Structural damage localization by outlier analysis of signal-processed mode shapes – Analytical and experimental validation

Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]

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Structural damage localization by outlier analysis of signal-processed mode shapes – Analytical and experimental validation M.D. Ulriksen n, L. Damkilde Department of Civil Engineering, Aalborg University, Niels Bohrs Vej 8, Esbjerg, Denmark

a r t i c l e i n f o

abstract

Article history: Received 14 March 2015 Received in revised form 22 June 2015 Accepted 20 July 2015

Contrary to global modal parameters such as eigenfrequencies, mode shapes inherently provide structural information on a local level. Therefore, this particular modal parameter and its derivatives are utilized extensively for damage identification. Typically, more or less advanced mathematical methods are employed to identify damage-induced discontinuities in the spatial mode shape signals, hereby, potentially, facilitating damage detection and/or localization. However, by being based on distinguishing damage-induced discontinuities from other signal irregularities, an intrinsic deficiency in these methods is the high sensitivity towards measurement noise. In the present paper, a damage localization method which, compared to the conventional mode shape-based methods, has greatly enhanced robustness towards measurement noise is proposed. The method is based on signal processing of a spatial mode shape by means of continuous wavelet transformation (CWT) and subsequent application of a generalized discrete Teager–Kaiser energy operator (GDTKEO) to identify damage-induced mode shape discontinuities. In order to evaluate whether the identified discontinuities are in fact damage-induced, outlier analysis is conducted by applying the Mahalanobis metric to major principal scores of the sensorlocated bands of the signal-processed mode shape. The method is tested analytically and benchmarked with other mode shape-based damage localization approaches on the basis of a free-vibrating beam and validated experimentally in the context of a residential-sized wind turbine blade subjected to an impulse load. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Structural health monitoring Damage localization Wavelet transformation Generalized discrete Teager–Kaiser energy operator Outlier analysis

1. Introduction Numerous vibration-based structural health monitoring (SHM) methods have been proposed for detection and/or localization of structural damages in aerospace, civil, and mechanical systems, see, for instance, [1–4]. Commonly, these methods are applied in such a way that a chosen damage feature, for example, direct kinematic response or modal parameters, from a current state is compared to the corresponding feature from the healthy reference state. In principle, the structure is potentially damaged if the chosen feature from the current state differs significantly from that of the reference state. The applicability of the proposed damage detection and/or localization methods has primarily been tested on the basis of analytical models, finite element (FE) simulations, and controlled laboratory tests; all in which significant simplifications n

Corresponding author. Tel.: þ45 52300716. E-mail address: [email protected] (M.D. Ulriksen).

http://dx.doi.org/10.1016/j.ymssp.2015.07.021 0888-3270/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

M.D. Ulriksen, L. Damkilde / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]

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and idealizations are made with regard to, among other things, geometric boundary conditions, applied loads, and environmental effects. Here, it has been found that the simple methods directly comparing pre- and post-damage vibrationbased quantities, such as modal parameters, exhibit only limited potential for damage detection and/or localization, see, for instance, [1,5]. Under more realistic conditions, these methods become completely inapplicable as the direct changes of the aforementioned quantities normally will be concealed by environmental effects and noise contamination. For instance, it is documented in [6,7] how environmental effects and general operational conditions can account for up to at least 5% shifts in eigenfrequencies, which, as documented in, for example, [7,8], cannot be expected to be exceeded by damage-induced eigenfrequency changes. Due to the general inadequacy of the simple methods, current research within the field of vibration-based SHM is mainly focused on developing more sophisticated and robust methods. Some of these methods are based on signal processing of spatial mode shape signals, which seems auspicious as this modal parameter, contrary to global modal parameters such as eigenfrequencies, inherently provides structural information on a local level. A common approach is to exploit that a damage will introduce mode shape discontinuities, albeit not always directly visible ones, which can be captured by use of signal processing techniques, for example, continuous wavelet transformation (CWT). CWT has been utilized extensively for localizing structural damages in both simple beam- and plate-like systems, see, for example, [9–11], and more complex structures such as wind turbine blades [8,12,13]. However, by being based on distinguishing damage-induced discontinuities from other signal irregularities, an intrinsic deficiency of these CWT-based methods is the rather low robustness towards measurement noise. In order to treat this issue, a method in which wavelet-transformed mode shapes are processed by use of the discrete Teager–Kaiser energy operator (DTKEO) is proposed in [14]. Here, it is shown, on the basis of analytical and experimental work with different beams, that by adding the DTKEO, the robustness towards noise is significantly enhanced and in general more unambiguous localization results are obtained. One of the major advantages of the method proposed in [14] is the lack of need for baseline mode shape signals, that is, signals from the healthy structural state. This, however, also means that the discrimination between potentially damaged locations and undamaged ones is based solely on direct, deterministic inspection of the signal-processed mode shapes from the current state; an approach that, as will be demonstrated in the present study, can be impossible in some particular cases. Consequently, the authors of the present paper have proposed a further development of the aforementioned method, in which the DTKEO is expanded to a generalized DTKEO (GDTKEO) by use of a lag parameter, κ, and where outlier analysis is added to yield the final discrimination between undamaged and damaged areas by use of signals from both the healthy (baseline) and the current state [15]. Regarding the constraint of needing baseline mode shape signals, it is noticed that, according to numerous studies, for instance, [16], it is problematic since a priori measurement in a healthy condition is affected greatly by environmental and operational variability. However, when statistical evaluation is applied, a baseline model representing the healthy state is derived on the basis of several measurements, hereby including the variability in the mode shape signals and, thus, making the baseline model applicable for classifying mode shape signals from the current structural state. By applying the method proposed in [15] to experimental setups, it has been found that the approach depends too much on the number of measurement points. Therefore, in the present paper, we propose a method that can be regarded as the final edition of the preliminary one documented in [15]. As such, the method proposed in the present paper constitutes a twofold refinement; firstly, the premise of selecting κ in the GDTKEO is changed in order to reduce the dependency on the amount of measurement points, and, secondly, the Mahalanobis metric is used in the statistical evaluation instead of T2statistics. The proposed method is composed of the following three steps: a filtered mode shape derivative is obtained of a spatial mode shape signal through CWT, and subsequently a generalized discrete Teager–Kaiser energy operator (GDTKEO) is applied to this derivative to form an energy-processed signal in which the damage-induced discontinuities are magnified. Finally, a statistical evaluation scheme based on the Mahalanobis metric is applied to principal scores of these energyprocessed, filtered mode shape derivatives in order to label the structure as healthy or damaged at each sensor location/ measurement point. To test the applicability of the method, and to justify this refinement of the preliminary method proposed in [15], two application examples of engineering interest, namely, an analytical beam model and experimental work with a residential-sized wind turbine blade, are treated. The paper is organized as follows: in Section 2, the proposed damage localization method is presented, and in the following section, the method is applied in the two application examples. Here, the localization performance is compared to those yielded by the application of well-known mode shape-based damage localization methods in order to benchmark the method. Finally, some concluding remarks are presented in Section 4. 2. Methodology Several aspects of the methodology have already been described by the authors in [15]. These aspects are, however, still included in the present section for the sake of completeness. 2.1. CWT Since the fundamentals of the CWT and its applicability in damage identification, primarily as a signal discontinuity scanner, is well documented in numerous publications, see, for instance, [8–13,17], only the most relevant CWT aspects will be presented in this paper. Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

M.D. Ulriksen, L. Damkilde / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]

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The spatial mode shape signal, f ðxÞ A L2 ðRÞ, is obtained as a signal-processed version of the original mode shape, u. The signal processing consists of signal interpolation, through cubic splines, and subsequent expansion in order to, respectively, smooth the mode shape signal, hereby removing artificial discontinuities arisen due to a sparse number of measurement points, and remove boundary distortions. For the signal expansion, several studies, see, for instance, [11], suggest using cubic spline extrapolation, but during testing, it has been found that the isomorphism approach described in [18] yields the best results, and therefore it is adopted here. Based on the spatial mode shape signal, f ðxÞ A L2 ðRÞ, and the wavelet function, γ ðxÞ A L2 ðRÞ, the CWT is defined as Z   Wf ða; bÞ ¼ a  1=2 f ðxÞγ n a  1 ðx  bÞ dx; ð1Þ R

where a 4 0 and b A R are wavelet scales and positions, respectively. Evidently, the CWT is obtained as the inner product of f and the complex conjugated, indicated with the superscript n, of the so-called wavelet family, which consists of functions constructed from dilations and translations of γ. In this way, Wf becomes a matrix with the number of rows and columns corresponding to the number of elements in a and f, respectively. The effectiveness of the CWT as a signal discontinuity scanner highly depends on the employed wavelet type; in particular, the amount of vanishing moments, m. Assume r A N þ , then Z 8 r om: γ ðxÞxr dx ¼ 0; ð2Þ R

which states that a wavelet with m vanishing moments is blind to polynomial trends up to degree m  1. Equivalently, when analyzing a spatial mode shape signal with a wavelet with m vanishing moments, a filtered m-derivative of the mode shape is obtained. This becomes evident by realizing that the expression in Eq. (1) also can be seen as the convolution of f with a scaled, flipped, and conjugated wavelet, that is,   x ; ð3Þ γ a ðxÞ ¼ a  1=2 γ n a thus ( θðxÞ ¼

Z R

γ ðxÞdm x 3 Wf ða; bÞ ¼

   m  m d d  f ðxÞγ  θa ðbÞ ¼ f nθa ðbÞ; a ðb  xÞ dx ¼ f n m m dx dx R

Z

ð4Þ

where θðxÞ is the so-called smoothing function, as described further in [11]. This convolution relation emphasizes the importance of choosing a wavelet with a certain minimum of vanishing moments when searching for discontinuities in a signal. Correspondingly, there is also a certain maximum for the proper amount of vanishing moments due to, among other things, magnification of signal noise and adverse border distortions. Clearly, the optimal choice of wavelet strongly depends on signal characteristics such as mode shape type and noise conditions and should therefore be drawn based on these. This is treated further in the application examples presented in Section 3. As stated in Section 1, CWTs have previously exhibited damage localization potential. However, as noise is added to the signal, this potential is significantly reduced since the discrimination between noise- and damage-induced CWT peaks becomes troublesome, especially at the lower wavelet scales. For the higher scales, at which noise is better filtered, the low spatial frequency reduces the potential for capturing damage-induced signal discontinuities, thus emphasizing the need for further enhancement of these. 2.2. GDTKEO Since the DTKEO was proposed, originally as a signal energy estimator [19], it has been utilized extensively in speech processing [20], image processing [21], and pattern recognition [22]. Recently, the DTKEO has been adopted to the field of damage localization; as in [14] where it is applied to post-damage wavelet-transformed mode shapes for damage localization in beams; in [23] where the method proposed in [14] is expanded to two-dimensional cases; in [16] where the DTKEO is applied to uniform load surface (ULS) curvatures; and in [15] where the method proposed in [14] is expanded by means of implementing statistical evaluation. As such, the method proposed here is a further refinement of the method proposed in [15]. With the notation utilized in the present paper, the DTKEO of a wavelet-transformed mode shape signal at scale al and spatial location i can be found as

ψ al ;i ¼ Wf 2al ;i Wf al ;i  1 Wf al ;i þ 1 :

ð5Þ

In order to enhance the performance of the DTKEO as a signal discontinuity magnifier, it is expanded intuitively by means of a lag parameter, κ A N þ , hence yielding the GDTKEO

Ψ al ;i ¼ Wf 2al ;i  Wf al ;i  κ Wf al ;i þ κ ;

ð6Þ

implying that Ψ al ¼ ψ al if κ ¼ 1. As can be realized from Eq. (6), the lag parameter serves directly to alleviate the adverse noise effects. Generally, κ should be chosen on the basis of signal characteristics such as the number of measurement points, that is, signal length, and noise conditions. In the present paper, a simple, yet very effective, approach based on the well-known modal assurance criterion Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

M.D. Ulriksen, L. Damkilde / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]

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Fig. 1. Restriction for κ with regard to wavelet cone of influence, with Υ being the number of data points between original measurement points j and j þ 1.

(MAC) is proposed. Here, the mean of the CWTs from the training state at scale amax is signal-processed by means of the GDTKEO with different κ-values. Then, the MACsp (with the subscript denoting MAC of signal-processed mode shapes) value between signal τ and τ þ 1 is calculated in accordance with  2 Wf amax ðf τ ÞT Wf amax ðf τ þ 1 Þ MACsp ðτ þ 1Þ ¼ A ½0; 1; ð7Þ Wf amax ðf τ ÞT Wf amax ðf τ ÞWf amax ðf τ þ 1 ÞT Wf amax ðf τ þ 1 Þ where it is noticed that the row vectors Wf amax ðf τ Þ and Wf amax ðf τ þ 1 Þ are the mean CWTs (evaluated at maximum scale, amax ) of, respectively, mode shape signal

τ and τ þ 1. In this way, κ is finally chosen as the value at which MACsp exceeds a threshold, χ,

which, in the present study, is set heuristically to χ ¼ 0:995. However, this approach is only useful when κ is sufficiently low to not cause disturbances between the cone of influences in the CWT, as illustrated in Fig. 1. In cases where 2κ Z Υ ,

Υ =2  1. The procedure of selecting κ is exemplified in Section 3.

κ should be set to

2.3. Statistical evaluation of signal-processed mode shapes In order to evaluate whether the localized mode shape discontinuities are in fact damage-induced, outlier analysis is conducted by applying the Mahalanobis metric to major principal scores of the sensor-located signal-processed mode shape bands. Consider the baseline, training matrix for signal band j 2 3 jΨ j;trainð1Þ jT 6 7 7; ⋮ Zj ¼ 6 ð8Þ 4 5 T jΨ j;trainðSÞ j whose S rows are constituted of CWT- and GDTKEO-processed signal bands at sensor/measurement point j from the healthy state of the structure. Zj is subjected to principal component analysis (PCA) to not only effectively reduce the data dimensionality, but also to discard minor components that do not contain useful information with regard to the damage. In general, PCA is a widely utilized unsupervised learning technique for dimensionality reduction and feature extraction. The principle, with reference to this particular study, is to project the data in Zj into a new set of coordinates, namely, principal component scores, Tj, which emphasize the variation in the original data (the first principal score accounts for most variance, the second one accounts for the second most, and so forth). This is done through the linear transformation T j ¼ Zj Pj

ð9Þ

in which Pj contains the principal components that constitute the axes in the new, principal coordinate system. With the aim of maximizing the variance, it can readily be shown, see, for instance, [24], that

Σ j P j ¼ P j Λj ;

ð10Þ

where Σj is the covariance matrix of Zj. Thus, the principal components, Pj, are found as the eigenvectors of Σj, while the variance associated with each principal component is the corresponding eigenvalue contained in Λ ¼ diagðλ1 ; λ2 ; …; λβ Þ, with β denoting the dimensionality of Σj. Generally, the Mahalanobis metric is a discordancy test for multivariate data and has, as such, been utilized extensively for damage detection, see, for example, [25]. With the notations applied in the present paper, the metric for each signal band located at a sensor/measurement point, j, is defined as  ~ ~ 1 ~T T D2j ¼ jΨ j jT  Z j P j Λj P j jΨ j jT Z j ; ~

ð11Þ

~

where P j D P j , Λj D Λj compose the dimensionality-reduced principal basis, Z j is the vector mean of Zj, and jΨ j j is the signal band to be classified. This classification is conducted on the basis of the following hypothesis: 9 H 0;j : D2j r ϑj = ; ð12Þ H 1;j : D2j 4 ϑj ; Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

M.D. Ulriksen, L. Damkilde / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]

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in which the null hypothesis, H 0;j , labels the signal band as an inlier, that is, no anomalies are present at the jth sensor/measurement point, whereas H 1;j ¼ :H 0;j labels the signal band as an outlier, thus declaring the structure damaged at this location. The threshold, ϑj, is found through the training data in an exclusive manner. Here, a Mahalanobis metric, in accordance 2 with Eq. (11), between each row in Zj and the remaining S 1 rows is calculated, and subsequently the S Dj -metrics are sorted in a descending order. Finally, ϑj is then chosen as the value exceeded by 1%. 3. Application examples In the present section, the proposed method is applied to two systems of engineering interest. The first system is an analytical beam model, while the second is a residential-sized wind turbine blade that is treated pseudo-experimentally; that is, experimentally obtained data are assigned Gaussian noise and subsequently duplicated to constitute a sufficiently large set of data for both the undamaged and damaged states. 3.1. Example I: free-vibrating Euler–Bernoulli beam This application example serves to demonstrate the fundamental applicability of the method, its robustness towards noise, the alterations to the method proposed in [15], and its performance compared to a well-established mode shapebased damage localization method. In [15], we use this particular example to demonstrate the improvement obtained by applying the GDTKEO instead of the DTKEO. Consequently, some aspects of the following example will be appear as repetition from [15]. 3.1.1. Analytical model of cracked beam The application example is based on free vibrations of the beam model depicted in Fig. 2a. The beam has a length of L, cross-sectional dimensions of v ¼ d ¼ 0:03L, and is assigned an isotropic material model with Young's modulus, E, corresponding to construction steel, that is, E ¼ 200 GPa. A structural damage with height s ¼ 0:15d is introduced as a crack at location x=L ¼ 0:4. The beam is treated analytically through Euler–Bernoulli beam theory, such transverse vibrations are governed only by the bending deformations. Assuming that the effect of the crack is apparent only in its area, the crack can be modelled as a linear rotational spring with stiffness [26] KR ¼

vEI ; 6π dFðζ Þ

ð13Þ

where EI is the flexural stiffness, while Fðζ Þ ¼ 1:86ζ  3:95ζ þ 16:37ζ þ 37:22ζ þ76:81ζ þ 126:9ζ þ 172:5ζ 144ζ þ 66:6ζ 2

3

4

5

6

7

8

9

10

ð14Þ

is a dimensionless local compliance function, see, for example, [26], with the independent variable ζ ¼ s=d. As seen in Fig. 2b, the spring divides the beam into two separate segments. The transverse motion of each beam segment, wi(x), is described spatially by the dimensionless governing equation x x 4  λ wi ¼ 0; i ¼ 1; 2; ð15Þ w⁗ i L L in which λ ¼ ω2 AL4 ρ=EI, with ω, A, I, and ρ being the eigenfrequency, the cross-sectional area, the cross-sectional moment of inertia, and the mass density. Solving Eq. (15) yields x   x  x  x  x ¼ c1;i cos λ þ c2;i sin λ þ c3;i cosh λ þ c4;i sinh λ ; i ¼ 1; 2; ð16Þ wi L L L L L 4

where the constants of integration are found by applying the boundary conditions of the cantilevered beam, that is, w1 ð0Þ ¼ 0

4

w01 ð0Þ ¼ 0

4

w″2 ð1Þ ¼ 0

4

w‴2 ð1Þ ¼ 0;

ð17Þ

Fig. 2. Cracked cantilevered beam. (a) General beam model. (b) Equivalent beam-spring model.

Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

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plus the following compatibility conditions in the crack area ðx ¼ xc Þ: x  x   x  x  x  c c c c c ¼ w2 4 K R w01 w02 ¼  EIw″1 4 w1 L L L L L

w″1

x  x  c c ¼ w″2 L L

4

w‴1

x  c

L

¼ w‴2

x  c : L

ð18Þ

When substituting Eq. (16) into Eqs. (17) and (18), the characteristic equation is obtained. From here, the eigenfrequencies and subsequently the constants of integration are derived and substituted into Eq. (16) to yield the mode shapes. 3.1.2. Damage localization 300 noise-contaminated (corresponding to a signal-to-noise ratio (SNR) of 65 dB) editions of both the first undamaged bending mode shape and the first damaged bending mode shape are analyzed with a spatial increment of Δx ¼ 0:0001L. 200 of the signal-processed mode shapes from the undamaged state have been randomly chosen to constitute the trained statistical baseline model against which the remaining 100 undamaged signal-processed mode shapes and the 300 damaged signal-processed mode shapes are tested. CWT settings: In the choice of wavelet type, a crucial parameter is the number of vanishing moments, cf. Eq. (2). A wavelet with m vanishing moments is blind to polynomials up to order m  1, hence implying that a wavelet with a minimum of four vanishing moments would, theoretically, provide the best results when analyzing the first bending mode shape (see [27] for a general and thorough study on this subject). However, extensive amount of differentiation, recall that the CWT yields pseudo-derivatives, would yield detrimental magnification of the border distortions, even when applying the isomorphism approach. Consequently, it is chosen to use a second-order Gaussian wavelet, because this has been found to provide a suitable trade-off between discontinuity enhancement and border distortion magnification. In this way, the wavelet analyzes the second-order derivative of the mode shape. In Fig. 3, CWTs of noise-contaminated editions of the first bending mode shape of the undamaged beam and the damaged one are presented for the wavelet scale interval a A ½1; 120. GDTKEO settings: In Section 2.2, it is described how κ is chosen as that particular value at which MACsp Z0:995. For the training state of the beam, the MACsp –κ-function is plotted in Fig. 4, where κ ¼ 340 is chosen as the optimal value (recall that κ should be as small as possible, while still fulfilling the threshold criterion). By using this setting for the GDTKEO and applying it to the wavelet transforms seen in Fig. 3, the results presented in Fig. 5a and b are obtained. Clearly, the damagedinduced discontinuity is located, albeit along with other irregularities, hence emphasizing the need for statistical discrimination. Outlier analysis: It is chosen to present outlier analysis results for four measurement points/sensor locations, namely, x=L ¼ 0:04, x=L ¼ 0:4 (damage location), x=L ¼ 0:5, and x=L ¼ 0:8. Here, the statistical model for each location have been trained by means of the relevant signal bands from the chosen 200 undamaged signals. It is found, for all four locations, that the two first principal directions contain above 90% of the variance, and therefore it is chosen only to use these two, that is, ~ ~ P j A R2002 and Λj A R22 . The corresponding thresholds are computed on the basis of the relevant bands from the 200 randomly chosen signal-processed mode shapes from the undamaged state, hence Z j A R200120 . With the training and threshold computation completed, the testing is conducted. First, the relevant bands from the remaining 100 undamaged signal-processed mode shapes are tested, followed by the corresponding bands from 300 damaged counterparts. In Fig. 6, the results are presented. Evidently, the data from the healthy state, that is, test number 1–100, are generally classified as inliers. Only a few type I errors, that is, false outlier classification, are obtained. This is, however, expectable due to the probabilistic nature of the threshold computation. For test number 101–400, at which the beam is damaged at x=L ¼ 0:4, the signal-processed mode shape bands at x=L ¼ 0:4 are all classified correctly as outliers, see Fig. 6b. For the three remaining locations, the signal-processed mode shape bands are generally classified as inliers, meaning that the method yields unambiguous localization of the crack. 3.1.3. Comparison with other mode shape-based damage localization methods In order to further validate and benchmark the proposed method, a comparison with other well-known data-driven (that is, methods not depending on mechanical models), mode shape-based localization techniques from the literature is given.

Fig. 3. CWT of noise-contaminated (65 dB) first bending beam mode shape. (a) Undamaged state. (b) Damaged state (crack at x=L ¼ 0:4).

Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

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Fig. 4. Selecting κ on the basis of MACsp.

Fig. 5. GDTKEO with κ ¼ 340 applied to CWT of noise-contaminated first beam mode shape. (a) Undamaged state. (b) Damaged state (crack at x=L ¼ 0:4).

Fig. 6. Mahalanobis metric on the basis of jΨ j j with κ ¼ 340 for undamaged (test number 1–100) and damaged (test number 101–400) beam states. (a) x=L ¼ 0:04. (b) x=L ¼ 0:4 (damage location). (c) x=L ¼ 0:5. (d) x=L ¼ 0:8. The dashed, horizontal lines mark the thresholds and the red circles designate type I errors. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

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2

Fig. 7. Tj -statistics on the basis of jΨ j j with κ ¼ 342 for undamaged (test number 1–100) and damaged (test number 101–400) beam states. (a) x=L ¼ 0:04. (b) x=L ¼ 0:4 (damage location). (c) x=L ¼ 0:5. (d) x=L ¼ 0:8. The dashed, horizontal lines mark the thresholds and the red circles designate type I errors. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 8. Absolute normalized outlier index, jNFj, based on changes in NULSC by use of 200 versions of the first mode shape from, respectively, the undamaged and damaged configuration. Damage is located at x=L ¼ 0:4. The horizontal dashed line marks the 2:8σ threshold.

Besides the preliminary method documented in [15], the method proposed in the present study is compared to the Normalized Uniform Load Surface Curvature (NULSC) method, which has proved to outperform other well-established mode shape-based methods, see [28]. The NULSC method is basically composed of statistical evaluation of NULSC changes between the healthy structural state and the damaged one; with each ULS being the displacement field due to a unit load acting in each measurement point/sensor. In Figs. 7 and 8, the results obtained by the use of the preliminary method [15] and the NULSC method [28] are presented for the same conditions as used to test the proposed method. It is clear, when comparing Figs. 6 and 7, that the preliminary method proposed in [15] yields just as good results as the final one proposed here. However, in the pseudo-experimental example in the coming section, it will be evidenced that the final method substantially outperforms the preliminary one presented in [15]. The results obtained by the use of the NULSC method are seen in Fig. 8, with a statistical basis composed of 200 signals of the first mode shape from the undamaged state and similarly 200 signals from the damaged counterpart. The threshold is, as done in [28], set to 2:8σ , where σ ¼ 1 due to the normalized nature of NF. Evidently, the method fails to localize the introduced crack on the basis of the first mode shape contaminated with 65 dB noise. Again, it is noticed that the NULSC method has outperformed well-established mode shape-based localization methods, thus the comparison between this method and the one proposed in the present paper emphasizes the robustness towards the noise of the latter.

Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

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Fig. 9. Experimental set-up for residential-sized blade. (a) Overall set-up. (b) Numbering of accelerometers and position of damage (hatched box between accelerometers 5 and 6). Table 1 Experimental mean findings for eigenfrequencies and corresponding MAC values of the first four modes of residential-sized blade (based on 10 experiments). The reference of each MAC is the corresponding mean mode shape of the intact blade. Modal results Mode 1 Mode 2 Mode 3 Mode 4

Frequency MAC Frequency MAC Frequency MAC Frequency MAC

(Hz) (Hz) (Hz) (Hz)

Intact blade

0.1 m crack

10.52 1.0000 32.50 1.0000 37.99 1.0000 74.65 1.0000

10.53 1.0000 32.50 1.0000 37.98 0.9991 74.29 0.9996

3.2. Example II: Pseudo-experimental work with residential-sized wind turbine blade The aim of this example is to verify the practical applicability of the proposed method and to demonstrate its improved performance compared to the preliminary version in [15]. The study is of pseudo-experimental nature, meaning that experimentally obtained data are assigned Gaussian noise and then duplicated to constitute a statistically sufficient set of data for both the undamaged and damaged states. 3.2.1. Experimental work The example is based on the approximately 2.2 m long residential-sized wind turbine blade depicted in Fig. 9, which was originally treated in [29] in the context of detecting damage based on shifts in modal parameters. As shown in Fig. 9a, the blade was set up in a constrained-free configuration. The excitation of the blade was achieved by monitored hammer hits in downwards ð zÞ direction, while the response was captured via 16 uni-axial piezoelectric (IEPE) accelerometers mounted along the blade edges, as marked in Fig. 9b. The experimental routine included tests with an intact blade and the same blade with a 0.1 m longitudinal edge crack. In Fig. 9b, the crack location is indicated as being between accelerometers 5 and 6. The acquisition system was set to sample in a frequency range of 0–500 Hz with a resolution of 1600 lines, thus yielding an increment of 0.3125 Hz. The obtained time domain responses of the hammer excitations and the accelerometer measurements were both filtered to reduce leakage and noise artifacts. For the excitations, a force window was used, while the accelerations were filtered with an exponential window. For more information regarding these signal windows, the reader is referred to, for instance, [30]. In the experimental modal tests of the blade, the excitation and the resulting dynamic response were captured by the use of data acquisition software and subsequently employed to derive the frequency response functions. From these functions, the modal parameters, that is, eigenfrequencies and mode shapes, were extracted in the commercial modal analysis software Brüel and Kjær (B&K) PULSE Reflex Advanced Modal Analysis. In Table 1, the found eigenfrequencies and MAC values (all as means of 10 experiments) are presented for the four first blade modes. Clearly, the damage induces very limited shifts in the modal parameters of the four presented modes. In Fig. 10, the appertaining mode shapes are shown. Here, it is noticed how 20 measurement points are assigned, although only 16 accelerometers are indicated in Fig. 9b. The four extra measurement points were placed at the constraint, thus no significant modal displacements occurred here. 3.2.2. Damage localization In the attempt to localize the 0.1 m edge damage, the leading edge components of the fourth mode shape, that is, those derived from accelerometers 1–8, are utilized in noise-contaminated configurations (corresponding to SNR¼ 65 dB). This Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

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Fig. 10. Experimentally obtained mean mode shapes of undamaged residential-sized blade. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4.

Fig. 11. CWT of noise-contaminated (65 dB) fourth blade mode shape. (a) Undamaged state. (b) Damaged state (crack between acc. no. 5 and 6).

particular mode shape is chosen on the basis of the MAC values presented in Table 1, where it is seen that damage-induced changes occur in this mode, albeit practically non-visible ones. CWT settings: As briefly mentioned in Section 2.1, it can be necessary to smooth the mode shape signal if only a sparse number of measurement points are available. In the present case, where only eight accelerometers are used, it is chosen to employ cubic spline interpolation to yield a spatial increment of Δx ¼ 0:001 m. Subsequently, this smoothed signal is expanded by use of the isomorphism approach. If the blade, as a simple analogy, is considered a beam, the fourth mode shape corresponds to the third bending mode shape. Due to the higher-order polynomial trends, a fourth order Gaussian wavelet is employed in the CWT (cf. the findings presented in [27]), yielding the results presented in Fig. 11. Here, the transforms are of the leading edge components of the mean mode shape depicted in Fig. 10d and the damaged counterpart. Clearly, the CWTs do not directly facilitate any damage localization. GDTKEO settings: By use of the κ-selection procedure described in Section 2.2, the settings indicated in Fig. 12 are found. In particular, κ ¼ 7 which, compared to the beam case in Section 3.1, is very low. This is due to the sparse number of employed measurement points. With the setting κ ¼ 7, application of the GDTKEO yields the results plotted in Fig. 13. Here, it is evident that the GDTKEO fails to visually pinpoint the damage location; again emphasizing the need for statistical evaluation. Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

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Fig. 12. Selecting κ on the basis of MACsp.

Fig. 13. GDTKEO with κ ¼ 7 applied to CWT of noise-contaminated fourth blade mode shape. (a) Undamaged state. (b) Damaged state (crack between acc. no. 5 and 6).

Fig. 14. Mahalanobis metric on the basis of jΨ j j with κ ¼ 7 for undamaged (test number 1–100) and damaged (test number 101–400) states. (a) Acc. 1. (b) Acc. 5. (c) Acc. 6. (d) Acc. 7. The dashed, horizontal lines mark the thresholds and the red circles designate type I errors. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

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Fig. 15. Selecting κ on the basis of RMS.

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Fig. 16. Tj -statistics on the basis of jΨ j j with κ ¼ 253 for undamaged (test number 1–100) and damaged (test number 101–400) states. (a) Acc. 1. (b) Acc. 5. (c) Acc. 6. (d) Acc. 7. The dashed, horizontal lines mark the thresholds and the red designate type I errors. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Outlier analysis: During the experiments, 10 measurement series were conducted for both the undamaged and damaged blade state. Therefore, a pseudo-experimental approach, in which Gaussian noise is added to each of the signals of the fourth mode shape, is carried out. This results in a total of 300 mode shape signals from the undamaged state and 300 ones from the damaged state; all with a SNR ¼65 dB compared to the original signals. It is found, for all accelerometer arrangements, that the 15 first principal directions contain above 90% of the variance, ~ ~ and therefore it is chosen to use these components in the outlier analysis, that is, P j A R20015 and Λj A R1515 . By conducting the outlier analysis, the results in Fig. 14 are obtained for accelerometers 1, 5, 6, and 7 (see Fig. 9b for accelerometer numbering). As the crack is located between accelerometers 5 and 6, the localization is evidently correct and unambiguous, albeit few type I errors are present.

3.2.3. Comparison with preliminary method Yet again, the outlier analysis results are compared to those obtained by use of the preliminary method proposed in [15]. Using this latter method, we find κ ¼ 253 by locating the first local minimum in an RMS-κ plot, as seen in Fig. 15, where RMS is the root mean square. This setting yields the outlier analysis results presented in Fig. 16, which clearly demonstrate the improvement obtained by the new, final method; an improvement primarily obtained because of the change in selecting κ. With the RMS-approach, κ is selected too high in cases where only few measurement points are available, thus disturbances arise between the cone of influences, in accordance with the principle depicted in Fig. 1. Please cite this article as: M.D. Ulriksen, L. Damkilde, Structural damage localization by outlier analysis of signalprocessed mode shapes – Analytical and experimental validation, Mech. Syst. Signal Process. (2015), http://dx.doi.org/ 10.1016/j.ymssp.2015.07.021i

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4. Conclusion The present paper introduces a noise-robust damage localization method whose principle is to statistically interrogate changes in signal-processed mode shapes with respect to the localization of damage. As such, the method is composed of two signal processing steps, namely, CWT and application of a GDTKEO, serving to localize and magnify damage-induced mode shape discontinuities, plus a subsequent statistical evaluation step, in which Mahalanobis metric-based outlier analysis is conducted of major principal scores of the processed mode shape signals to discriminate between damageinduced discontinuities and other signal irregularities. The fundamental applicability of the method and its robustness towards measurement noise has been demonstrated in the context of analytical work with a free-vibrating Euler–Bernoulli beam. Here, it is found for the damaged state that the signal bands at the damage location are consistently labeled as outliers, whereas the Mahalanobis metrics derived for the remaining locations are generally classified correctly as inliers. For the undamaged state, only three outliers occur, meaning that the crack is unambiguously localized. In order to verify the practical applicability of the proposed method, experimental work with a residential-sized wind turbine blade has been conducted. Through the same procedure as for the beam, it is found that the introduced crack is unambiguously localized. Therefore, the findings in this study suggest that the damage localization method generally possesses the ability to validly localize critical structural damages. Three major advantages of the method are, respectively, the lack of need for measuring the excitation input, the independence of mechanical models, and the direct possibility of extending the dimensionality such, for example, two-dimensional measurement grids can be treated. The major shortcoming of the method is the need for a relatively fine measurement density. However, with the ongoing development of fine-grid measurement equipment, such as wireless accelerometers and laser vibrometers, the proposed method becomes more and more realistic and promising for the employment in aerospace, civil, and mechanical systems under noisy conditions. References [1] O.S. Salawu, Detection of structural damage through changes in frequency: a review, Eng. Struct. 19 (9) (1997) 718–723, http://dx.doi.org/10.1016/ S0141-0296(96)00149-6. [2] S.W. Doebling, C.R. Farrar, M.B. Prime, D.W. Shevitz, A review of damage identification methods that examine changes in dynamic properties, Shock Vib. 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