Keywords

1 Introduction

Analysis of damaged structure, from nano size to macro size, is an interesting topic for researchers [1,2,3,4]. Generally, damaged structures can be analyzed using two approaches: forward analysis and inverse analysis. In forward analysis, the damage is modeled, and modal characteristics of the structure are determined. In inverse analysis, the damage is determined using the modal characteristics of the structure [5, 6]. For many decades, damage detection techniques have been investigated by many researchers [7,8,9,10,11]. The researchers have been used different strategies for damage detection of the structures.

Some strategies are optimization-based methods and an objective function is set to be minimized or maximized in order to localize damage [12,13,14,15]. Dinh-Cong et al. [16] employed the Jaya optimization algorithm to localize damages in plates. Gomes et al. [17] used the SFOA algorithm to detect damages in laminate composites. Mishra et al. [18] proposed the ALO algorithm for identifying damages via vibration data. Maity et al. [19] suggested the GA algorithm for detecting damages of structures via shifts in natural frequencies. They demonstrated that the GA algorithm could localize damage with high precision. Also, Alexandrino et al. [20] proposed a multiobjective optimization plan to identify damages to the plates. This optimization method revealed that using the multiobjective method, artificial neural networks (ANNs) and fuzzy inference systems can act as a proper means to identify damages. In [21], the effectiveness of the PSO and ACO algorithms to localize damages in 3D trusses was considered. The objective function suggested was established using mode shapes and corresponding natural frequencies. Gerist et al. [22] employed the IC optimization Algorithm to localize damage in engineering structures based on a novel objective function. In [23], the DE optimization algorithm was used to determine the damage's location and severity in structures.

Also, many techniques have been created in the inverse analysis in the research. One of the most contemporary inverse methods is the artificial neural network (ANN) for localizing damage’s location and severity [24]. ANN is a superior means for localizing damages in the structures. Investigators usually utilized modal characteristics such as natural frequency and mode shape as the input of ANNs to identify the level of damage in various structures [25,26,27,28].

Another method of damage detection in structures is signal processing-based methods. In these methods, signals are decomposed to detect the damage’s location and level using the existed information in the signal acquired for the damaged structure [29]. Wavelet transform is a signal processing-based method. In this method, a signal is transformed into sub-signals to show the damaged signal's discontinuity [30]. The singularities and discontinuities detected by wavelet transform in signals indicate damages in a specific location or time [31]. Yang et al. utilized two-dimensional wavelets for detecting damage of laminate composites [32]. Cao et al. proposed a new approach named integrated wavelet transforms to detect damages on composite beams. This method combined the continuous wavelet transforms (CWTs) and stationary wavelet transforms (SWTs) to create a powerful wavelet-based damage detection model [33]. Sohn et al. presented a wavelet-based approach for detecting delaminations [34]. Mitra et al. introduced an element-based model combined with wavelet transform to detect damages on composite beams [35]. The reviewed investigations on wavelet-based damage detection methods show that they are the simple and effective methods for damage detection in various structures since they only need signals. In this paper, Damage detection in engineering structures by wavelet transforms for various structures are reviewed.

2 One Dimensional Continuous Wavelet Transforms (1D-CWT)

The signals obtained for cables, bars, beams, and all beam-like structures are one-dimensional; therefore, a one-dimensional wavelet transform is applied to decompose these signals. Wavelet transform can be continuous or discrete. Assume the one-dimensional signal A(t) having a subtle discontinuity or damage. In order to identify this discontinuity, it is required to transform its values into another coordinate system using a wavelet function called ψ(t). The following expression indicates a family of wavelets [23]:

For a signal \(\mathrm{f}(\mathrm{x})\), , family of wavelets in real space at the Hilbert space (\(\uppsi \left(\mathrm{x}\right)\upepsilon {\mathrm{L}}^{2}({\mathbb{R}}))\) is defined as follows:

$${\uppsi \left(\mathrm{x}\right)}_{\mathrm{u},\mathrm{s}}=\frac{1}{\sqrt{\mathrm{s}}}\uppsi \left(\frac{\mathrm{x}-\mathrm{u}}{\mathrm{s}}\right)$$
(1)

where \(\mathrm{x}\) indicates the space variable related to the original signal \(\mathrm{f}(\mathrm{x})\), \(\mathrm{u}\) is the shifting factor, and \(\mathrm{s}\) denotes the scaling factor. The continuous wavelet coefficients \(\mathrm{C}\left(\mathrm{u},\mathrm{s}\right)\) can be calculated as follows [36]:

$$\mathrm{C}\left(\mathrm{u},\mathrm{s}\right)=\langle \mathrm{f},{\uppsi \left(\mathrm{x}\right)}_{\mathrm{u},\mathrm{s}} \rangle =\frac{1}{\sqrt{\mathrm{s}}}{\int }_{-\infty }^{+\infty }\mathrm{f}(\mathrm{x})\uppsi \left(\frac{\mathrm{x}-\mathrm{u}}{\mathrm{s}}\right)\mathrm{dx}$$
(2)
$$\mathrm{C}\left(\mathrm{u},\mathrm{s}\right)=\frac{{\mathrm{s}}^{\mathrm{n}}}{\sqrt{\mathrm{s}}}{\int }_{-\infty }^{+\infty }\mathrm{f}\left(\mathrm{x}\right)\frac{{\mathrm{d}}^{\mathrm{n}}}{{\mathrm{dx}}^{\mathrm{n}}}\uptheta \left(\frac{-\left(\mathrm{u}-\mathrm{x}\right)}{\mathrm{s}}\right)\mathrm{dx}=\frac{{\mathrm{s}}^{\mathrm{n}}}{\sqrt{\mathrm{s}}}\frac{{\mathrm{d}}^{\mathrm{n}}}{{\mathrm{du}}^{\mathrm{n}}}\mathrm{f}*\uptheta \left(\frac{-\mathrm{u}}{\mathrm{s}}\right)={\mathrm{s}}^{\mathrm{n}}\frac{{\mathrm{d}}^{\mathrm{n}}}{{\mathrm{du}}^{\mathrm{n}}}(\mathrm{f}*{\uptheta }_{\mathrm{s}})(\mathrm{u})$$
(3)

Figure 1 shows an example of a one-dimensional signal obtained from a beam structure and its continuous wavelet transform.

Fig. 1.
figure 1

An example of a one-dimensional signal obtained from a beam structure and its continuous wavelet transform.

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In the continuous wavelet transform results, the brightest position horizontal axis of the wavelet coefficients diagram shows the location of the damage. Therefore, the damage is located at X = 17.

Janeliukstis et al. [37] conducted an experimental study for damage localization in beam-like structures based on spatial mode shape signals. Montanari et al. [38] used continuous wavelet transform for detecting cracks in beam structures. Rucka et al. [39] investigated the application of one-dimensional wavelet transforms for damage detection of beams.

Figure 1 shows the continuous wavelet transform created from the signal obtained from the finite element method; however, the number of data points in experimental data is high. Figures 2 and 3 indicate the continuous wavelet transform created from the experimental velocity signals of an intact structure and a damaged structure, respectively. Figures 4 and 5 show the continuous wavelet transform created from the experimental displacement signals of an intact structure and a damaged structure, respectively.

Fig. 2.
figure 2

Experimental velocity signal for an intact structure and its continuous wavelet transform

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Fig. 3.
figure 3

Experimental velocity signal for a damaged structure and its continuous wavelet transform

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Fig. 4.
figure 4

Experimental displacement signal for an intact structure and its continuous wavelet transform

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Fig. 5.
figure 5

Experimental displacement signal for a damaged structure and its continuous wavelet transform

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3 One Dimensional Discrete Wavelet Transforms (1D-DWT)

Another type of wavelet transform is one-dimensional discrete wavelet transform. The 1D-DWT for a signal \(\mathrm{f}(\mathrm{x})\) is defined as follows [40]:

$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{A}}_{\mathrm{j}}\left(\mathrm{x}\right)+\sum_{\mathrm{j}<\mathrm{J}}{\mathrm{D}}_{\mathrm{j}}(\mathrm{x})$$
(4)

where \({\mathrm{A}}_{\mathrm{j}}\) are approximation signals at level j, \({\mathrm{D}}_{\mathrm{j}}\) show detail signals at level j.

The approximation signals at level j are calculated as follows:

$${\mathrm{A}}_{\mathrm{j}}\left(\mathrm{x}\right)=\sum_{\mathrm{k}= -\infty }^{+\infty }{\mathrm{cA}}_{\mathrm{j},\mathrm{k}}{\upphi }_{\mathrm{j},\mathrm{k}}(\mathrm{x})$$
(5)

where \({\mathrm{cA}}_{\mathrm{j},\mathrm{k}}\) indicate approximation coefficients at level j. \({\upphi }_{\mathrm{j},\mathrm{k}}(\mathrm{x})\) are scaling functions at level j.

The detail signals at level j are expressed as follows:

$${\mathrm{D}}_{\mathrm{j}}\left(\mathrm{x}\right)=\sum_{\mathrm{k\epsilon Z}}{\mathrm{cD}}_{\mathrm{j},\mathrm{k}} {\uppsi }_{\mathrm{j},\mathrm{k}}(\mathrm{x})$$
(6)

where \({\mathrm{cD}}_{\mathrm{j},\mathrm{k}}\) denote detail coefficients at level j. \({\uppsi }_{\mathrm{j},\mathrm{k}}(\mathrm{x})\) indicate wavelet functions.

Figure 2 shows an example of a one-dimensional signal obtained from a delaminated composite beam structure and its discrete wavelet transform. Note that damage causes a local jump in the signal; therefore, damage can be detected in the detail signal. As shown in Fig. 2, two jumps are observed in the detail signal. These two jumps are the boundaries of the damage delamination in a laminated composite beam.

Fig. 6.
figure 6

An example of a one-dimensional signal obtained from a beam structure and its discrete wavelet transform.

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Along with this, there are many types of research; for instance, Yang et al. [41] identified a crack in aluminum beams using discrete wavelet transform. The experimental findings showed that the suggested discrete wavelet transform could effectively determine the damage locations and be applied to more complicated structures.

Figure 6 shows the discrete wavelet transform created from the signal obtained from the finite element method; however, as mentioned, the number of data points in experimental data is high. Figures 7 and 8 indicate the discrete wavelet transform created from the experimental velocity signals of an intact structure and a damaged structure, respectively. Figures 9 and 10 show the discrete wavelet transform created from the experimental displacement signals of an intact structure and a damaged structure, respectively.

Fig. 7.
figure 7

Experimental velocity signal for an intact structure and its discrete wavelet transform

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Fig. 8.
figure 8

Experimental velocity signal for a damaged structure and its discrete wavelet transform

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Fig. 9.
figure 9

Experimental displacement signal for an intact structure and its discrete wavelet transform

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Fig. 10.
figure 10

Experimental displacement signal for a damaged structure and its discrete wavelet transform

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4 Discussion

Damage detection using one-dimensional wavelet transform is a powerful and proper method based on the structure’s signal. There is no need for having other modal characteristics such as natural frequencies. Signals from structures such as cables, bars, and beams are one-dimensional signals. Thus, these structures can be detected by the one-dimensional wavelet transforms. The one-dimensional continuous wavelet transforms (1D-CWT) and one-dimensional discrete wavelet transforms are two main types of one-dimensional wavelet transforms. These two different methods decompose the original signals using two different ways. The 1D-CWT shifts and scales the original signal at various continuous levels, and the 1D-DWT decomposes the original signals into approximation and detail sub-signals at various discrete levels. Both methods are powerful tools for detecting damages and faults in one-dimensional signals.

5 Conclusions

In this study, a review of damage detection methods in structures is performed. The emphasis of the research is on the introduction of one-dimensional damage detection methods using wavelet transform. Two common types of one-dimensional wavelet transform are introduced. Their formulation and how to evaluate the results of these two continuous and discrete types of damage detection based on one-dimensional wavelet transform are discussed. It has been proven that damage detection by the one-dimensional wavelet transform method can be a powerful tool for locating damage in cable structures, rods, isotropic beams, and laminated composite beams.