Vibration-Based Damage Features from Parametric

Methods

1.3.4.1

Modal Parameters and Model Coefficients

The stochastic subspace identification (SSI) technique was introduced in 1996 by Van Overschee and De Moor [50]. It is a time domain method that uses output-only mea- surements and describes a dynamic system as a state-space model. The approach is based on the assumption of white-noise excitation and linearity. SSI can be either covariance-driven or data-driven. Covariance-driven SSI makes use of the output cor- relations for building the Hankel matrix. The factorization property might then be used to obtain the controllability and observability matrices and to extract the modal properties. In the data-driven approach, the weighted Hankel matrix is built directly from measured signals. Then, the factorization property might be used to obtain the observability matrix and the Kalman filter state sequence. One of the advantages of the SSI technique is that many model orders can be derived by a single block matrix, resulting in a low computation time. In [51], both SSI schemes are used for the op- erational modal analysis of an aerospace vehicle. An application of SSI in footbridge monitoring can be found in [52]. In [17], the triangulation-based extraction of modal parameters (TEMP) is proposed for reducing the solutions obtained from data-driven SSI. The combination of data-driven SSI is verified on tower data from a wind turbine of the alpha ventus offshore wind farm. This combination is also employed in [18] for identifying the modal parameters of a 34 m rotor blade.

In signal processing, autoregressive (AR) models use time series of structural re- sponses to build a model described by model coefficients. This model is used to predict the present time step based on past values of the signal and subsequently to identify the model coefficients of the system. Precisely, the present output is expressed as a linear combination of past outputs. Many variations of AR models are used for sys- tem identification, such as vector autoregressive models (VAR), moving average (MA) models, autoregressive moving average (ARMA) models and autoregressive models with exogenous (or extra) input (ARX). The nature of all the aforementioned models in the context of system identification is presented thoroughly in [53]. VAR mod- els use more than one channel. MA models estimate the present output as a linear combination of only past inputs, that refer to a stochastic term. ARMA models are combinations of AR and MA models, and offer higher accuracy than MA models for a much lower model order. ARX models predict present outputs based on past output values and values of an exogenous source, which is usually the excitation. While AR,

MA and ARMA models are stochastic time series models and assume stationarity, the ARX model is not based on this assumption [54].

Bodeux et al. employ an ARMA vector model in order to identify the modal pa- rameters and perform damage detection on the benchmark of a testing steel building [55]. In [56], a VAR model is used to identify the modal parameters of an offshore wind turbine. In [57], SSI is used to identify the natural frequencies of the Z-24 bridge, which are regarded as a time series of measurement values. At the next step, an ARX model is built using temperature measurements as input and the natural frequency time series as output, creating thereby a model that includes the thermal dynamics of the bridge.

Model coefficients can also be directly used as damage features. In [58], AR parameters are used as damage-sensitive features for the LANL three-story frame structure. In [59], the Mahalanobis squared distances of AR coefficients are used for detecting damage on concrete beams.

1.3.4.2

Transmissibility Measures

When ARX models are used as output-only models by solely considering system re- sponses, they are essentially transformed to transmissibility function models. In [60], an ARX model, which involves only acceleration signals is considered for damage identification. Based on numerical examples, the model residual error, i.e., the dif- ference between the measured signal of the actual state and the predicted signal, as well the residual error standard deviation are shown to be damage-sensitive features. For a multi-DOF (MDOF) system, the standard deviation of the residual error can also serve for damage localization, since higher values are observed close to damage. Moreover, for an undamped system, the model parameters are expressed in terms of stiffness. A similar formulation is provided in [61], where the parameters of an AR- MAX model are expressed as a function of stiffness and damping. In [62], an ARMA model is treated as an ARX model and two damage localization indices are defined based on the AR coefficients of the model. The two indices are tested on a numerical simulation of the American Society of Civil Engineers (ASCE) benchmark structure. The results show that one of them is capable of localizing even minor damage, while the other localizes significant damage, but is non-conclusive for minor damage.

A frequency domain ARX model, which is able to distinguish between nonlinear and linear damage, is used in [63]. Despite the fact that the model is a transmissi- bility model in frequency domain, it falls into the category of parametric methods, since the model is described by the coefficients of the AR and the exogenous part. The AR coefficients are used to characterize nonlinear damage, whereas the exoge- nous coefficients are used to characterize linear damage. The approach is verified on a 4-DOF simulation model and on the LANL three-story building structure. The same frequency domain ARX model is used in [64] to detect damage on the same three-story structure based on the impedance of piezoelectric (PZT) sensors that are used as both sensors and actuators.

1.3.4.3

Migration of Transfer Function Poles

Transfer functions have the system output in the numerator and the system input in the denominator. The roots of the denominator characteristic equation are called poles. Poles constitute the solutions which maximize the transfer function, since for these values the transfer function tends to infinity. In structural dynamics, poles correspond to the natural frequencies and the damping ratios of the structure. On the other hand, the roots of the numerator are called zeros. Zeros are the solutions for which the transfer function is equal to zero, i.e., the values for which the system attenuates the input. In control systems engineering, poles and zeros are a means of designing and controlling the response characteristics of a system. Hence, also in structural engineering, pole migration on the complex plane can serve as an indicator of structural changes, since poles encapsulate information on both natural frequencies and damping ratios.

The concept of transfer function pole migration for SHM is introduced by Lynch in [65]. Pole migration in ARX models is employed in a cantilever aluminum plate to distinguish between healthy and damaged poles. The plate is damaged by hack saw cuts and is actively sensed by PZT pads. In [66], the same concept is used for detecting damage on the IASC-ASCE benchmark structure. Poles from each state are clustered for all states. It is shown that pole cluster migration is sensitive to damage and that the degree of pole cluster migration is affected by the extent of damage. In [67], transfer function pole migration is successfully applied for damage characterization on the Z24 bridge data. A damage index is built as a weighted sum of three indices, which are based on three different types of classifiers: (i) a classifier based on the nearest mean, (ii) the perceptron classifier and (iii) the mean separation distance criterion. The classifiers are used to distinguish between healthy and damaged poles.

1.3.4.4

Residues

As an alternative to the classical modal parameters, residues from the covariance- driven SSI technique are used to detect changes in the system’s behavior. The most commonly used SSI residue was introduced by Baseville [68]. First, the Hankel matrix of output covariances is built and SVD is performed to estimate the left nullspace of the Hankel matrix. Residues can be formed by multiplying the nullspace matrix of the reference Hankel matrix by the Hankel matrix of the current dataset. This residual damage feature shows sensitivity to damage but also to varying EOCs. Therefore, data normalization is necessary prior to its deployment. In [56], the aforementioned residue is employed along with manual data clustering for monitoring an onshore wind turbine. Another residual vector, which is supposed to be robust to changes in excitation is introduced by Döhler [69]. This formulation is based on the fact that the Hankel matrix has the same nullspace matrix as the matrix containing

the left singular vectors. Thus, a new residue is built by multiplying the left singular vectors by the Hankel matrices of the new datasets. Both residues are used for detecting damage and ice on rotor blades [18] and for detecting structural changes on the LANL three-story test structure [16].

Residual damage features can also be calculated from VAR models. For this purpose, the time series of a measured response is compared to the time series estimated by the autoregressive (AR) coefficients. One residue is the fitted coefficient of determination, which makes use of the actual time series values (e.g., acceleration signals), the model estimation and the mean value of the measured signal [70]. Another residue makes use of the model error covariance matrix and is based on Box’s M-Test, which compares the error covariance matrix of the current dataset with that of the reference dataset [71, 56]. The residual error of a two-stage AR-ARX model, similar to an ARMA model, is used in [27] to identify damage on an 8-DOF mass-spring system. The prediction error of the AR model is calculated and is used as input in an ARX model for the prediction of the system responses. The residual error of the ARX model is then used as a damage-sensitive feature.