Shock and Vibration

Volume 2018, Article ID 4892428, 13 pages

https://doi.org/10.1155/2018/4892428

## Structural Damage Identification Based on the Transmissibility Function and Support Vector Machine

^{1}School of Civil Engineering, Qingdao University of Technology, Qingdao, 266033, China^{2}Collaborative Innovation Center of Engineering Construction and Safety in Shandong Blue Economic Zone, Qingdao 266033, China

Correspondence should be addressed to Yansong Diao; moc.361@syoaid

Received 15 March 2018; Revised 23 April 2018; Accepted 8 May 2018; Published 11 June 2018

Academic Editor: Radoslaw Zimroz

Copyright © 2018 Yansong Diao et al. 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

A novel damage identification method based on transmissibility function and support vector machine is proposed and outlined in this paper. Basically, the transmissibility function is calculated with the acceleration responses from damaged structure. Then two damage features, namely, wavelet packet energy vector and the low order principal components, are constructed by analyzing the amplitude of the transmissibility function with wavelet packet decomposition and principal component analysis separately. Finally, the classification algorithm and regression algorithm of support vector machine are employed to identify the damage location and damage severity respectively. The numerical simulation and shaking table model test of an offshore platform under white noise excitation are conducted to verify the proposed damage identification method. The results show that the proposed method does not need the information of excitation and the data from undamaged structure, needs only small size samples, and has certain antinoise ability. The detection accuracy of the proposed method with damage feature constructed by principal component analysis is superior to that constructed by wavelet packet decomposition.

#### 1. Introduction

Vibration-based damage detection and structural health monitoring (SHM) have been developed in recent years [1, 2]. The goal of SHM is to detect damage that may exist before it becomes critical for the structure’s integrity.

Among vibration-based damage detection approaches, transmissibility function (TF) analysis attracts extensive interest because TF defined as the ratio between two outputs is independent of excitation [3]. Up to now, methods based on TF have been developed to detect damage or assess the structural damage severity. Steenackers et al. [4] used the TFs instead of frequency response functions (FRFs) in model updating. The authors concluded that the finite element model updated with TF was equivalent to the model updated with FRFs or operational modes. Maia et al. [5] proposed a method to detect and quantify structural damage of a beam, using response TFs measured along the structure. Some numerical simulations were presented and a comparison was made with results using FRFs. Li et al. [6] proposed a damage detection approach based on TF to identify the damage of shear connectors in slab-on-girder bridge structures with or without reference data from the undamaged structure. Kong et al. [7] derived the TF of a vehicle-bridge coupled system consisting of a simply supported beam and a single-degree-of-freedom vehicle, and a numerical study was conducted to investigate the feasibility of detecting bridge damage using TF. Zhou et al. [8] defined a sensitive damage indicator based on TF coherence, which was used to detect and identify damage severity. The proposed approach was validated on data from a physics-based numerical model as well as experimental data from a three-story aluminum frame structure. Chesné and Deraemaeker [9] reviewed the state-of-the-art using TF for damage detection and localization and discussed the extension of these results to more general dispersive systems such as beams or plates. Zhu et al. [3] proposed a decentralized damage detection approach using TF, which was ideal for using limited number of sensors upon a large-scale structure. Zhou et al. [10] proposed a new approach for detecting structural damage using TF together with hierarchical clustering and similarity analysis. Chen et al. [11] tried to predict damage with neural networks (NNs) trained by TF; the test results showed that the NN classifiers clearly delivered the diagnostic indications of the faults introduced into the structural systems. In [12], a TF based damage detection method using artificial intelligence was proposed for detecting the structural damage and predicting its severity.

Broadly, the damage detection methodologies can be mainly classified into two categories: model-driven approaches [13, 14] or data-driven approaches [15, 16]. Model-driven approaches concentrate on the understanding of the structure from its numerical model, where finite element analysis is often employed. In contrast, data-driven approaches construct a model by learning from measured data and then make a comparison between the model and measured responses to detect damage.

The machine learning techniques have been raising increasing attentions among the data-driven approaches. There are a variety of machine learning techniques, including Bayesian networks [17], artificial NNs [18], and support vector machines (SVMs) [19].

SVM is a supervised learning technique with strong theoretical foundations based on the Vapnik-Chervonenkis theory [20]. This method promises to overcome the shortcomings of conditional NNs such as the local minimization and inadequate statistical capabilities. Furthermore, SVM can unify different types of discriminant functions such as linear, polynomial, and radial basic functions in the same framework [21]. SVM is especially suited in the case of small size samples [22]. SVM has been widely accepted as an effective tool for characteristic extraction and damage detection. Kim et al. [23] attempted to systematically integrate the wavelet transform, the autoregressive model, the damage-sensitive energy feature, and the SVM into a single SHM framework for damage detection of smart structures equipped with time-varying nonlinear hysteretic control devices; the damage was identified using the SVM based on the differences of AR parameters between the damaged and undamaged structures. Khatibinia et al. [24] employed SVM with a wavelet kernel for seismic reliability assessment of existing reinforced concrete structures with consideration of soil-structure interaction effects in accordance with performance-based design. Chong et al. [25] proposed a nonlinear multiclass SVM based SHM system for smart structures. Based on the wavelet filtered signals, several wavelet based autoregressive models were constructed. Damage-sensitive features were extracted from the wavelet based autoregressive coefficients and then the nonlinear multiclass SVM was employed to estimate the damage severity. Dushyanth et al. [26] developed a new multilevel SVM and stated that the proposed SVM method required much less training data as compared to other methods. Gui et al. [27] presented three optimization algorithm based SVMs for damage detection. The optimization algorithms, including grid search, partial swarm optimization, and genetic algorithm, were used to optimize the penalty parameters and Gaussian kernel function parameters; two types of feature extraction methods in terms of time series data were selected to capture effective damage characteristics.

Nevertheless, little research has been reported on damage identification using SVM combined with TF analysis.

For damage identification using SVM, the core issue is to seek some features from the structural responses that are sensitive to structural damage. Since features hold a fundamental role in results performance, one should pay special attention to the feature extraction. In this study, wavelet packet decomposition (WPD) and principal component analysis (PCA) are employed to extract damage features from TF; the aim is to compare the effect of the proposed method using damage features extracted from time-frequency domain and time domain.

WPD can be considered as an extension of the discrete wavelet transform, which decomposes not only the approximation but also the detail coefficients at each level of decomposition. WPD creates the same frequency bandwidths in every resolution. The wavelet packet component energies extracted from structural dynamic responses have been used as a damage feature to detect damage. Hou et al. [28] presented a brief background of wavelet analysis and its SHM applications. Sun and Chang [29] proposed a WPD-based damage assessment method of structures. Dynamic signals measured from a structure were first decomposed into wavelet packet components. Component energies were then calculated and used as inputs into NN models for damage assessment. Ding et al. [30] put forward a workable realization procedure for damage alarming of frame structures based on energy variations of structural dynamic responses decomposed by WPD. Ravanfar et al. [31] presented a two-step damage identification approach based on wavelet multiresolution analysis and genetic algorithm in beam structures. The location of the crack was identified in the first step by defining the damage index called relative wavelet packet entropy. Then, the damage severities at the identified locations were assessed in the second step using genetic algorithm.

TF has high dimension with high redundancy and correlation. This can slow the damage detection procedure and make the underlying patterns more complicated to classify. Also, computation on high dimensional data can be time-consuming and costly. PCA is a popular technique for dimensionality reduction, which accelerates the detection procedure and makes the underlying patterns more easy to classify; so far, some research works have been done which use PCA for dimensionality reduction in damage detection. Sohn et al. [32] used PCA with the statistical process control for damage detection. Also, Worden and Manson [33] applied linear and nonlinear PCA for reducing the data dimension. Then, Mahalanobis distance was utilized to find anomaly in the reduced dimensional data. In [18], PCA was employed to compress FRF data for damage detection.

In this study, a novel damage identification method based on TF and SVM is proposed and outlined. Basically, the TF is calculated with the acceleration responses from damaged structure. Then two damage features, namely, wavelet packet energy vector (WPEV) and the low order principal components (PCs), are constructed by analyzing the amplitudes of TF with WPD and PCA separately. Finally, each of the two damage features (WPEV, PC) is used as input for SVM; the classification algorithm and regression algorithm of SVM are employed to identify the damage location and damage severity, respectively. The numerical simulation and shaking table model test of an offshore platform under white noise excitation are conducted to verify the proposed damage identification method. The main contribution of this study is the introduction of SVM in the context of the development of transmissibility-based damage identification methods.

#### 2. Transmissibility Function

In this paper, TF will be used as primary data to perform the damage identification. A TF is the ratio of FRF at two spatial locations on a structure, and it can be estimated in a nonparametric preprocessing step.

By assuming a single force that is located in the input degree of freedom (DOF) , the TF is given by [34]where and are the Fourier transform of the responses measured at DOF and DOF* j*, respectively, and and are the elements of FRF at DOF and DOF , respectively.

From (1), it can be seen that when the structure is subjected to a single point excitation, the TF only depends on the location of the force, which is not dependent on either the nature of the force signal or its amplitude. The influence of the input signal (excitation force) is eliminated.

The TF is very sensitive to damage because it is a ratio of functions with many peaks and valleys. Since the dynamic responses of structure change when damage occurs, it is natural to expect that the TF will also change if damage occurs. A change in the TF represents a change in the structural properties, and it is inferred that the change of TF indicates the damage of structure.

#### 3. Damage Feature Based on WPD

The wavelet analysis is a multiresolution analysis method in the time and frequency domain of a signal, and it has the ability to describe the local characteristic of a signal. WPD can be considered as an extension of the discrete wavelet analysis, which does not require a priori knowledge of the frequency bands, but it filters the signal in adaptive way. Discrete wavelet analysis decomposes just the previous approximation coefficients through high and low pass filters, whereas WPD decomposes both the detail and approximation coefficients, which keeps the important information available in higher frequency components. More details can be found in the textbook by Mallat [35].

The original signal can be expressed as a summation of WPD components aswhere is time lag and is the WPD component signal that can be represented by a linear combination of wavelet packet functions.

To examine the structural health condition, it is essential to achieve an index that is sensitive to structural damage. Usually, the measured vibration signals are decomposed by WPD into component signals and then component energies are calculated. The wavelet packet component energy is a suitable tool for identifying and characterizing a specific phenomenon of signal in the time-frequency domain. It has been shown by Yen and Lin [36] that the energy stored in a specific frequency band at a certain level of WPD provides a greater potential for signal feature than the coefficients alone.

The wavelet packet component energy of a signal is defined as

The total signal energy can be expressed as the summation of wavelet packet component energies when the mother wavelet is orthogonal.

Then, the energy ratio of each wavelet coefficient can be written as

The values correspond to a ratio of the energy of a particular coefficient to the total energy.

The normalized wavelet packet component energy vector of a signal is defined as

When the structure is damaged at different location, the values of and WPEV become different; therefore, WPEV can be used as damage feature to identify damage of the structure in this study.

#### 4. Damage Feature Based on PCA

PCA is a multivariate analysis method that can be considered as a linear data compression technique, and it is widely used in the fields of image processing, flow visualization, and the reduced order models. Using an orthogonal projection, the original set of variables in an* N*-dimensional space is transformed into a new set of uncorrelated variables, the so-called principal components, in a* P*-dimensional space such that . A summary of the method will be given here for the sake of completeness [37].

At the assumption of stationary stochastic process, the amplitude of the TF is used as the analysis signal, and it is divided into groups, each with frequency points, forming a matrix . Hence the centralized matrix can be obtained by using where is the th column of matrix and is the mean value of .

The correlation matrix can be defined as

By definition, the principal components are the eigenvalues and associated eigenvectors of the correlation matrix where and are the th eigenvalue and eigenvectors, respectively. The principal component of the normalized matrix is given by where is the th principal component. Any two principal components are uncorrelated, and the variance of is . Because of , the first principal component, i.e., the highest eigenvalue and its associated eigenvector, represents the direction and amount of maximum variability in the original data. The next principal component, which is orthogonal to the first component, represents the next most significant contribution from the original data, and so on.

The contribution rate is given by

The greater the contribution rate value is, the stronger the ability of the corresponding principal component to synthesize the information of original data is.

The accumulative contribution rate is given by

In order to increase the reliability of the selected principal component, the accumulative contribution rate is more than 85%. The damage feature PC, composed of the first* m*-order principal components, could be formed as followed:

Generally, the first several principal components contain the most of the information; the entire variation characteristics of the original data can be reflected by analysis of the first several principal components. The damage identification can be implemented in lower dimensional space constituted by the first several principal components. Therefore, the difficulty of the analysis is reduced.

#### 5. Support Vector Machines

SVM is initially developed for classification and then it is successfully extended to the regression analysis by Vapnik [38]. By applying kernel function (e.g., linear, polynomial, or radial basis function) as the inner product of functions mapping the data to a high dimensional feature space, SVM can also be applied in a case of nonlinear classification and regression analysis. A brief introduction of SVM is presented here for completeness. Readers are referred to the tutorials on SVM [39, 40] for details.

##### 5.1. Classification Algorithm of SVM

The whole process of classification of data by SVM is illustrated in Figure 1. Consider a training data set as , where each denotes the input space of the sample and has a corresponding target value for , where corresponds to the size of the training data.