Mathematical Problems in Engineering

Volume 2016, Article ID 4280704, 15 pages

http://dx.doi.org/10.1155/2016/4280704

## Application of Blind Source Separation Algorithms and Ambient Vibration Testing to the Health Monitoring of Concrete Dams

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

Received 13 August 2016; Revised 5 October 2016; Accepted 20 November 2016

Academic Editor: Yuri Vladimirovich Mikhlin

Copyright © 2016 Lin Cheng and Fei Tong. 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

In this work, using AVT data, a health monitoring method for concrete dams based on two different blind source separation (BSS) methods, that is, second-order blind identification (SOBI) and independent component analysis (ICA), is proposed. A modal identification procedure, which integrates the SOBI algorithm and modal contribution, is first adopted to extract structural modal features using AVT data. The method to calculate the modal contribution index for SOBI-based modal identification methods is studied, and the calculated modal contribution index is used to determine the system order. The selected modes are then used to calculate modal features and are analysed using ICA to extract some independent components (ICs). The square prediction error (SPE) index and its control limits are then calculated to construct a control chart for the structural dynamic features. For new AVT data of a dam in an unknown health state, the newly calculated SPE is compared with the control limits to judge whether the dam is normal. With the simulated AVT data of the numerical model for a concrete gravity dam and the measured AVT data of a practical engineering project, the performance of the dam health monitoring method proposed in this paper is validated.

#### 1. Introduction

Dam health monitoring [1] is a process based on the structural static/dynamic response and environmental variables measured by sensors installed on a dam body, foundation, and reservoir to evaluate structural health. It is an effective way to prevent the occurrence of catastrophic dam failure that would result in great loss of human life and property. Dam health monitoring is a special case of structural health monitoring (SHM) [2]. Among all SHM methods that have been proposed, the vibration-based SHM method [3] has obvious advantages. In conventional static dam health monitoring methods, the commonly measured dam responses are displacement, seepage flow, temperature, and so on, which can indicate only the local damage of a structure where the instruments are installed, and it is usually very difficult to evaluate the global state of a structure. Unlike static dam response quantities, many structural dynamic features are global, for example, changes in the natural frequencies of the structure. The advantage of global features is that damage can be detected remotely from a sensor [4]. Other dynamic features, such as modal shape vectors, can also provide information about the damage location. Thus, using the vibration-based method to monitor the health of large concrete dams is feasible and meaningful. Moreover, with the construction of some vibration monitoring systems, such as the dam strong earthquake monitoring system, the ambient vibration testing (AVT) [5–8] data of concrete dams under certain ambient excitation can be acquired to diagnose structural health conveniently. Therefore, research on dam health monitoring methods for concrete dams based on ambient AVT at full scale has received much attention in the last few years.

To diagnose dam health using AVT data, two key steps, that is, feature extraction and feature classification, are necessary. Feature extraction is the process of identifying damage-sensitive properties, derived from the measured dynamic response, which allows one to distinguish between undamaged and damaged structures. The area of SHM that receives the most attention in the technical literature is feature extraction [9]. Although many different types of features have been reported to be used as structural damage indictors, modal features, for example, natural frequencies, modal shape vectors, and coordinate modal assurance criteria (COMAC) [10], are still the most commonly used features in SHM. The advantage of using modal features is that they are independent of external ambient excitation and reflect the structural state directly. Using AVT data to identify the modal parameters of concrete dams is the operation modal analysis (OMA) [11] problem of structures. Recently, one area of research focused on OMA methods is the use of the blind source separation (BSS) technique to implement output-only modal identification. Among all proposed BSS-based OMA methods, the SOBI-based method, which has been studied in our previous work [12], shows many advantages, such as high identification accuracy, high computation efficiency, and strong robustness. The main difficulty in applying system identification for OMA of large-scale structures is the selection of the model order and the corresponding system poles. The most commonly used method is the stabilization diagram method. A conventional stabilization diagram using identified natural frequencies, damping ratios, and modal assurance criterion (MAC) is used as an index to select stable modes. Recently, a new index, modal contribution [13], has been proposed and shows better performance in discriminating structural models from mathematical poles. Whereas Cara’s work [13] shows us how to calculate the modal contribution index for the stochastic subspace identification (SSI) method [14], the procedure to calculate the modal contribution index for the SOBI-based method has not been studied.

Concrete dams are large-scale engineering structures with complicated operation environments and are characterized by the structure-water interaction effect. Environmental variables, such as temperature, water level, and rainfall, cannot be ignored directly in practical engineering as in numerical and experimental investigations. Therefore, for dam health monitoring based on AVT, how to remove the variability due to environmental changes is an important problem. To monitor the health of a structure under a varying environment, a commonly used method that seeks to remove the variability due to environment without measuring the environmental variables is the latent variable-based method. For example, Sohn et al. [15], Yan et al. [16], Deraemaeker et al. [17], and Ni et al. [18] have used the principle component analysis (PCA) method for bridge structure health monitoring. In our previous work [19], the kernel component analysis (KPCA) method is adopted to monitor the health state of concrete dams. Recently, some literature has reported on the good performance of independent component analysis (ICA) [20–22], which is a typical BSS algorithm and can be deemed an extension of PCA, in state monitoring of complex systems. PCA can impose independence only up to second-order statistical information (mean and variance), whereas ICA involves higher-order statistics; that is, it not only decorrelates the data (second-order statistics) but also reduces higher-order statistical dependencies. Therefore, an ICA-based SHM method can give more sophisticated results than a PCA-based one because ICA can extract the essential independent components that drive a process.

In this work, we first present a health monitoring method for concrete dams based on AVT data. Using the measured AVT data, the SOBI-based modal identification method is adopted to extract the basic modal parameters of a dam. For the SOBI-based modal identification method, the method to calculate the modal contribution index is proposed for the first time. The modal contribution index is then introduced into the stabilization diagram method to remove spurious modes of ambient excitation. The identified or calculated modal features are then analysed using ICA, another BSS method, to extract some ICs, which may represent the effect of environmental variables. The square prediction error (SPE) control charts and the contribution plots are then used to detect structural abnormalities and the locations of such abnormalities. At the end of this work, a numerical example and an engineering example are used to verify the proposed dam health monitoring method.

#### 2. Modal Identification of Concrete Dams Based on SOBI and Modal Contribution

##### 2.1. SOBI-Based Modal Identification

Based on McNeill’s derivation [23], an analytic expression of the structural acceleration response can be constructed using Hilbert transformation (HT). After considering observation noise , the expression of an analytic signal is as follows:where is the physical acceleration of channels and is its HT. The HT is performed to identify imaginary parts of modal shape vectors. is the acceleration response selection matrix; and are the real and imaginary parts of complex mode shapes, respectively. and are the real and imaginary parts of the modal response.

If all structural modal responses are independent of each other, (1) shows a BSS problem indeed. The structural vibration response under ambient excitation can be regarded as a linear mixture of each modal response. Each modal response can then be regarded as a virtual source signal [24], and the mixing matrix is . If the BSS problem can be solved, the modal parameters can be extracted from the estimated mixing matrix and modal response . To solve the BSS problem shown in (1), the SOBI algorithm [25, 26] is adopted and a modal identification is proposed in our previous work.

After the discrete sampling of the physical acceleration of channels at identical time intervals, the SOBI-based modal identification method forms a vector , which consists of the measured seismic response of channels and its time-lagged data.where , is the measured physical acceleration of channels at time interval* k*, is its Hilbert transformation, and . If , then parameter , and no time-lagged data are needed; otherwise, the time delay should satisfy . is the system order, which is equal to the number of structural active modal orders multiplied by 2.

The Hankel matrix with time delay can then be defined asIf , and the Hankel matrix becomes the covariance function matrix . is the expectation operator. For a structure under white-noise-like support excitation, the covariance function matrix of the measured acceleration is related only to the initial state and system parameters (modal parameters) of the structure, which is similar to that of the structural free vibration response and the impulse response when time is big enough.

By further deduction, the Hankel matrix can be expressed as where , , and are the eigenvalues of the system matrix . ; is the output selection matrix of acceleration. Superscript represents Hermitian (complex conjugate and transpose).

In order to realize the diagonalization of the Hankel matrix , a robust algorithm called the joint approximate diagonalization (JAD) [27] technique is adopted. Instead of diagonalizing only one Hankel matrix, a group of Hankel matrices with different time delays are diagonalized simultaneously in the JAD technique. For the JAD technique, the corresponding optimization problem is expressed as follows: The whitening matrix is obtained using the PCA method.

After solving the optimization problem shown above, the generalized diagonalization matrix can be obtained. The mixing matrix , separation matrix , and source signal, that is, the modal response , can then be expressed as The submatrix of contains the real parts and the imaginary parts of the observed complex mode shape. Using the identified real modal response and the modal identification method of the single-degree-of-freedom (SDOF) system, the natural frequencies () and damping ratios () can be obtained. Some details of the above modal identification method can be found in our previous work.

##### 2.2. Modal Contribution and System Order Determination

In the modal identification process presented above, activated system order* n* is an important parameter. However, it is usually unknown previously. Moreover, practical ambient excitation may have some dominant frequency components and may not be like white noise support excitation at all. Therefore, it is of great importance to determine the system order and discard the dominant frequencies of ambient excitation for the OMA of concrete dams. Recently, Cara et al. proposed a new index, that is, modal contribution, to select the proper order for the state space model in the SSI method. Here, we extend this method to the SOBI-based modal identification procedure and propose an automated operation modal analysis procedure for concrete dams.

Let be the measured physical acceleration response of a dam at different times; can then be expressed as where is the acceleration due to identified modes, is the acceleration response due to the th mode, and is the error term of the measured acceleration.

According to (1) and (2), the observed acceleration response of a structure can be expressed by where is the acceleration due to the th mode; and are the th columns of matrices and , respectively; and and are the th real and imaginary modal response coordinates, respectively, at the th time interval.

For the SOBI-based modal identification method mentioned in Section 2.1, , , , and can be calculated using (6). The acceleration due to each mode can then be calculated as shown in (8).

The covariance matrix can be calculated as The operator means that is a matrix consisting of the diagonal elements of matrix and zeros elsewhere. Normalizing the above equation,

For the diagonal elements of the diagonal matrices, the following expression can be obtained:where and are the diagonal elements of the diagonal matrices and , respectively.

The two metrics and can then be calculated using the diagonal elements of the above two matrices as

As noted by Cara et al., indicates the importance of the th identified mode; the modal contributions of these spurious modes and weak modes are almost zero. In the vector , the greater the number of nonzero elements is, the more the activated modes are included in the vibration response and the more abundant the information therein is.

##### 2.3. The Flowchart of Modal Identification

After extracting the modal parameters, the MAC and COMAC could be calculated using the following equations: where is the th modal shape vector of an undamaged structure; is the th modal shape vector of a structure with an unknown state; and and are the th components of modal shape vectors and , respectively.

Many researchers have proved that MAC and COMAC are more sensitive to structural damage than natural frequencies and modal shape vectors. In addition, COMAC can also provide damage location information of a structure.

According to the SOBI-based modal identification methods, stabilization diagram, and proposed modal contribution index, an OMA procedure for concrete dams using AVT data is proposed. The flowchart of the integrated OMA method is shown in Figure 1.