it involves complicated mathematical calculations. Equation (18) shows that the transfer force variance can be directly obtained from the random variables’ numerical characteristics. The calculations can thus be simplified because the sample is not included in the mathematical operations. The correlations between parameters are merely modified in (18) and do not increase the number of calculations required. The transmissibility is defined as the ratio of the amplitudes between the transfer force and the vibration source excitation force: 𝛽_𝑖=𝐹_𝑖𝐹₀. (20)

Neglecting the disturbance of the excitation force 𝐹₀, based on the random variable algebra synthesis method, the expectation, variance, and transfer coefficient of the transmissibility can be written as𝐸_𝛽𝛽(𝑎=𝐸𝐹=𝐸)^𝑡𝐸𝐹₀𝜎_2𝛽,=𝐸𝐹_𝑖²𝐸𝐹₀²Var𝐹_𝑖𝐸𝐹_𝑖²,𝐸𝜃=^𝛽𝜎_𝛽. (21a)(21b)(21c)

The result of a random variable that follows a normal distribution divided by a constant also follows a normal distribution. 𝐸[𝐹₀] and 𝐸[𝐹_𝑖] are known in the deterministic system, so the variance of the transmissibility will follow a normal distribution. The transfer coefficient is the transfer efficiency of the excitation force. The gradient sorting of the vibration path transmissibility in the frequency domain can be obtained by changing the frequency of the excitation force.

A solution of the vibration transfer path for the two types of disturbances and correlations is proposed based on the above methods. The path transmissibility and its probabilistic characteristic are provided. In this study, the solution only involves the first-order sensitivity of the random parameter and the probabilistic characteristic of the random variable. Additionally, the introduction of the two types of correlations does not involve excessive calculations. Therefore, the calculation accuracy is improved for practical problems.

4. Example Analysis

The main structure of the umbrella unit in a large hydropower station is shown in Figure 2, and the simplified model of transfer path is shown in Figure 1. In this example, the effect of the spiral case and substructure is ignored [41, 42] because their effect is far less than that of the upper structure. The excitation is assumed to be a simple harmonic excitation. The mean values of the random parameters can be obtained from the hydropower station design diagrams. These mean values are 𝑚₁=8.28×10⁴, 𝑚₂=1.042×10⁶, 𝑚₃=3.29×10⁵, 𝑚₄=9×10⁵, 𝑚₅=1.2×10⁵, 𝑚₆=1.15×10⁵, 𝑚₇=1.39×10⁵ and 𝑚₈=8.92×10⁵, in units of kg; 𝑘₁=7.26×10¹⁰, 𝑘₃=5.72×10¹⁰, 𝑘₄=2.32×10¹⁰, 𝑘₅₁=2.20×10¹², 𝑘₅₂=9.41×10⁹, 𝑘₆₁=1.73×10⁸, 𝑘₆₂=1.73×10¹⁰, 𝑘₇=7.70×10⁹, and 𝑘₈=4.26×10⁸, where the unit of the stiffness 𝑘 is N/m; and 𝑐₁=5.48×10⁶, 𝑐₃=4.11×10⁶, 𝑐₄=1.02×10⁷, 𝑐₅₁=2.57×10⁷, 𝑐₅₂=7.51×10⁵, 𝑐₆₁=2.23×10⁵, 𝑐₆₂=9.99×10⁴, 𝑐₇=1.64×10⁶, and 𝑐₈=9.74×10⁵, where the unit of the dampness 𝑐 is N⋅s/m. Each parameter comprises two disturbances that follow a normal distribution. The multiplicative disturbance is related to the difficulty of obtaining the parameters. The multiplicative variance coefficients of 𝑘₅₁ (including the vertical stiffness of the thrust bearing) and 𝑘₆₁ (the sealing equivalent vertical stiffness between the head cover and runner) are set to 0.10. The equivalent bending rigidity of 𝑘₆₂s multiplicative variance coefficient is set to 0.075 because there are many uncertain factors in the control components and other attached components on the head cover. The multiplicative random variance coefficients of the other parameters are set at 0.05. The additive disturbance is related to the measurement range. The standard deviations of the additive random variable with respect to mass and stiffness are set at 10⁴ and 10⁸, respectively, according to each parameter’s mean value. The variance coefficient of damping is similar to that of the mass and stiffness. In this example, the parameters’ disturbances are generated by a function in the MATLAB software, and the sample size is 10000 for each disturbance. All errors are determined to be smaller than 0.1% by comparing the variances and mean values of the generated samples with the corresponding set values. Therefore, the generated sample variances and mean values are used in the study. The correlation function is determined by the generated function. The partial derivative is calculated by the software Mathematica. The coefficient correlation of the generated sample is 0.0329.

Figure 2 

Cutaway view of the main structure of hydropower.

Figure 3 displays the transmissibility variances of the head-cover transfer path obtained using different methods. Method 1 does not involve the two types of correlations. The Monte-Carlo simulation is not used in method 1 (due to the inclusion of 26 random variables and because the calculation time is too long, this method only focuses on how each group of variables changes with time, that is, the number of calculations is equal to the sample size). Method 2 (following the method developed by W. J. Vetter in 1973) only involves the correlation with respect to one parameter, which involves the correlations between the disturbances of the parameters. Method 3 (similar to method 2) involves the correlations between parameters instead of the correlations between the disturbances of the parameters. Method 4 involves both types of correlations. The results for method 4 are obtained from (18). Figure 3 displays the characteristic curve of the head-cover system transmissibility variances in the frequency band of 0–1000 rad/s. Figure 4 presents the optimal curve of the two types of correlations for the transmissibility variance range as the correlation coefficient changes. This figure contains two curves: the disturbance curve, which describes the mean values of the results of method 2 subtracted from method 1 and the results of method 4 subtracted from method 3, and the parameter curve, which is similar to the disturbance curve.

Figure 3 

Head-cover transmissibility variances in the frequency domain as predicted by the four methods.

Figure 4 

Optimization range of the correlation coefficient characteristic curve.

Figures 3 and 4 show the following results: (1) compared to the other methods, the variance range for the method that contains two types of correlations is the smallest. This finding indicates that this method significantly decreases the disturbance range under theoretical calculations. The path transmissibility variance decreases by 72.51%. (2) In optimizing the method considering correlations between the disturbances and variances, the range merely decreases by 0.644%. This small decrease may have arisen because the difference between the additive and multiplicative disturbances is large, and the optimizing function is weak. (3) The optimization effect is significant for the method considering the correlations between parameters. The variance range decreases by 75.71% when the correlation coefficient is 0.0085. Thus, neglecting this type of correlation leads to an inaccurate analysis. (4) As the correlation coefficient increases, the optimization of the transmissibility variance is more efficient, and the effect on the disturbance range is more significant. This result demonstrates that the relationship between the optimization range and the correlation coefficients for the parameters is linear and that the relationship between the optimization range and the correlation coefficients for the disturbances is proportional. However, the latter relationship does exhibit some fluctuations.

Figure 5 compares the characteristic curve for each path transmissibility, 𝛽_𝑖, to the excitation frequency. This curve shows the ratio between the forces that are transmitted to the machine pier by the corresponding paths and the excitation force of the water turbine hydraulic source. This ratio is equivalent to the amplification coefficient. The curve indicates two results: (1) resonance occurs when the frequency of the excitation force is the same as the natural frequency. The transmissibility reaches its maximum value for each path at this frequency. This result indicates that the path has a large effect on the structural vibration at this frequency. (2) Considering the whole process of vertical vibration transfer, the transmissibility of the head cover is significantly smaller than that of the shafting system. Throughout the entire frequency range, the mean value of the path transmissibility ratio is 32.37. This value indicates that the vibration effect of the head-cover system can be neglected in accurate calculations.

Figure 5 

Path transmissibility of the excitation frequency curve.

Figure 6 shows that the characteristic curve of the transmissibility variance varies with the excitation frequency for the two types of correlations. This curve reflects the diversion of the transmissibility and indicates the following findings. (1) The path transmissibility variance of the shaft system is greater than that of the head-cover system due to the large number of components and the randomness of these components’ parameters. Thus, the variance of a structure with a high level of parameter randomness is large in the transfer path. (2) The variance maximum occurs at the natural frequency. This frequency is determined by the structural parameter, which accounts for the parameter contribution of every transfer path, enabling the transfer force to be obtained from the expectation and variance of the transmissibility.

Figure 6 

Variance of the path transmissibility of the excitation frequency characteristic curve.

Figure 7 shows that the characteristic curve of the transfer coefficients varies with the excitation frequency for the two types of correlations. This curve reflects the transfer efficiency of the excitation force. The figure shows that the transfer efficiency of the head-cover system is more efficient than that of the shafting system. Therefore, more attention should be given to the assigned vibration proportion of the shaft and head-cover system in a design.

Figure 7 

Transfer coefficient of the paths of the excitation frequency characteristic curve.

5. Conclusions

(1)The multivibration source, multipath vibration model, is improved with respect to the coupling effect between hydropower station units and powerhouses by the introduction of a head-cover system. The scalar expression of each vibration transfer path is provided in this study.(2)A perturbation analysis of the transfer path is carried out by using a test signal. A complex situation with two types of correlations is considered. Kronecker algebra, the Hadamard product, and probabilistic statistics are employed to develop an analysis method for the transfer path. In the solution process, excessive additional computations are not required because only the numerical characteristics of the random variables are used. The application of the stochastic perturbation method is expanded, and a method for analyzing the vibration transfer path is developed. The analysis method efficiently decreases the disturbance range of the path’s contribution. The disturbance ranges of the path transmissibility and the contribution rate are efficiently reduced by considering the two types of correlations. This reduction is significant for the optimization function with respect to the results.(3)The simulation results indicate that when the effect of the spiral case and substructure is ignored, the influence on the disturbance range continually increases as the correlation coefficient increases in the hydropower station model. The optimization function of the correlations between parameters is too important to be neglected when calculating the disturbance range.(4)In conclusion, the analysis of the proposed model shows that the effect of the water turbine head cover is not evident in the vertical vibration transfer, but its transfer efficiency is significant.

The analysis of vibration transfer paths of hydropower station units and powerhouses is complicated. Based on the complex disturbances and parameters, the contribution of each transfer path can be calculated by analyzing the sensitivity of the transfer force and the transmissibility of the vibration model in the frequency domain. This study provides a reference for future comprehensive research on the transfer paths of hydropower station units and powerhouses.

Acknowledgment

This research was sponsored by the National Natural Science Foundation of China (51079020).