Data-driven damage identification based on measurements of the structural health monitoring (SHM) system is a hot issue. In this study, based on the intrinsic mode functions (IMFs) decomposed by the empirical mode decomposition (EMD) method and the trend term fitting residual of measured data, a structural damage identification method based on Mahalanobis distance cumulant (MDC) was proposed. The damage feature vector is composed of the squared MDC values and is calculated by the segmentation data set. It makes the changes of monitoring points caused by damage accumulate as “amplification effect,” so as to obtain more damage information. The calculation method of the damage feature vector and the damage identification procedure were given. A mass-spring system with four mass points and four springs was used to simulate the damage cases. The results showed that the damage feature vector MDC can effectively identify the occurrence and location of the damage. The dynamic measurements of a prestress concrete continuous box-girder bridge were used for decomposing into IMFs and the trend term by the EMD method and the recursive algorithm autoregressive-moving average with the exogenous inputs (RARMX) method, which were used for fitting the trend term and to obtain the fitting residual. By using the first

The structural health monitoring (SHM) system provides a wealth of measurements making the data-driven method effective and rapid for damage identification [

The first issue of damage identification via measurements of the SHM system is that we need to obtain the real structural responses to identify whether the damage happened or not. It is well known that in monitoring data, the variations induced by temperature, wind, and vehicle are always fused together so that the changes caused by damage are submerged which leads the damage-induced variation to drown out. Many studies have focused on the relationship between temperature and monitoring structural responses and how to eliminate the effects of noise and temperature influences [

Another key issue of damage identification using the data-driven method is to establish an applicable and effective damage feature index or vector to evaluate the occurrence and location of the damage. The Mahalanobis distance method has been widely used in structural damage identification due to its good applicability and sensitivity to damage. The advantages of Mahalanobis distance are as follows: (1) data normalization: it can make the scale of monitoring data independent, so that a variety of data can be integrated into the same scale; (2) it can eliminate the interference of related variables when the monitoring data are multidimensional and the correlation of each dimension variable is large. For the measurement of the SHM system, such as displacement and strain, there are significant correlations among them, and the Mahalanobis distance method can fuse these multisource dimensional data and obtain more damage information from these different types of data.

Luo et al. [

This study proposed a new structural damage identification method based on Mahalanobis distance. The methodology of Mahalanobis distance, empirical mode decomposition, and recursive algorithm autoRegresive-moving average with exogenous inputs was proposed. A damage feature vector sensitive to the damage based on Mahalanobis distance was established. The calculation method of the damage feature vector and the damage identification procedure were given. A numerical simulation with a four sensor system for verifying the damage identification availability was carried out. Finally, the strain measurements of a real bridge SHM system were used to identify whether the damage occurred. The results proved that the damage vector MDC has great potential for structural damage identification.

The Mahalanobis distance calculation method was proposed by an Indian statistician Mahalanobis P. C. in 1936 [_{i} are the elements of the cluster

Huang et al. [

The core of the EMD method is to decompose the signal into a set of intrinsic mode functions. No information about the original signal is required. The decomposition is completely adaptive, and IMFs are arranged from high frequency to low frequency. The specific decomposition method is as follows:

First, the maximum and minimum values of _{1}(

If _{1}(_{1}(_{1k}(_{1}(_{1}, which is the first-order IMF. For IMF, the signal frequency is considered as the “filter,” and _{1} is considered to contain the highest signal frequency component:

where _{1k} (_{1(k−1)} (_{1k}(

A new signal _{l} (_{1}(

_{l}(

_{2}(

_{3}(

_{n}(

_{i}(

_{n}(

From the EMD method, it can be seen that the signal can be decomposed into IMFs and a residual, which are arranged in the order of high frequency to low frequency. However, the damage usually affects some interval frequencies, causing the energy redistribution of each IMF, although the redistribution law is unknown [

The recursive algorithm autoregressive-moving average with the exogenous inputs (RARMX) method is a model training based on the Gaussian Newton method [

NNRARMX model structure.

The regression vector and the fitting vector of the RARMX model are expressed as_{a} is the number of past fitting value, _{b} is the number of fitting influencing factors, _{c} is the fitting error, and _{k} is the time delay equal to 1.

The RARMX neural network (neural network, NN) model is shown in Figure

A neural network model.

The NNRARMX method can provide an effective way for signal fitting, which makes the signal have great nonlinearity [

In this study, a damage feature vector was defined based on the segmentation summation of Mahalanobis distance square. The main purpose is that the segmentation of data can make damage-induced changes of each monitoring point accumulate as “amplification effect,” so as to obtain more damage information [

A cluster can be defined as _{i} and

The sensor number of the system is _{1} (_{2} (_{p} (

In each group of IMFs, the first

According to formula (_{k} and damage feature vector MDC are established using formula (

EMD is used to decompose the measured signal into a group of IMF. The first _{k} values and vector MDC can also be calculated.

By comparing the damage feature vector MDC, the damage location of the structure is determined.

If the measured signals have a residual decomposed by the EMD method, the NNRARMX method can be used for fitting the residual, and a new fitting residual can be obtained. Then, the new fitting residual is considered as a cluster to be used while performing Step

Figure _{1} = _{2} = _{3} = _{4} = 100 kg and _{1} = _{2} = _{3} = _{4} =5 × 10^{5} kN/m.

Mechanics model of the quality-spring system.

The Motion equation of this system is_{4} for 0.01 second. Four mass points are making freedom movement. Their acceleration time histories are recorded. The sampling frequency

The mass-spring numerical simulation system was established using Matlab Software Simulink Toolbox. Four acceleration signals of mass points can be computed. In order to simulate the actual condition affected by noise, 5% white noise was added to the measured signals. G. Rilling EMD toolbox [_{3} was reduced by 10% and 20%, respectively. The acceleration signal of each mass point can be decomposed into three clusters to compare the damage identification results. There are three cases as follows:

Original acceleration signal

First order of IMFs

First three orders of IMFs

Figures _{3} with 10% and 20%, respectively. It can be found that the acceleration time history curve has little change after damage. However, the vibration amplitude of the first three orders of IMFs has little changes after the damage. For example, the amplitude of the second-order IMF of _{1} has increased when the damage increased from 10% to 20%; moreover, the amplitude of the third-order IMF has changed slightly. When damage increased from 10% to 20%, the amplitude of the second-order IMF of _{3} changed, and the second-order IMF of _{4} decreased with the increase of damage. However, with these changes in IMFs, it is difficult to identify the locations of the damage.

Figure

Figure _{1} and _{2} did not change significantly. This is because that mass points _{1} and _{2} may be far away from the damage. The variation of MDC values of particle _{3} increased in a ladder shape. It can be clearly seen that the subsections changes with the increase in damage. In the undamaged state, the range of MDC values of _{4} is large. When the damage is accumulated to 20%, the variation of MDC values increased significantly.

Figure _{1} and _{2} changed slightly after damage. However, after 10% damage, the MDC values of _{3} present an overall upward trend. When the damage reaches 20%, the trend is significant. It can be clearly determined that the spring _{3} has been damaged. The reason for the variation of stiffness is that IMFs are redistributed when the damage occurred; in addition, the Mahalanobis distance cumulant method can consider the correlation variation among the elements in clusters, which makes the small variation accumulate and make the damage easier to identify.

_{1} acceleration signal and the first three orders of IMFs before and after damage: (a) undamaged, (b) 10% damage, and (c) 20% damage.

_{2} acceleration signal and the first three orders of IMFs before and after damage: (a) undamaged, (b) 10% damage, and (c) 20% damage.

_{3} acceleration signal and the first three orders of IMFs before and after damage: (a) undamaged, (b) 10% damage,and (c) 20% damage.

_{4} acceleration signal and the first three orders of IMFs before and after damage: (a) undamage, (b) 10% damage, and (c) 20% damage.

The variation of MDC vector for Case

The variation of MDC vector for Case

The variation of MDC vector for Case

A real bridge structural health monitoring system was used to provide measurement data for damage identification by the method mentioned above. This prestressed concrete continuous box-girder bridge is located in Heilongjiang, China (N47°14′40″, E131°58′11″). The total length of the bridge is 1170 m, which consists of six main spans of 150 m and two side spans of 85 m.

In order to evaluate the long-term static and dynamic performances of the bridge under the influences of traffic and ambient environment, a long-term SHM system was implemented on the bridge. The sensor location of the SHM system is shown in Figure

(a) Sensor distribution of the SHM system on Fu Sui bridge. (b) Sensor distribution on section B of Fu Sui bridge.

The SHM system elevation of Fu Sui bridge.

Generally, the residual of each strain sensor signal can be considered as a temperature trend term. The EMD method was used to extract the temperature trend-term signals of the strain sensors DS1 and DS2 at D-section on May 21. According to the NNRARMX method, the simulated trend term for temperature trend signals was fit as shown in Figure

Fitting results of the strain trend terms. (a) DS1 strain trend term and fitting results. (b) DS2 strain trend term and fitting results.

ACF values and 95% confidence interval of fitting error of DS1. (a) ACF values of error when _{a} = 3. (b) ACF values of error when _{a} = 6. (c) ACF values of error when _{a} = 9. (d) ACF values of error when _{a} = 12.

ACF values and 95% confidence interval of fitting error of DS2. (a) ACF values of error when _{a} = 3. (b) ACF values of error when _{a} = 6. (c) ACF values of error when _{a} = 9. (d) ACF values of error when _{a} = 11. (e) ACF values of error when _{a} = 13. (f) ACF values of error when _{a} = 15.

The self-correlation function values of the fitting error of the strain trend items calculated according to the ACF criteria are shown in Figures _{a} in the NNRARMX model is the number of parameters that have been fitted according to formula (_{a} from 1 to 12 were calculated. For DS2 strain trend term, the fitting error of autocorrelation function values of the model for _{a} from 1 to 15 was calculated. It can be seen that the error of autocorrelation fitting values of the model NNRARMX(12) for DS1 nearly is almost within 95% confidence interval, which means that there is no information loss and the model fitted well [

The strain data of sensor DS1 from May 16 to May 23 of the next year were selected as analysis clusters. Due to the construction around the bridge, the cable transmission was interrupted and the data from November 15 to January 8 of the next year were lost. The original strain measurements of sensor DS1 is shown in Figure

Strain monitoring results of DS1 from May to the next May.

Strain trend term of DS1.

DS1 signal after separating the trend term.

According to the NNRARMX method, the NNRARMX(25) model was used to fit the trend term. The ^{2} value was 0.96, indicating that the NNRARMX(25) model fitted well (Figures

Fitting results of the NNRARMX model (May to the next February).

Fitting residual results.

The samples without trend item and a fitting residual signal from May to February of the next year were selected as two reference clusters. The samples without trend item and a fitting residual signal from March to May were considered as two unknown clusters. The variations of MDC values were calculated by the reference clusters and the testing clusters, as shown in Figures

Variation of MDC vector of the signal after separating the trend term.

Variation of MDC values of fitting residuals.

The first seven orders of IMFs of DS1 monthly strain monitoring data in June and May of are shown in Figures

First seven orders of IMFs (June).

First seven orders of IMFs (next year May).

The first order, first three orders, first five orders, and first seven orders of the IMFs were selected from the whole IMFs of DS1 strain of June and May of the next year as four reference and testing clusters, respectively. The damage feature vector of each reference cluster and each testing cluster were calculated, respectively. The selected cluster from IMFs was used to determine whether the structural damage is identified more easily by some high-frequency parts or some low-frequency parts. As can be seen from Figure

Variation of MDC vector of different IMFs in the next May.

The strain trend terms of DS1 monitoring data of June and May of the next year were fit by the NNRARMX method, respectively. The fitting residual is shown in Figure

Fitting residual signal.

Variation of MDC vector of fitting residuals.

In this study, a structural damage identification method based on Mahalanobis distance cumulant with IMFs and the fitting residual was proposed. The damage feature vector MDC calculation method and the damage identification procedure were given. Through the numerical simulation of the mass-spring system and the analysis of real bridge monitoring data, the sensitivity of the damage feature vector and the effectiveness of the method were verified. Additionally, the SHM system usually needs to monitor a variety of data, such as displacement, strain, and acceleration. This method can provide a potential way for different types of monitoring data fusion for damage identification.

Before the damage identification, a series of solving processes are needed to make the data more valid and obtain more damage information. In this study, EMD and NNRARMX methods were applied for data preprocessing. The EMD method is used to decompose the original monitoring data into IMF and trend term, and the NNRARMX method is used to obtain the trend fitting residual. In this way, original monitoring data can work as a global judgment to roughly judge whether the damage has occurred; however, these data cannot analyze the exact location of the damage. IMFs of monitoring data contain the damage-induced structural response variation which is caused by the redistribution of the frequency energy, and the fitting residual of the trend term can help to find the long-term damage-induced structural changes. All of these can provide more damage information for microdamage identification and low signal-to-noise data for damage identification.

The data used to support this study are available from the first author via e-mail (

The authors declare that there are no conflicts of interest.

This research was funded by the National Natural Science Foundation of China, under grant number 51908497.