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Multiple damage detection in laminated composite beams by data fusion of Teager energy operator-wavelet transform mode shapes
Journal Pre-proofs Multiple damage detection in laminated composite beams by data fusion of Teager energy operator-wavelet transform mode shapes Ganggang Sha, Maciej Radzienski, Rohan Soman, Maosen Cao, Wieslaw Ostachowicz, Wei Xu PII: DOI: Reference:
S0263-8223(19)31514-4 https://doi.org/10.1016/j.compstruct.2019.111798 COST 111798
To appear in:
Composite Structures
Received Date: Accepted Date:
26 April 2019 9 December 2019
Please cite this article as: Sha, G., Radzienski, M., Soman, R., Cao, M., Ostachowicz, W., Xu, W., Multiple damage detection in laminated composite beams by data fusion of Teager energy operator-wavelet transform mode shapes, Composite Structures (2019), doi: https://doi.org/10.1016/j.compstruct.2019.111798
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© 2019 Published by Elsevier Ltd.
Multiple damage detection in laminated composite beams by data fusion of Teager energy operator-wavelet transform mode shapes Ganggang Shaa,b , Maciej Radzienskib , Rohan Somanb , Maosen Caoa,∗, Wieslaw Ostachowiczb , Wei Xua a
b
Department of Engineering Mechanics, Hohai University, Nanjing 210098, China Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Gdansk 80-231, Poland
Abstract Mode shapes have been widely used for structural damage detection. The basic premise of this method is that damage occurring in a structure causes singularities in mode shapes, which in turn reveal damage. However, singularities induced by small damage are insignificant and susceptible to noise. To address these deficiencies, the Teager energy operator (TEO) together with wavelet transform (WT) is introduced to process mode shapes, producing TEO-WT mode shapes. It is noted that each TEO-WT mode shape has its specific sensitivity to damage at a certain location, which means that multiple damage may not be identified simultaneously from a single TEO-WT mode shape. Thus data fusion of multiple TEO-WT mode shapes is used to create an overall TEOWT mode shape. A damage indicator (DI) is then obtained by integrating the overall TEO-WT mode shape in the scale domain. The DI features distinctive capability to suppress noise, intensify singularities caused by damage, and improve the reliability of damage detection. The efficacy of the method is verified numerically and then validated experimentally on cracked laminated composite beams. The numerical and experimental results demonstrate the capability of the method to detect multiple damage in laminated composite beams under noisy conditions. Keywords: Multiple damage detection, Laminated composite beam, Mode shape, Wavelet transform, Teager energy operator, Data fusion
1. Introduction Composite laminates have been extensively utilized in civil, mechanical, military, and aerospace industries owing to their high specific stiffness and strength, low weight, corrosion resistance, and non-conductivity [1]. These composite laminates are exposed to various types of damage. Damage detection in composite laminates is important to avoid structural failure and has been a research focus during the last few decades [2–5]. Vibration-based methods have been widely studied, as they are non-destructive, inexpensive, and expedient [6–8]. In particular, mode shapes have been widely used for damage detection as they carry local information about the damage and have high ∗
corresponding author Email address: [email protected] (Maosen Cao)
Preprint submitted to Composite Structures
November 7, 2019
resistance to environmental effects, such as temperature [9, 10]. The premise of the technique is that damage, such as surface cracks in a composite laminate, causes singularities in mode shapes at damage positions and in turn these singularities can indicate damage positions [11]. Generally, damage-induced singularities in mode shapes are easily masked by noise and visible only for relatively large damage [12]. For small damage, the singularities in mode shapes, which are less pronounced and even invisible, can be characterized using wavelet transform (WT), because singularities in mode shapes lead to substantial variations of wavelet coefficients in the neighborhood of damage [13–15]. Moreover, WT mode shapes exhibit the intrinsic multi-scale property of wavelets. That multi-scale property entails decay of the noise-related wavelet coefficients with an increase in the scale. This is highly advantageous for characterizing damage by tolerating noise. Thus, WT has been widely used in mode shape-based damage detection methods and many different wavelets have been exploited, such as Mexican hat wavelets [16], Gabor wavelets [17], Haar wavelets [18], and Debauchies wavelets [19]. Zhu et al. [20] used WT mode shape to detect crack in functionally graded beams. The crack locations were determined from the damage index defined according to the position of the wavelet coefficient modulus maxima in the scale domain. An intensity factor is calculated by the Lipschitz regulation of the wavelet coefficient to evaluate damage severity. Abdulkareem et al. [21] adopted WT mode shape difference to solve the border distortion problem in detecting damage in plate structures. Xu et al. [22] proposed a delamination detection method by characterizing singularities in mode shapes of carbon fiber-reinforced polymer (CFRP) laminated plates. Zhou et al. [23] used a two-dimensional WT to process mode shape curvature for damage detection in composite sandwich panels. Despite their popularity in characterizing damage, WT mode shapes have a noticeable drawback: they inherit greater global fluctuation trend from mode shapes and this global fluctuation trend easily masks damage features. It has been demonstrated that local singularities can be intensified while the global fluctuation trend is removed when a signal is processed by the Teager energy operator (TEO) [24]. The TEO was originally used as an effective nonlinear tool for handling the varying characteristics of speech signals from an energy viewpoint [25]. The TEO was introduced to the field of damage detection as described in the authors’ previous work [26, 27]. To take advantage of the merits of WT mode shapes in suppressing noise, while counteracting their disadvantage in portraying damage, the TEO can be adopted to enhance the WT mode shapes, producing TEO-WT mode shapes. Ideally, a single TEO-WT mode shape can characterize damage by singular peaks. However, when damage occurs near the node of one mode shape, causing an insignificant singularity to the mode shape, the TEO-WT mode shape usually fails to identify that damage [28]. Moreover, measurement error in a certain mode shape can also impair the capacity of the TEO-WT mode shape to portray damage. All these factors result in failure to identify multiple damage simultaneously from a single TEO-WT mode shape [29]. As an alternative to a single TEO-WT mode shape, the concept of data fusion of multiple TEO-WT mode shapes is introduced here to create an overall TEO-WT mode shape. A damage indicator (DI) is then obtained by integrating the overall TEO-WT mode shape in the scale domain. The DI inherits distinct capabilities from WT, TEO, and data fusion to suppress noise, intensify singularities caused by damage, and improve the reliability of damage detection. With these distinctive features, the DI can be used to identify multiple damage in laminated composite beams under noisy conditions. The rest of the paper is organized as follows. Section 2 presents the basic steps of the formu2
lation of the DI. Section 3 provides numerical proof of the method using a finite element model (FEM). Section 4 describes experimental validation of the applicability of this method by detecting multiple cracks in a CFRP beam. Conclusions are presented in Section 5. 2. Method Formulation 2.1. WT mode shape For a measured mode shape φ(x), its WT is defined as Z +∞ 1 ∗ x−u W(u, s) = √ φ(x)ψ dx, s s −∞
(1)
where W(u, s) is the wavelet coefficient of the wavelet ψu,s (x), u and s indicate translation and scale parameters, respectively, '∗ ' denotes a complex conjugation. Equation 1 provides a definition of the WT mode shape. In this study, a second-order Gaussian wavelet ('gaus2' in Matlab, Fig. 1) is selected as the analyzing wavelet. It is seen from Eq. 1 that the WT mode shape is defined as the convolution of φ(x) with wavelet ψu,s (x). For a finite-length signal φ(x), when the convolution operation is executed near its border, the wavelet window extends outside the interval of the mode shape, so that the wavelet coefficients achieve extremely high value near the border (border distortion) and the real damage features are consequently masked [30, 31]. To handle this border distortion problem, two extrapolation methods are considered: (i) linear extrapolation, which can extrapolate mode shape linearly; (ii) isomorphism extrapolation [32], which extrapolates mode shape either through a mirror symmetry when its first derivative tends to zero or a polar-like symmetry when its second derivative tends to zero. The smallest length of the extrapolated signal is estimated as a half-width of the wavelet window at the highest scale. A detailed description of removing border distortion is provided in Section 3.
Figure 1: Second-order Gaussian wavelet.
3
2.2. TEO-WT mode shape Let W[n, s] be the discrete form of the WT mode shape. Implementation of the TEO on W[n, s] is expressed as [26] T [n, s] = (W[n, s])2 − W[n − 1, s]W[n + 1, s]. (2) In Eq. 2, T [n, s] is defined as the TEO-WT mode shape (discrete form). 2.3. Overall TEO-WT mode shape by data fusion Data fusion methods such as Bayesian theory, Dempster-Shafer evidence theory, and averaging have been successfully used in structural damage identification [33–35]. In this study, the simplest data fusion method of averaging is adopted as it can achieve sound results. For a clearer presentation of singularities, the moduli of the WT mode shape and TEO-WT mode shape are used and denoted as |W(u, s)| and |T (u, s)|, respectively. To perform direct averaging between different modes, each mode of |W(u, s)| and |T (u, s)| is normalized as e s)| = |W(u,
|W(u, s)| , max(|W(u, s)|)
|T (u, s)| . max(|T (u, s)|) Considering m mode shapes, the overall TEO-WT mode shape is defined as |Te(u, s)| =
(3) (4)
m
1X e |T¯ (u, s)| = |T i (u, s)|, m i=1
(5)
where subscript i is the mode number. 2.4. DI Although damage-induced singularities can be identified in the overall TEO-WT mode shape at fine scales, the selection of these fine scales is user-dependent. To remove the user interface associated with the selection of fine scales, the overall TEO-WT mode shape is integrated in the scale domain as Z s1 P(u) = |T¯ (u, s)|ds, (6) s0
where s0 and s1 are the lower and upper limits of the scale parameter. Z-score normalization is then applied to filter out lower amplitude peaks that do not indicate damage: Z-score =
P − mean(P) , SD(P)
where SD is standard deviation. The final DI is then defined as Z-score, i f Z-score ≥ 0 DI = 0, i f Z-score < 0 Damage locations are pinpointed by the peaks of DI. 4
(7)
(8)
3. Numerical demonstration 3.1. Modeling of a laminated composite beam A laminated composite beam consisting of five layers is modeled using the FE software ANSYS. The dimensions of the beam are length (L) 300 mm, width (W) 10 mm, and height (H) 1.5 mm. The material properties employed in the FE analysis are given in Table 1. Table 1: Material properties employed in FE analysis.
E11 (GPa) 134
E22 (GPa) 10.3
G12 (GPa) 5
ν12 0.33
ρ (kg/m3 ) 1480
A 3D layered solid element (Solid 185) is employed and the layer information of the beam is shown in Fig. 2. The beam model is meshed using 300 elements along the length, 10 elements along the width, and one element for each layer of the laminate. Three through-width cracks are created by removing the length-wise 61 st , 151 st , and 218th elements of the first layer. The dimensionless coordinates of the three cracks are ζ1 = 0.202, ζ2 = 0.502, and ζ3 = 0.725 as shown in Fig. 3. Four cases of boundary conditions are considered: simply-supported (SS), free-free (FF), clamped-free (CF), and clamped-clamped (CC). The first six mode shapes of each case are obtained via modal analysis. These mode shapes are displayed in Fig. 4, in which damage-induced singularities are invisible.
Figure 2: Layer information of the beam model.
Figure 3: Beam model with three through-width cracks.
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Figure 4: First six mode shapes of the SS, FF, CF, and CC beams.
3.2. Damage detection results For the first mode shapes of the SS, FF, CF, and CC beams, the associated WT mode shapes and TEO-WT mode shapes are shown in Fig. 5 and Fig. 6, respectively. It is clear from Fig. 5 that border distortion occurs in WT mode shapes, especially for the SS, FF, and CF beams. According to Eq. 2, border distortion in WT mode shapes is inevitably introduced into TEO-WT mode shapes (Fig. 6) making damage-induced singularities disappear for the SS, FF, and CF beams. To weaken border distortion, the first mode shapes are extended outside their original parts by linear (Fig. 7a) and isomorphism extrapolation (Fig. 7b), respectively. The improved TEO-WT mode shapes are displayed in Fig. 8. For the SS and FF beams, border distortion in TEO-WT mode shapes is completely eliminated by both linear and isomorphism extrapolation. For the CF beam, border distortion at the free end is removed by these two methods, but there are still relatively high values at the clamped end, which can only be partially reduced by isomorphism extrapolation. For the CC beam, high values at the clamped ends can only be partially reduced by isomorphism extrapolation. In accordance with the foregoing analysis, the isomorphism extrapolation is adopted in the following section. 6
(a)
(b)
(c)
(d)
Figure 5: The first WT mode shapes of the SS (a), FF (b), CF (c), and CC (d) beams.
(a)
(b)
(c)
(d)
Figure 6: The first TEO-WT mode shapes of the SS (a), FF (b), CF (c), and CC (d) beams.
7
(a)
(b)
Figure 7: The first mode shapes of the SS, FF, CF, and CC beams extended by linear (a) and isomorphism (b) extrapolation.
8
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 8: The first TEO-WT mode shapes of the SS, FF, CF, and CC beams improved by linear (a, c, e, g), and isomorphism (b, d, f, h) extrapolation.
The first six WT mode shapes of the SS beam are displayed as contour plots in Fig. 9. The 9
WT is implemented for scales 1-10 with step 0.1. From Fig. 9 it is evident that damage-induced singularities are seriously disturbed by the greater global fluctuation of the WT mode shapes. In contrast, damage-induced singularities are intensified while the global fluctuation trend is removed in TEO-WT mode shapes, as shown in Fig. 10. As seen from Fig. 10, a single TEO-WT mode shape cannot always account for three cracks at the same time. This requires the use of data fusion to produce an overall TEO-WT mode shape (Fig. 11a) and the DI (Fig. 11b), in which the actual damage positions are marked with red dashed lines. The results show that the DI displays three prominent peaks, explicitly pinpointing damage locations. The overall TEO-WT mode shapes and the DIs of the FF and CF beams are plotted in Fig. 12, giving good damage detection results.The DI of the CF beam produces high values at the clamped end, which do not indicate damage because border distortion at the clamped end cannot be entirely removed. It is seen from Fig. 12c that when the scale is lower, the region of high values is narrower. To reduce the high value region at the clamped end, the wavelet transform scales are reset as 1-5 with step 0.1 for the CF and CC beams. The associated DIs are plotted in Fig. 13. It is clear that the high value regions at the clamped ends of the CF and CC beams are largely reduced and the resolution of damage detection results increases.
Figure 9: First six WT mode shapes of the SS beam.
10
Figure 10: First six TEO-WT mode shapes of the SS beam..
(a)
(b)
Figure 11: Overall TEO-WT mode shape (a) and DI (b) of the SS beam
3.3. Noise immunity test To investigate the performance of the DI in noisy environments, white Gaussian noise is added to the mode shapes of the SS beam using the Matlab function 'awgn' . Noise level is defined by signal-to-noise ratio (SNR) and thus a higher value of SNR means a lower noise level. SNRs across a broad range (80 dB, 70 dB, 60 dB, and 50 dB) are considered. The corresponding DIs 11
(a)
(b)
(c)
(d)
Figure 12: Overall TEO-WT mode shape (a) and DI (b) of the FF beam; overall TEO-WT mode shape (c) and DI (d) of the CF beam.
(a)
(b)
Figure 13: DIs of the CF (a) and CC (b) beams for scales 1-5 with step 0.1.
are shown in Fig. 14. At the noise levels of 80 and 70 dB, the DIs appear slightly noisier than the results without noise in Fig. 11b, and the crack-induced peaks still clearly indicate three cracks in the beam. At the noise level of 60 dB, noise interference becomes more intense, but the peaks are still identified. At the noise level of 50 dB, noise interference becomes severe, and the peaks are just distinguishable. These results illustrate the capability of the method for locating damage under noisy conditions.
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Figure 14: DIs of the SS beam at different noise levels.
4. Experimental validation 4.1. Experimental description An experiment on a cantilever CFRP beam is taken from the authors’ previous work [36], in which only the fourth and fifth mode shapes were used. The test beam consists of five layers [0◦ /45◦ /90◦ /-45◦ /0◦ ], each 0.3 mm thick. The geometrical dimensions of the beam are length 490 mm, width 10 mm, and thickness 1.5 mm. Two damage scenarios, Scenarios I and II, are considered. In Scenario I, three through-width cracks of about 0.5 mm in depth (being cut through till the depth between the first and second interfaces) are fabricated. In Scenario II, these three cracks are extended to 0.9 mm (being cut through three layers of laminates) in depth. The three cracks are located at positions 103 mm, 211 mm, and 355 mm from the clamped end. The first six mode shapes are measured using an electromechanical shaker as an exciter and a scanning laser Doppler vibrometer (SLDV, Polytec PSV-400) as a sensor. The SLDV scanning area spans 486 mm from the clamped end, and thus the dimensionless coordinates of the first, second, and third cracks in the scanning length are ζ1 = 0.212, ζ2 = 0.434, and ζ3 = 0.730, respectively (Fig. 15). A sketch of the experimental setup is shown in Fig. 16. 4.2. Experimental results The first six mode shapes of Scenario I are displayed individually in Fig. 17, where the damage features are insignificant. The WT is implemented on these mode shapes for scales 1-25 with step 13
Figure 15: CFRP beam with three cracks.
Figure 16: Sketch of the experimental setup.
0.1. Fig. 18 presents the first six WT mode shapes improved by isomorphism extrapolation for Scenario I. It is clear that damage-induced singular peaks are largely overwhelmed by the global fluctuation trend of WT mode shapes. In comparison, the singular peaks in TEO-WT mode shapes (Fig. 19) are much more predominant. However, different modes of TEO-WT mode shapes have different capability of portraying damage. For higher reliability of damage detection, the first six TEO-WT mode shapes are fused to form an overall TEO-WT mode shape, shown in Fig. 20. The magnitudes of the overall TEO-WT mode shape at scales 5, 10, 15, and 20 are individually plotted in Fig. 21. It is seen that the results of the crack detection are scale-dependent. To remove the user interface in the selection of fine scales, the overall TEO-WT mode shape is integrated in the scale domain, producing the DI in Fig. 22a. The identification results are in good agreement with the actual crack positions marked with red dashed lines. Similarly, the DI improved by linear extrapolation for Scenario I are shown in Fig. 22b, demonstrating good damage detection results. For Scenario II with deeper cracks, the DIs improved by both isomorphism and linear extrapolation are shown in Fig. 23, where three peaks stand out obviously without high values in the clamped end, clearly pinpointing three cracks. 14
Moreover, the DIs for Scenarios I and II improved by both isomorphism and linear extrapolation are plotted together in Fig. 24. It is observed from Fig. 24 that the DIs for Scenario II are less disturbed by border distortion than those for Scenario I. The DIs for Scenario II are higher than those for Scenario I and can be used for the prediction of damage severity qualitatively.
Figure 17: First six mode shapes for Scenario I.
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Figure 18: First six WT mode shapes for Scenario I.
Figure 19: First six TEO-WT mode shapes for Scenario I.
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Figure 20: Overall TEO-WT mode shape for Scenario I.
Figure 21: Magnitudes of overall TEO-WT mode shape at different scales for Scenario I.
(a)
(b)
Figure 22: DIs improved by isomorphism (a) and linear (b) extrapolation for Scenario I.
17
(a)
(b)
Figure 23: DIs improved by isomorphism (a) and linear (b) extrapolation for Scenario II.
(a)
(b)
Figure 24: Comparison between DIs for Scenarios I and II: (a) isomorphism and (b) linear extrapolation.
5. Conclusions A novel DI formulated by data fusion of multiple TEO-WT mode shapes is proposed for damage detection in laminated composite beams. From WT, TEO, and data fusion, the DI inherits distinct capabilities of suppressing noise, intensifying singularities caused by damage, and improving the reliability of damage detection. The performance of the proposed method is numerically demonstrated by detection of damage in a laminated composite beam with three cracks under different boundary conditions. The applicability of the method is experimentally validated on a cracked CFRP beam. The numerical and experimental results show that the proposed method can pinpoint the locations of multiple damage in noisy environments, without a healthy reference of the structure, and without knowledge of either the geometrical parameters or the material properties of the structure. Aside from cracks, delamination is another common type of damage occurring in laminated composite beams. Delamination affects mode shapes, and hence the proposed method is theoretically potential to detect location and size of delamination. In practice, it is foreseeable that detection of the delamination interface might pose additional challenges due to the complex mechanism that determines how the delamination in different interfaces affects mode shapes. We intend to improve the method for delamination detection in our future work.
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Acknowledgements The authors are grateful for the partial support from the National Natural Science Foundation of China (11772115), the National Key Research and Development Program of China (2018YFF0214700), the Scientific and Technological Transform Project of Jiangsu Transport Department (2017Y02), and a scholarship from the China Scholarship Council (201706710025). Particularly, the authors acknowledge the early contribution of Dr. Wei Xu to formulating the concept of the TEO-WT mode shape for structural damage detection. Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. References [1] K. Wang, B. Young, S. T. Smith, Mechanical properties of pultruded carbon fibre-reinforced polymer (CFRP) plates at elevated temperatures, Engineering Structures 33 (7) (2011) 2154–2161. [2] G. F. Gomes, Y. A. D. Mendéz, P. d. S. L. Alexandrino, S. S. da Cunha, A. C. Ancelotti, The use of intelligent computational tools for damage detection and identification with an emphasis on composites-a review, Composite Structures 196 (2018) 44–54. [3] E. Manoach, J. Warminski, L. Kloda, A. Teter, Numerical and experimental studies on vibration based methods for detection of damage in composite beams, Composite Structures 170 (2017) 26–39. [4] S. Sikdar, P. Kudela, M. Radzienski, A. Kundu, W. Ostachowicz, Online detection of barely visible low-speed impact damage in 3d-core sandwich composite structure, Composite Structures 185 (2018) 646–655. [5] Z. Zhang, C. Zhan, K. Shankar, E. V. Morozov, H. K. Singh, T. Ray, Sensitivity analysis of inverse algorithms for damage detection in composites, Composite Structures 176 (2017) 844–859. [6] M. Cao, G. Sha, Y. Gao, W. Ostachowicz, Structural damage identification using damping: a compendium of uses and features, Smart Materials and Structures 26 (4) (2017) 043001. [7] B. Li, Z. Li, J. Zhou, L. Ye, E. Li, Damage localization in composite lattice truss core sandwich structures based on vibration characteristics, Composite Structures 126 (2015) 34–51. [8] G. Sha, M. Radzie´nski, M. Cao, W. Ostachowicz, A novel method for single and multiple damage detection in beams using relative natural frequency changes, Mechanical Systems and Signal Processing 132 (2019) 335– 352. [9] W. Fan, P. Qiao, Vibration-based damage identification methods: a review and comparative study, Structural health monitoring 10 (1) (2011) 83–111. [10] J. Ciambella, F. Vestroni, The use of modal curvatures for damage localization in beam-type structures, Journal of Sound and Vibration 340 (2015) 126–137. [11] K. Roy, S. Ray-Chaudhuri, Fundamental mode shape and its derivatives in structural damage localization, Journal of Sound and Vibration 332 (21) (2013) 5584–5593. [12] H. Hu, B.-T. Wang, C.-H. Lee, J.-S. Su, Damage detection of surface cracks in composite laminates using modal analysis and strain energy method, Composite Structures 74 (4) (2006) 399–405. [13] C.-C. Chang, L.-W. Chen, Detection of the location and size of cracks in the multiple cracked beam by spatial wavelet based approach, Mechanical Systems and Signal Processing 19 (1) (2005) 139–155. [14] Q. Wang, X. Deng, Damage detection with spatial wavelets, International journal of solids and structures 36 (23) (1999) 3443–3468. [15] V. Shahsavari, L. Chouinard, J. Bastien, Wavelet-based analysis of mode shapes for statistical detection and localization of damage in beams using likelihood ratio test, Engineering Structures 132 (2017) 494–507.
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S0263-8223(19)31514-4 https://doi.org/10.1016/j.compstruct.2019.111798 COST 111798
To appear in:
Composite Structures
Received Date: Accepted Date:
26 April 2019 9 December 2019
Please cite this article as: Sha, G., Radzienski, M., Soman, R., Cao, M., Ostachowicz, W., Xu, W., Multiple damage detection in laminated composite beams by data fusion of Teager energy operator-wavelet transform mode shapes, Composite Structures (2019), doi: https://doi.org/10.1016/j.compstruct.2019.111798
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Ltd.
Multiple damage detection in laminated composite beams by data fusion of Teager energy operator-wavelet transform mode shapes Ganggang Shaa,b , Maciej Radzienskib , Rohan Somanb , Maosen Caoa,∗, Wieslaw Ostachowiczb , Wei Xua a
b
Department of Engineering Mechanics, Hohai University, Nanjing 210098, China Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Gdansk 80-231, Poland
Abstract Mode shapes have been widely used for structural damage detection. The basic premise of this method is that damage occurring in a structure causes singularities in mode shapes, which in turn reveal damage. However, singularities induced by small damage are insignificant and susceptible to noise. To address these deficiencies, the Teager energy operator (TEO) together with wavelet transform (WT) is introduced to process mode shapes, producing TEO-WT mode shapes. It is noted that each TEO-WT mode shape has its specific sensitivity to damage at a certain location, which means that multiple damage may not be identified simultaneously from a single TEO-WT mode shape. Thus data fusion of multiple TEO-WT mode shapes is used to create an overall TEOWT mode shape. A damage indicator (DI) is then obtained by integrating the overall TEO-WT mode shape in the scale domain. The DI features distinctive capability to suppress noise, intensify singularities caused by damage, and improve the reliability of damage detection. The efficacy of the method is verified numerically and then validated experimentally on cracked laminated composite beams. The numerical and experimental results demonstrate the capability of the method to detect multiple damage in laminated composite beams under noisy conditions. Keywords: Multiple damage detection, Laminated composite beam, Mode shape, Wavelet transform, Teager energy operator, Data fusion
1. Introduction Composite laminates have been extensively utilized in civil, mechanical, military, and aerospace industries owing to their high specific stiffness and strength, low weight, corrosion resistance, and non-conductivity [1]. These composite laminates are exposed to various types of damage. Damage detection in composite laminates is important to avoid structural failure and has been a research focus during the last few decades [2–5]. Vibration-based methods have been widely studied, as they are non-destructive, inexpensive, and expedient [6–8]. In particular, mode shapes have been widely used for damage detection as they carry local information about the damage and have high ∗
corresponding author Email address: [email protected] (Maosen Cao)
Preprint submitted to Composite Structures
November 7, 2019
resistance to environmental effects, such as temperature [9, 10]. The premise of the technique is that damage, such as surface cracks in a composite laminate, causes singularities in mode shapes at damage positions and in turn these singularities can indicate damage positions [11]. Generally, damage-induced singularities in mode shapes are easily masked by noise and visible only for relatively large damage [12]. For small damage, the singularities in mode shapes, which are less pronounced and even invisible, can be characterized using wavelet transform (WT), because singularities in mode shapes lead to substantial variations of wavelet coefficients in the neighborhood of damage [13–15]. Moreover, WT mode shapes exhibit the intrinsic multi-scale property of wavelets. That multi-scale property entails decay of the noise-related wavelet coefficients with an increase in the scale. This is highly advantageous for characterizing damage by tolerating noise. Thus, WT has been widely used in mode shape-based damage detection methods and many different wavelets have been exploited, such as Mexican hat wavelets [16], Gabor wavelets [17], Haar wavelets [18], and Debauchies wavelets [19]. Zhu et al. [20] used WT mode shape to detect crack in functionally graded beams. The crack locations were determined from the damage index defined according to the position of the wavelet coefficient modulus maxima in the scale domain. An intensity factor is calculated by the Lipschitz regulation of the wavelet coefficient to evaluate damage severity. Abdulkareem et al. [21] adopted WT mode shape difference to solve the border distortion problem in detecting damage in plate structures. Xu et al. [22] proposed a delamination detection method by characterizing singularities in mode shapes of carbon fiber-reinforced polymer (CFRP) laminated plates. Zhou et al. [23] used a two-dimensional WT to process mode shape curvature for damage detection in composite sandwich panels. Despite their popularity in characterizing damage, WT mode shapes have a noticeable drawback: they inherit greater global fluctuation trend from mode shapes and this global fluctuation trend easily masks damage features. It has been demonstrated that local singularities can be intensified while the global fluctuation trend is removed when a signal is processed by the Teager energy operator (TEO) [24]. The TEO was originally used as an effective nonlinear tool for handling the varying characteristics of speech signals from an energy viewpoint [25]. The TEO was introduced to the field of damage detection as described in the authors’ previous work [26, 27]. To take advantage of the merits of WT mode shapes in suppressing noise, while counteracting their disadvantage in portraying damage, the TEO can be adopted to enhance the WT mode shapes, producing TEO-WT mode shapes. Ideally, a single TEO-WT mode shape can characterize damage by singular peaks. However, when damage occurs near the node of one mode shape, causing an insignificant singularity to the mode shape, the TEO-WT mode shape usually fails to identify that damage [28]. Moreover, measurement error in a certain mode shape can also impair the capacity of the TEO-WT mode shape to portray damage. All these factors result in failure to identify multiple damage simultaneously from a single TEO-WT mode shape [29]. As an alternative to a single TEO-WT mode shape, the concept of data fusion of multiple TEO-WT mode shapes is introduced here to create an overall TEO-WT mode shape. A damage indicator (DI) is then obtained by integrating the overall TEO-WT mode shape in the scale domain. The DI inherits distinct capabilities from WT, TEO, and data fusion to suppress noise, intensify singularities caused by damage, and improve the reliability of damage detection. With these distinctive features, the DI can be used to identify multiple damage in laminated composite beams under noisy conditions. The rest of the paper is organized as follows. Section 2 presents the basic steps of the formu2
lation of the DI. Section 3 provides numerical proof of the method using a finite element model (FEM). Section 4 describes experimental validation of the applicability of this method by detecting multiple cracks in a CFRP beam. Conclusions are presented in Section 5. 2. Method Formulation 2.1. WT mode shape For a measured mode shape φ(x), its WT is defined as Z +∞ 1 ∗ x−u W(u, s) = √ φ(x)ψ dx, s s −∞
(1)
where W(u, s) is the wavelet coefficient of the wavelet ψu,s (x), u and s indicate translation and scale parameters, respectively, '∗ ' denotes a complex conjugation. Equation 1 provides a definition of the WT mode shape. In this study, a second-order Gaussian wavelet ('gaus2' in Matlab, Fig. 1) is selected as the analyzing wavelet. It is seen from Eq. 1 that the WT mode shape is defined as the convolution of φ(x) with wavelet ψu,s (x). For a finite-length signal φ(x), when the convolution operation is executed near its border, the wavelet window extends outside the interval of the mode shape, so that the wavelet coefficients achieve extremely high value near the border (border distortion) and the real damage features are consequently masked [30, 31]. To handle this border distortion problem, two extrapolation methods are considered: (i) linear extrapolation, which can extrapolate mode shape linearly; (ii) isomorphism extrapolation [32], which extrapolates mode shape either through a mirror symmetry when its first derivative tends to zero or a polar-like symmetry when its second derivative tends to zero. The smallest length of the extrapolated signal is estimated as a half-width of the wavelet window at the highest scale. A detailed description of removing border distortion is provided in Section 3.
Figure 1: Second-order Gaussian wavelet.
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2.2. TEO-WT mode shape Let W[n, s] be the discrete form of the WT mode shape. Implementation of the TEO on W[n, s] is expressed as [26] T [n, s] = (W[n, s])2 − W[n − 1, s]W[n + 1, s]. (2) In Eq. 2, T [n, s] is defined as the TEO-WT mode shape (discrete form). 2.3. Overall TEO-WT mode shape by data fusion Data fusion methods such as Bayesian theory, Dempster-Shafer evidence theory, and averaging have been successfully used in structural damage identification [33–35]. In this study, the simplest data fusion method of averaging is adopted as it can achieve sound results. For a clearer presentation of singularities, the moduli of the WT mode shape and TEO-WT mode shape are used and denoted as |W(u, s)| and |T (u, s)|, respectively. To perform direct averaging between different modes, each mode of |W(u, s)| and |T (u, s)| is normalized as e s)| = |W(u,
|W(u, s)| , max(|W(u, s)|)
|T (u, s)| . max(|T (u, s)|) Considering m mode shapes, the overall TEO-WT mode shape is defined as |Te(u, s)| =
(3) (4)
m
1X e |T¯ (u, s)| = |T i (u, s)|, m i=1
(5)
where subscript i is the mode number. 2.4. DI Although damage-induced singularities can be identified in the overall TEO-WT mode shape at fine scales, the selection of these fine scales is user-dependent. To remove the user interface associated with the selection of fine scales, the overall TEO-WT mode shape is integrated in the scale domain as Z s1 P(u) = |T¯ (u, s)|ds, (6) s0
where s0 and s1 are the lower and upper limits of the scale parameter. Z-score normalization is then applied to filter out lower amplitude peaks that do not indicate damage: Z-score =
P − mean(P) , SD(P)
where SD is standard deviation. The final DI is then defined as Z-score, i f Z-score ≥ 0 DI = 0, i f Z-score < 0 Damage locations are pinpointed by the peaks of DI. 4
(7)
(8)
3. Numerical demonstration 3.1. Modeling of a laminated composite beam A laminated composite beam consisting of five layers is modeled using the FE software ANSYS. The dimensions of the beam are length (L) 300 mm, width (W) 10 mm, and height (H) 1.5 mm. The material properties employed in the FE analysis are given in Table 1. Table 1: Material properties employed in FE analysis.
E11 (GPa) 134
E22 (GPa) 10.3
G12 (GPa) 5
ν12 0.33
ρ (kg/m3 ) 1480
A 3D layered solid element (Solid 185) is employed and the layer information of the beam is shown in Fig. 2. The beam model is meshed using 300 elements along the length, 10 elements along the width, and one element for each layer of the laminate. Three through-width cracks are created by removing the length-wise 61 st , 151 st , and 218th elements of the first layer. The dimensionless coordinates of the three cracks are ζ1 = 0.202, ζ2 = 0.502, and ζ3 = 0.725 as shown in Fig. 3. Four cases of boundary conditions are considered: simply-supported (SS), free-free (FF), clamped-free (CF), and clamped-clamped (CC). The first six mode shapes of each case are obtained via modal analysis. These mode shapes are displayed in Fig. 4, in which damage-induced singularities are invisible.
Figure 2: Layer information of the beam model.
Figure 3: Beam model with three through-width cracks.
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Figure 4: First six mode shapes of the SS, FF, CF, and CC beams.
3.2. Damage detection results For the first mode shapes of the SS, FF, CF, and CC beams, the associated WT mode shapes and TEO-WT mode shapes are shown in Fig. 5 and Fig. 6, respectively. It is clear from Fig. 5 that border distortion occurs in WT mode shapes, especially for the SS, FF, and CF beams. According to Eq. 2, border distortion in WT mode shapes is inevitably introduced into TEO-WT mode shapes (Fig. 6) making damage-induced singularities disappear for the SS, FF, and CF beams. To weaken border distortion, the first mode shapes are extended outside their original parts by linear (Fig. 7a) and isomorphism extrapolation (Fig. 7b), respectively. The improved TEO-WT mode shapes are displayed in Fig. 8. For the SS and FF beams, border distortion in TEO-WT mode shapes is completely eliminated by both linear and isomorphism extrapolation. For the CF beam, border distortion at the free end is removed by these two methods, but there are still relatively high values at the clamped end, which can only be partially reduced by isomorphism extrapolation. For the CC beam, high values at the clamped ends can only be partially reduced by isomorphism extrapolation. In accordance with the foregoing analysis, the isomorphism extrapolation is adopted in the following section. 6
(a)
(b)
(c)
(d)
Figure 5: The first WT mode shapes of the SS (a), FF (b), CF (c), and CC (d) beams.
(a)
(b)
(c)
(d)
Figure 6: The first TEO-WT mode shapes of the SS (a), FF (b), CF (c), and CC (d) beams.
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(a)
(b)
Figure 7: The first mode shapes of the SS, FF, CF, and CC beams extended by linear (a) and isomorphism (b) extrapolation.
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(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 8: The first TEO-WT mode shapes of the SS, FF, CF, and CC beams improved by linear (a, c, e, g), and isomorphism (b, d, f, h) extrapolation.
The first six WT mode shapes of the SS beam are displayed as contour plots in Fig. 9. The 9
WT is implemented for scales 1-10 with step 0.1. From Fig. 9 it is evident that damage-induced singularities are seriously disturbed by the greater global fluctuation of the WT mode shapes. In contrast, damage-induced singularities are intensified while the global fluctuation trend is removed in TEO-WT mode shapes, as shown in Fig. 10. As seen from Fig. 10, a single TEO-WT mode shape cannot always account for three cracks at the same time. This requires the use of data fusion to produce an overall TEO-WT mode shape (Fig. 11a) and the DI (Fig. 11b), in which the actual damage positions are marked with red dashed lines. The results show that the DI displays three prominent peaks, explicitly pinpointing damage locations. The overall TEO-WT mode shapes and the DIs of the FF and CF beams are plotted in Fig. 12, giving good damage detection results.The DI of the CF beam produces high values at the clamped end, which do not indicate damage because border distortion at the clamped end cannot be entirely removed. It is seen from Fig. 12c that when the scale is lower, the region of high values is narrower. To reduce the high value region at the clamped end, the wavelet transform scales are reset as 1-5 with step 0.1 for the CF and CC beams. The associated DIs are plotted in Fig. 13. It is clear that the high value regions at the clamped ends of the CF and CC beams are largely reduced and the resolution of damage detection results increases.
Figure 9: First six WT mode shapes of the SS beam.
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Figure 10: First six TEO-WT mode shapes of the SS beam..
(a)
(b)
Figure 11: Overall TEO-WT mode shape (a) and DI (b) of the SS beam
3.3. Noise immunity test To investigate the performance of the DI in noisy environments, white Gaussian noise is added to the mode shapes of the SS beam using the Matlab function 'awgn' . Noise level is defined by signal-to-noise ratio (SNR) and thus a higher value of SNR means a lower noise level. SNRs across a broad range (80 dB, 70 dB, 60 dB, and 50 dB) are considered. The corresponding DIs 11
(a)
(b)
(c)
(d)
Figure 12: Overall TEO-WT mode shape (a) and DI (b) of the FF beam; overall TEO-WT mode shape (c) and DI (d) of the CF beam.
(a)
(b)
Figure 13: DIs of the CF (a) and CC (b) beams for scales 1-5 with step 0.1.
are shown in Fig. 14. At the noise levels of 80 and 70 dB, the DIs appear slightly noisier than the results without noise in Fig. 11b, and the crack-induced peaks still clearly indicate three cracks in the beam. At the noise level of 60 dB, noise interference becomes more intense, but the peaks are still identified. At the noise level of 50 dB, noise interference becomes severe, and the peaks are just distinguishable. These results illustrate the capability of the method for locating damage under noisy conditions.
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Figure 14: DIs of the SS beam at different noise levels.
4. Experimental validation 4.1. Experimental description An experiment on a cantilever CFRP beam is taken from the authors’ previous work [36], in which only the fourth and fifth mode shapes were used. The test beam consists of five layers [0◦ /45◦ /90◦ /-45◦ /0◦ ], each 0.3 mm thick. The geometrical dimensions of the beam are length 490 mm, width 10 mm, and thickness 1.5 mm. Two damage scenarios, Scenarios I and II, are considered. In Scenario I, three through-width cracks of about 0.5 mm in depth (being cut through till the depth between the first and second interfaces) are fabricated. In Scenario II, these three cracks are extended to 0.9 mm (being cut through three layers of laminates) in depth. The three cracks are located at positions 103 mm, 211 mm, and 355 mm from the clamped end. The first six mode shapes are measured using an electromechanical shaker as an exciter and a scanning laser Doppler vibrometer (SLDV, Polytec PSV-400) as a sensor. The SLDV scanning area spans 486 mm from the clamped end, and thus the dimensionless coordinates of the first, second, and third cracks in the scanning length are ζ1 = 0.212, ζ2 = 0.434, and ζ3 = 0.730, respectively (Fig. 15). A sketch of the experimental setup is shown in Fig. 16. 4.2. Experimental results The first six mode shapes of Scenario I are displayed individually in Fig. 17, where the damage features are insignificant. The WT is implemented on these mode shapes for scales 1-25 with step 13
Figure 15: CFRP beam with three cracks.
Figure 16: Sketch of the experimental setup.
0.1. Fig. 18 presents the first six WT mode shapes improved by isomorphism extrapolation for Scenario I. It is clear that damage-induced singular peaks are largely overwhelmed by the global fluctuation trend of WT mode shapes. In comparison, the singular peaks in TEO-WT mode shapes (Fig. 19) are much more predominant. However, different modes of TEO-WT mode shapes have different capability of portraying damage. For higher reliability of damage detection, the first six TEO-WT mode shapes are fused to form an overall TEO-WT mode shape, shown in Fig. 20. The magnitudes of the overall TEO-WT mode shape at scales 5, 10, 15, and 20 are individually plotted in Fig. 21. It is seen that the results of the crack detection are scale-dependent. To remove the user interface in the selection of fine scales, the overall TEO-WT mode shape is integrated in the scale domain, producing the DI in Fig. 22a. The identification results are in good agreement with the actual crack positions marked with red dashed lines. Similarly, the DI improved by linear extrapolation for Scenario I are shown in Fig. 22b, demonstrating good damage detection results. For Scenario II with deeper cracks, the DIs improved by both isomorphism and linear extrapolation are shown in Fig. 23, where three peaks stand out obviously without high values in the clamped end, clearly pinpointing three cracks. 14
Moreover, the DIs for Scenarios I and II improved by both isomorphism and linear extrapolation are plotted together in Fig. 24. It is observed from Fig. 24 that the DIs for Scenario II are less disturbed by border distortion than those for Scenario I. The DIs for Scenario II are higher than those for Scenario I and can be used for the prediction of damage severity qualitatively.
Figure 17: First six mode shapes for Scenario I.
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Figure 18: First six WT mode shapes for Scenario I.
Figure 19: First six TEO-WT mode shapes for Scenario I.
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Figure 20: Overall TEO-WT mode shape for Scenario I.
Figure 21: Magnitudes of overall TEO-WT mode shape at different scales for Scenario I.
(a)
(b)
Figure 22: DIs improved by isomorphism (a) and linear (b) extrapolation for Scenario I.
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Figure 23: DIs improved by isomorphism (a) and linear (b) extrapolation for Scenario II.
(a)
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Figure 24: Comparison between DIs for Scenarios I and II: (a) isomorphism and (b) linear extrapolation.
5. Conclusions A novel DI formulated by data fusion of multiple TEO-WT mode shapes is proposed for damage detection in laminated composite beams. From WT, TEO, and data fusion, the DI inherits distinct capabilities of suppressing noise, intensifying singularities caused by damage, and improving the reliability of damage detection. The performance of the proposed method is numerically demonstrated by detection of damage in a laminated composite beam with three cracks under different boundary conditions. The applicability of the method is experimentally validated on a cracked CFRP beam. The numerical and experimental results show that the proposed method can pinpoint the locations of multiple damage in noisy environments, without a healthy reference of the structure, and without knowledge of either the geometrical parameters or the material properties of the structure. Aside from cracks, delamination is another common type of damage occurring in laminated composite beams. Delamination affects mode shapes, and hence the proposed method is theoretically potential to detect location and size of delamination. In practice, it is foreseeable that detection of the delamination interface might pose additional challenges due to the complex mechanism that determines how the delamination in different interfaces affects mode shapes. We intend to improve the method for delamination detection in our future work.
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Acknowledgements The authors are grateful for the partial support from the National Natural Science Foundation of China (11772115), the National Key Research and Development Program of China (2018YFF0214700), the Scientific and Technological Transform Project of Jiangsu Transport Department (2017Y02), and a scholarship from the China Scholarship Council (201706710025). Particularly, the authors acknowledge the early contribution of Dr. Wei Xu to formulating the concept of the TEO-WT mode shape for structural damage detection. Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. References [1] K. Wang, B. Young, S. T. Smith, Mechanical properties of pultruded carbon fibre-reinforced polymer (CFRP) plates at elevated temperatures, Engineering Structures 33 (7) (2011) 2154–2161. [2] G. F. Gomes, Y. A. D. Mendéz, P. d. S. L. Alexandrino, S. S. da Cunha, A. C. Ancelotti, The use of intelligent computational tools for damage detection and identification with an emphasis on composites-a review, Composite Structures 196 (2018) 44–54. [3] E. Manoach, J. Warminski, L. Kloda, A. Teter, Numerical and experimental studies on vibration based methods for detection of damage in composite beams, Composite Structures 170 (2017) 26–39. [4] S. Sikdar, P. Kudela, M. Radzienski, A. Kundu, W. Ostachowicz, Online detection of barely visible low-speed impact damage in 3d-core sandwich composite structure, Composite Structures 185 (2018) 646–655. [5] Z. Zhang, C. Zhan, K. Shankar, E. V. Morozov, H. K. Singh, T. Ray, Sensitivity analysis of inverse algorithms for damage detection in composites, Composite Structures 176 (2017) 844–859. [6] M. Cao, G. Sha, Y. Gao, W. Ostachowicz, Structural damage identification using damping: a compendium of uses and features, Smart Materials and Structures 26 (4) (2017) 043001. [7] B. Li, Z. Li, J. Zhou, L. Ye, E. Li, Damage localization in composite lattice truss core sandwich structures based on vibration characteristics, Composite Structures 126 (2015) 34–51. [8] G. Sha, M. Radzie´nski, M. Cao, W. Ostachowicz, A novel method for single and multiple damage detection in beams using relative natural frequency changes, Mechanical Systems and Signal Processing 132 (2019) 335– 352. [9] W. Fan, P. Qiao, Vibration-based damage identification methods: a review and comparative study, Structural health monitoring 10 (1) (2011) 83–111. [10] J. Ciambella, F. Vestroni, The use of modal curvatures for damage localization in beam-type structures, Journal of Sound and Vibration 340 (2015) 126–137. [11] K. Roy, S. Ray-Chaudhuri, Fundamental mode shape and its derivatives in structural damage localization, Journal of Sound and Vibration 332 (21) (2013) 5584–5593. [12] H. Hu, B.-T. Wang, C.-H. Lee, J.-S. Su, Damage detection of surface cracks in composite laminates using modal analysis and strain energy method, Composite Structures 74 (4) (2006) 399–405. [13] C.-C. Chang, L.-W. Chen, Detection of the location and size of cracks in the multiple cracked beam by spatial wavelet based approach, Mechanical Systems and Signal Processing 19 (1) (2005) 139–155. [14] Q. Wang, X. Deng, Damage detection with spatial wavelets, International journal of solids and structures 36 (23) (1999) 3443–3468. [15] V. Shahsavari, L. Chouinard, J. Bastien, Wavelet-based analysis of mode shapes for statistical detection and localization of damage in beams using likelihood ratio test, Engineering Structures 132 (2017) 494–507.
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