Shock and Vibration

Volume 2017, Article ID 2046345, 11 pages

https://doi.org/10.1155/2017/2046345

## Computation of Rayleigh Damping Coefficients for the Seismic Analysis of a Hydro-Powerhouse

State Key Laboratory Base of Eco-Hydraulic Engineering in Arid Area, Xi’an University of Technology, Xi’an 710048, China

Correspondence should be addressed to Zhiqiang Song; moc.621@4002qihzs

Received 12 April 2017; Accepted 27 July 2017; Published 30 August 2017

Academic Editor: Xing Ma

Copyright © 2017 Zhiqiang Song and Chenhui Su. 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

The mass and stiffness of the upper and lower structures of a powerhouse are different. As such, the first two vibration modes mostly indicate the dynamic characteristics of the upper structure, and the precise seismic response of a powerhouse is difficult to obtain on the basis of Rayleigh damping coefficients acquired using the fundamental frequencies of this structure. The damping ratio of each mode is relatively accurate when the least square method is used, but the accuracy of the damping ratios that contribute substantially to seismic responses is hardly ensured. The error of dynamic responses may even be amplified. In this study, modes that greatly influence these responses are found on the basis of mode participation mass, and Rayleigh damping coefficients are obtained. Seismic response distortion attributed to large differences in Rayleigh damping coefficients because of improper modal selection is avoided by using the proposed method, which is also simpler and more accurate than the least square method. Numerical experiments show that the damping matrix determined by using the Rayleigh damping coefficients identified by our method is closer to the actual value and the seismic response of the powerhouse is more reasonable than that revealed through the least square method.

#### 1. Introduction

With the exploitation of water resources in southwest China, large-scale hydropower stations have been established in regions frequently hit by earthquakes. The antiseismic safety of powerhouses is essential for water turbine generators and operators in powerhouses. The seismic responses of powerhouse structures are often analyzed with time history method. For example, structural dynamic time history analysis includes modal superposition and step-by-step integration methods. Multi-degree-of-freedom vibration is decoupled in multiple single-degree-of-freedom vibrations in modal superposition method. Single-degree-of-freedom systems can be subjected to a precise dynamic response analysis through Duhamel’s integration. With this method, input damping is accurate and high-speed calculation is achieved. However, modal superposition time history analysis method is only applicable to linear elastic structure systems because a structure possesses different decomposition modes at various times due to nonlinear structural material, nonlinear contact state, and other parameters. Therefore, a step-by-step integration dynamic time history analysis method can be used for a nonlinear system, which requires damping matrix establishment.

Unlike the formation of stiffness and mass matrixes, the formation mechanism of damping is complicated; that is, a damping matrix can be calculated by using a construction method but cannot be directly determined by identifying the material, size, and characteristics of structures [1]. Consequently, different damping matrix construction theories have been proposed [2–9]. For instance, a Rayleigh damping model is widely used because of its excellent advantages [10–13]. (1) In this model, the damping matrix of a structure is a linear combination of mass and stiffness matrixes. As such, Rayleigh damping models can provide a clear physical meaning and present a convenient expression. Thus, these models can be easily applied. (2) A Rayleigh damping matrix must be orthogonal to mode shapes. Consequently, decoupling the dynamic equations of multiple degree-of-freedom systems via mode superposition become convenient. Mode damping ratios can be directly used in single degree-of-freedom systems (generated by decoupling) dynamic response calculation. Therefore, damping input shows an enhanced accuracy and a reduced calculation scale. (3) Rayleigh damping coefficients can be determined by the orthogonality of a damping matrix for a modal shape. (4) With appropriate Rayleigh damping coefficients, results of a dynamic response analysis of a multi-degree-of-freedom system are the same as experimental data. A damping model is also embedded in commercial finite element software, and Rayleigh damping models are considered as a basis for damping matrix construction commonly utilized in the seismic time history analysis of hydraulic structures.

The damping matrix of a structure is the linear combination of the mass and stiffness matrixes of a Rayleigh damping model:where and , respectively, represent the mass and stiffness proportional damping coefficients, which are collectively known as Rayleigh damping coefficients. , , and are the mass, damping, and stiffness matrixes, respectively. In traditional methods, two reference vibration modes (- and -order) are selected, and their damping ratios and obtained through measurement or reliable test data estimation and their frequencies and are used to calculate and :This equation can be simplified as follows when = = :

Two orders of the reference frequency can be easily and appropriately selected to determine Rayleigh damping coefficients when the degree of freedom of a structure is low or the dynamic response of this structure is controlled by some low-order modes. For example, the first two orders are generally obtained as reference frequencies in traditional methods. For complex structures and structures with a number of modes that contribute greatly to dynamic responses, difficulties in selecting two orders of reference frequencies to obtain reasonable Rayleigh damping coefficients and are encountered. If damping coefficients are chosen inappropriately, a slight difference in damping may seriously distort the calculation of the seismic response of a given structure [14–17].

Many scholars investigated the calculation of Rayleigh damping coefficients. Pan et al. [18] proposed a constrained optimization method to determine Rayleigh damping coefficients for the accurate analysis of complex structures. An objective function is defined as a complete quadratic combination of the modal errors of a peak base reaction evaluated through response spectral analysis. An optimization constraint is enforced to determine the exact damping ratio of modes that contribute greatly to dynamic responses. This method is based on Duhamel’s integral formula, which is suitable for linear elastic systems. Yang et al. [19] studied the application of a multi-mode-based computation method in single-layer cylindrical latticed shells because the traditional two-mode Rayleigh damping method is unsuitable. Yang et al. [19] also suggested that the multi-mode-based computation method is preferable when many dominant modes are distributed loosely and found in a wide range of frequencies under some ground motions. Jehel et al. [20] comprehensively compared the initial structural stiffness and updated tangent stiffness of Rayleigh damping models to allow a practitioner to objectively choose the type of Rayleigh damping models that satisfy his needs and be provided with useful analytical tools for the design of these models with good control on their damping ratios during inelastic analysis. Erduran [21] evaluated the effects of a Rayleigh damping model based on the engineering demand parameters of two steel moment-resisting frame buildings. Rayleigh damping models, which combine mass and stiffness proportional components, are anchored at reduced modal frequencies, which create reasonable damping forces and floor acceleration demands for both buildings but do not suppress higher-mode effects. Zhe et al. [22] developed an improved method to calculate Rayleigh damping coefficients for the seismic response time history analysis of powerhouse structures by considering the spectrum characteristics of the ground motion and the frequency characteristics of these structures. Using the improved method, Zhe et al. [22] obtained calculation results that are consistent with experimental findings.

Hongshi [23] initially compared and analyzed several methods of calculating Rayleigh damping coefficients and subsequently proposed the least square method to minimize the difference between the calculated damping ratio and the actual damping ratio within the cutoff frequency. Li et al. [24] then established the corresponding method for the seismic response analysis of powerhouse structures by calculating the proportional damping coefficient through the weighted least square method.

Analyzing the seismic dynamic response of long-span arch bridges and super high structures with long periods, Lou [15, 25] found that Rayleigh damping coefficients determined by traditional methods involving the first two orders of frequencies as reference frequencies inaccurately reflect the actual damping effect of long-period structures in a dynamic process and even create a large deviation. Hence, calculation methods of the Rayleigh damping coefficients of long-period structures in dynamic processes should be further discussed.

The definition of long- and short-period structures is a relative concept depending on the relationship between the basic period of a structure and the characteristic period of dynamic loads. A short-period structure is characterized as a structure whose basic period is less than or close to the characteristic period of external loads. Otherwise, a given structure is called a long-period structure. The basic characteristic period is usually 10^{−1} s or higher because the stiffness of the upper frame structure is relatively weak. Therefore, this weak structure is also described as a long-period structure because of the characteristic period of seismic waves. The mass and stiffness of the upper and lower structures of powerhouses are different. As such, the first two modes of vibration often indicate the dynamic characteristics of the upper structure. The damping matrix obtained by traditional methods of calculating Rayleigh damping coefficients does not easily reveal the actual damping of the whole powerhouse structure. Hence, the influence of the calculation methods of Rayleigh damping coefficients on the dynamic response of a powerhouse under seismic actions should be investigated. In this study, the calculation of Rayleigh damping coefficients is examined to analyze the seismic responses of a powerhouse.

To calculate Rayleigh damping coefficients, Chopra [1] suggested that “in dealing with practical problems, it is reasonable to select the modes of vibrations and with specific damping ratios to ensure that the damping ratios of all modes of vibration that contribute greatly to the dynamic response are reasonable.” Differences in the mass and stiffness of the upper and lower structures of a powerhouse remarkably create the dynamic characteristics of a powerhouse structure. The first two vibration modes often involve the relatively soft upper structure of the powerhouse, whose mode participation mass is quite smaller than that of the whole powerhouse. The damping of these buildings under seismic actions is mainly due to various interior frictions and deformations of components and the ones between them. Therefore, mode participation mass should be considered as a key factor affecting the calculation of damping. The modes that contribute greatly to dynamic responses are found on the basis of mode participation mass. In this study, the Rayleigh damping coefficient is calculated.

This paper is organized as follows. Section 2 introduces the finite element models of a powerhouse and several methods of calculating Rayleigh damping coefficients. The two modes that remarkably affect the dynamic responses are determined on the basis of mode participation mass, and the Rayleigh damping coefficients are calculated. Section 3 presents the results of the dynamic response obtained by different methods. Numerical results show that the proposed method accurately reveals the two modes contributing to the dynamic response of the powerhouse, and the calculated Rayleigh damping coefficients are consistent with actual results. The calculated results are also closer to the exact solutions and even higher than those acquired by the least square method. Section 4 provides the conclusion.

#### 2. Seismic Analysis Model of Powerhouses and Calculation Method of Rayleigh Damping Coefficients

##### 2.1. Three-Dimensional Finite Element Model and Seismic Inputs

In Figure 1, the three-dimensional finite element model of a typical unit of the hydropower station is established. The depth of the foundation is about twice the height of the powerhouse, and the upstream and downstream and the left and right sides extend twice the height of the powerhouse because of the elastic coupling effect of the foundation. The foundation boundary condition is that the bottom is fixed and the four boundaries are normally constrained. Turbine generators, cranes, roof loads, and hydrodynamic pressures are simulated as the additional mass at corresponding locations. A linear elastic model is used in the structure, and the local strengthening effects of the volute steel plate and the seating ring are neglected. The damage cracks of the concrete around the volute are also disregarded. Zhang et al. [26] observed that cracks on thin parts of the concrete of a volute greatly change the local stiffness of the volute but slightly alter the displacement, speed, and acceleration of the upper structure compared with those without cracks. The nonmass foundation, which does not affect the response peaks of the structure, is adopted for calculation convenience.