Multi-damage identification based on joint approximate diagonalisation and robust distance measure

Mode shapes or operational deflection shapes are highly sensitive to damage and can be used for multi-damage identification. Nevertheless, one drawback of this kind of methods is that the extracted spatial shape features tend to be compromised by noise, which degrades their damage identification accuracy, especially for incipient damage. To overcome this, joint approximate diagonalisation (JAD) also known as simultaneous diagonalisation is investigated to estimate mode shapes (MS’s) statistically. The major advantage of JAD method is that it efficiently provides the common Eigen-structure of a set of power spectral density matrices. In this paper, a new criterion in terms of coefficient of variation (CV) is utilised to numerically demonstrate the better noise robustness and accuracy of JAD method over traditional frequency domain decomposition method (FDD). Another original contribution is that a new robust damage index (DI) is proposed, which is comprised of local MS distortions of several modes weighted by their associated vibration participation factors. The advantage of doing this is to include fair contributions from changes of all modes concerned. Moreover, the proposed DI provides a measure of damage-induced changes in ‘modal vibration energy’ in terms of the selected mode shapes. Finally, an experimental study is presented to verify the efficiency and noise robustness of JAD method and the proposed DI. The results show that the proposed DI is effective and robust under random vibration situations, which indicates that it has the potential to be applied to practical engineering structures with ambient excitations.


Introduction
Vibration-based structural damage identification and health monitoring attract more and more attention recently. This is an inverse problem and the basic approach to addressing this issue is to monitor damage-induced changes in vibration responses [1][2]. Basically, modal parameters and vibration response statistics which are extracted from vibration responses are used to identify the damage. However, a noticeable drawback of detecting damage through the estimated damage features is that they are easily compromised by noise, which generally comes from four sources: operational, environmental, measurement and computational [3]. To overcome this noise problem, a current trend is to estimate spatial modal parameters based on the common Eigen-structure of correlation/covariance matrices or power spectral density (PSD) matrices, which are functions of vibration responses [4][5]. From a statistical point of view, the identified averaged spatial modal parameters will be much more accurate and robust for structural damage identification. For a standard second-order blind identification (SOBI), two steps are traditionally adopted: vibration data is pre-whitened and then joint approximate diagonalisation (JAD) is applied. It is noticed that the pre-whitening procedure is possible to introduce bias or error to SOBI, which cannot be corrected in the JAD step [6]. Moreover, SOBI has limitations in detecting the presence of spatially close modes or repeated frequencies. In order to solve these problems, applying the JAD algorithm to PSD matrices for mode shape (MS) estimation is investigated. The PSD matrix possesses the auto-and cross-spectral correlations of output vibration responses and provides an average energy distribution of a random process. Different from traditional frequency domain decomposition (FDD) method, the proposed JAD diagonalises a set of PSD matrices to obtain the common eigenvectors, which minimises the effect of the leakage error and statistically increase the noise robustness of estimated mode shapes. In this paper, MS's are estimated individually based on JAD method using a narrow frequency band around each resonant frequency without the pre-whitening procedure.
The mathematical approach of applying JAD to PSD matrices was presented by the authors in [5]. In this study, a new damage index (DI) is proposed as the squared Euclidean distance of MS local distortions and the distortions of different MS's are weighted by their corresponding vibration participation factors. In addition, this damage index has the ability to indicate the relative damage severity. In the numerical study, two linear cracks are modelled using fracture mechanics approach [7][8] to validate the feasibility of the proposed DI and coefficient of variation (CV) of MS's over 1000 Gaussian noise realisations is adopted to better demonstrate the noise robustness of JAD method. Then an experimental study of a beam with two cracks tested by PSV500 Scanning Laser Vibrometer is presented to verify noise robustness of MS estimation and the proposed DI. Moreover, the physical meaning and the roles of the real part and imaginary part of MS's in damage identification are discussed.

Robust MS estimation based on JAD method
The stochastic vibrations of structures under operational conditions are essentially uncertain in nature and statistical models should be utilised to analyse the random processes. In this study, measured random velocity responses ( ) ∈ ℝ × collected at locations with samples are used to formulate PSD matrices and estimate MS's according to JAD method. In terms of modal expansion theorem of structural response, measured vibration responses can be expressed using mode shapes and modal coordinates as where ( ) ∈ ℝ is the modal coordinate vector (the total number of modes is ) , y ∈ ℝ × represents the mode shape matrix at the measured degrees of freedom and ( ) ∈ ℝ denotes a vector of measurement noise. Under the general assumption that ( ) and y ( ) are uncorrelated, correlation matrix should be computed as where (= 0,1, ⋯ , − 1) implies time delay. qq ∈ ℝ × and νν ∈ ℝ × indicate correlation matrices of modal coordinates and noise, respectively. Taking discrete Fourier transform of equation (2), the PSD matrix is obtained as where ( ) is a Hermitan and positive definite matrix. According to FDD method, the singular value decomposition (SVD) of ( ) is The columns of ( ) are complex unitary singular vectors and ( ) is a real nonnegative diagonal matrix with its diagonal entries in a descending order: 1 ≥ 2 ≥ ⋯ ≥ o ≥ 0 , which represent the energy contribution factors of corresponding singular vectors in ( ) to vibration responses at this frequency. At a resonant frequency , the estimated mode shapes of measured degrees of freedom from ( ) could be biased due to the SVD orthogonal/unitary criterion. Fortunately, the mode shapes of weak modes at this resonant frequency are mainly affected by the bias whilst the estimated dominant mode shape is still good. Thus, the mode shapes are obtained individually using their corresponding resonant frequencies.
In FDD method, the dominant singular vector in ( ) is taken as the corresponding MS. From a statistical point of view, it is not accurate and robust just using a single PSD matrix at each resonant frequency. Moreover, the dominant mode shape of each resonant frequency does not change much around the resonant frequency. Therefore, the PSD matrices of several adjacent frequencies at each resonant frequency can be decomposed simultaneously for robust MS estimation. Here, a simultaneous diagonalisation technique is proposed by applying JAD method, which estimates the MS's based on a PSD matrix set according to the least squares criterion or maximum likelihood approach. Equation (6) demonstrates the problem of diagonalising a set of PSD matrices around a resonant frequency to find the common unitary matrix ( ): where indicates the resonant frequency , (= − , − + 1, ⋯ , ) denotes the adjacent frequency around and represents the error matrix. The least-squares criterion is used and the over determined diagonalisation is now equivalent to a minimization problem of variables ( ) and (ω + ): It is worth noting that the identified MS's are unitary complex vectors. According to the Hermitain properties of PSD matrix, the real part of MS's corresponds to the in-phase vibration whilst their imaginary part is related to the out-phase vibration. For damage identification, the damage-induced local stiffness reduction can be detected from the amplitude change of in-phase vibration or absolute value of MS's whereas the damage-induced phase changes are sensitively reflected by the imaginary part of MS's.

Damage identification index
With the estimated MS's d = ( d , d , ⋯ , d ) of damaged structures, damage identification is traditionally accomplished by comparing with the MS's = ( 1 , 2 , ⋯ , ) of healthy structures [9]. The difference or distance between and d at measurement point is measured by where denotes the weighting coefficient matrix and indicates the -th mode shape such as . In the case that mode shape matrix of healthy structures is not available, is obtained by polynomial smoothing approach of d based on the assumption that the mode shapes of healthy structures are smooth [10]. In this study, gapped smoothing method (GSM) is used to estimate the undamaged mode shape matrix ̂ ̂ = 3 3 + 2 2 + 1 + 0 where indicates the location of point and = [ 0 , 1 , 2 , 3 ] are coefficients of the cubic polynomial. GSM is sensitive to the local peaks and valleys of MS's associated with higher frequencies. However, the higher modes are normally more sensitive to local damage. Hence, for different mode shapes, a compromise between damage identification accuracy and sensitivity should be made when using GSM. The proposed damage index (DI) is computed as Here, = diag( 1 , 2 , ⋯ , ) is a diagonal matrix and the diagonal terms are defined by normalised vibration contribution factors of MS's in terms of The basic steps to calculate the proposed DI are summarised as: (1)     In order to compare the noise robustness of computed MS's and DI between JAD method and FDD method, Gaussian white noise is generated to contaminate the velocity responses in terms of ̂= + level ( ), = 1,2, ⋯ ,

Numerical study
where d ∈ ℝ 1× is a vector containing normally distributed random values with a zero mean and variance being 1, level is the noise level range of   It is demonstrated in figure 4 that the two open cracks can be both identified using the real part and signed absolute value of MS's and the overall damage identification performance of JAD method is better than FDD method. Furthermore, the real part of MS's of JAD method provides the most noise robust damage identification results. To further validate the efficiency and feasibility of the proposed DI, an experimental study is conducted next.

Experimental study
Experimental test is performed on a cantilever beam of dimensions 700 × 20 × 20 mm 3 with two open cracks. Detailed material information about this beam is given in table 1. Figure 5(a) shows the experimental set-up using a PSV-500 Scanning Laser Vibrometer. Pseudo-random excitation in the frequency range of 0-800 Hz is generated by the PSV-500 system and applied to the cantilever beam by a shaker (LDS V406) as shown in figure 5(b). and are marked as blue lines on the front surface in figure 5(b). Surface velocity data at prescribed 21 are collected using Scanning Laser Vibrometer for damage identification.  The damage identification results are presented in Figure 7 and a comparison of damage identification performance between JAD method and FDD method is conducted as well.  x 10 correctly. Figure 7(b) presents the DI values calculated using the signed absolute value of MS's, which implies that the absolute value of MS's can effectively improve the damage identification accuracy of FDD method. However, the enhancement for JAD method by using signed absolute value is not obvious which even degrades the damage identification results at measurement point 10 as shown in figure 7(c). Another conclusion from Figure 7 is that overall DI values calculated by JAD method are much more accurate and noise robust than those of FDD method.

Conclusions
This study proposes a mode shape-based damage identification method without requiring the baseline information of healthy structures. Two aspects of improvements have been made. First, a new algorithm based on joint approximate diagonalisation is proposed to statistically estimate the dominant mode shape at each resonant frequency by diagonalising a set of PSD matrices. Secondly, local polynomial fitting method is used to extract the damage-caused local shape distortions of MS's and several MS's are combined according to their vibration participation factors to compute the proposed damage index. Numerical simulation with Gaussian white noise is used to demonstrate the noise robustness of MS's and damage index of JAD method. Furthermore, form an experimental study, the obtained MS's and damage index of JAD method are illustrated to be more accurate and noise robust than those of FDD method. Consequently, JAD method is demonstrated to be effective in addressing the noise effects in MS estimation and damage identification. In addition, the proposed PSD-based damage index has a good potential to be applied to practical engineering applications with ambient excitations. Another conclusion is that JAD is effective to enhance the accuracy and noise robustness of mode shapes extracted from experiments. As a result, it should be used to improve accuracy of the PSD based modal analysis.