1 Introduction

Finding initial cracks and other kinds of incipient damage in structures is the main issue of various industrial applications. As structural damage detection requirements become increasingly strict and rigorous, monitoring or testing of operational structures must be done regularly. In such inspections, non-destructive testing (NDT) is usually carried out. Therefore, these approaches aim to be sensitive to different kinds of structural damage, assess the locations and severities of such losses as early as possible to prevent collapses of structural performance, and plan maintenance operations in advance (Zhu et al. 2011). In recent years, numerous NDT techniques have been developed and further refined to meet the requirements of the current maintenance philosophies (Abdel-Latif 2009; Grandt 2011). Among the variety of non-destructive tests available today, optical methods provide non-destructive, contactless, and structural examination with a relatively high degree of sensitivity to both exterior and interior damage. Shearography is one of the most prominent methods, exhibiting high sensitivity levels thanks to its interferometric NDT technology (Abdel-Latif 2009; Grandt 2011; Lopes et al. 2012). In part, this method has a high level of sensitivity because it does not require any contact. Furthermore, unlike contact measurement systems, this type of method significantly reduces the time and cost of data acquisition (Feng and Feng 2018). Shearography allows the detection of defects and faults in a structure by direct measurement of the gradient of the modal displacement field. Although shearography was developed at the beginning of the 1970s (dos Santos and Lopes 2018), its advantages still make it an effective non-destructive testing method. In addition, shearography is recommended and approved for inspection of aircraft structures by the European Aviation Safety Agency (EASA) and has also been recommended for use by the Federal Aviation Administration (FAA) (Katunin et al. 2021).

Due to the ability to measure structural response in a field environment without contact requirements, the NDT shearographic approach has also been found helpful in the construction industry for the inspection and identification of cracks and delaminations in concrete and steel structures (Bagheri et al. 2019; Jahangir and Esfahani 2020a, b; Jahangir et al. 2021a). Signal processing techniques (Jahangir et al. 2021b) improve the sensitivity of shearography (Zhao et al. 2018) to various forms of structural damage, which is one of the subjects of numerous research studies conducted by the authors. Prior research studies have focused on measuring differences in modal rotation using shearography (Reis Lopes et al. 2011; Lopes et al. 2014) and applying optimal spatial sampling to these measurements (Mininni et al. 2016; Katunin et al. 2018), which increased the sensitivity of the proposed approaches to structural damage. Several case studies in the authors’ research focus on transforming algorithms on wavelets (Araujo Dos Santos et al. 2019; Katunin et al. 2019; Yazdanpanah et al. 2021) that helped to identify in some cases the boundaries of damaged areas. The results of previous studies show that the sensitivity and accuracy of the proposed damage indicator are highly dependent on the selected wavelet family and mother wavelets (Jahangir and Esfahani 2012, 2013; Katunin and Holewik 2013; Seyedi et al. 2015; Yazdanpanah et al. 2020), so it is crucial to make the right choice.

The continuous wavelet transform (CWT) gives a time–frequency representation of a signal. It relies on wavelet basis functions, also known as mother wavelets, that can be chosen based on satisfying the admissibility condition. Any mother wavelet is continuous in both time and frequency domains and is the source of construction of other basis functions. Several research studies have confirmed the effectiveness of CWTs in 1D and 2D damage detection in beams (Gentile and Messina 2003; Douka et al. 2003; Loutridis et al. 2004; Rucka and Wilde 2006), trusses (Mousavi et al. 2021) and plates (Douka et al. 2004; Rucka and Wilde 2006; Huang et al. 2009; Fan and Qiao 2009; Janeliukstis et al. 2017; Yang and Oyadiji 2017; Zhou et al. 2018), respectively. CWTs have been successfully applied in all these studies, regardless of their potential lack of baseline information. It is possible to further improve the detectability of structural damage through a suitable baseline, which may include experimental or numerical data acquisition in an undamaged state, as noted by Solís et al. (2013, 2018). Additionally, Solís et al. (2013, 2018) found a reduction of the boundary effect and reduced importance of noise in the post-processing of baseline and damaged data with wavelets since they are highly sensitive to disturbances (Abdulkareem et al. 2019). The boundary effect reduction was also observed by dos Santos et al. (2021) in the damage identification with differences in modal rotations.

2 Research Significance

Most previous studies in the damage detection field utilized sensors connected to the surface of the structural element to measure the vibration of responses. In this case, any small surface crack at the location of the sensors caused their detachment and stopped the measuring process. To solve this problem, non-contact sensing approaches such as shearography are beneficial, which is the core and novelty of the current study.

In this study, systematic investigations were conducted to detect multiple structural damages in aluminium beams using new methods applied to differences in modal rotations after a 1D CWT is applied. Modal rotations of one undamaged beam and two damaged beams were measured with shearography. As opposed to some previous studies conducted on the detection of damage, including a large reduction in thickness (Rucka and Wilde 2006), smaller damage cases were investigated in the present study. Indeed, one of the aluminium beams experimentally examined has an area with a thickness reduction of 7%. Additionally, the beams have multiple damage scenarios, which are still a challenge to identify because such cases are prone to false positive indications of damage locations (Nayyar et al. 2021). Although in the literature, more wavelets, which were applied to a similar class of problems, can be found, all the available wavelet families in Matlab software are selected to be evaluated in the current study. In this paper, a novel statistical method as a new metric is utilized to select the best wavelet family and related mother wavelet based on the frequency of the number of times the selected mother wavelet can detect the exact location of the damage scenarios as slots. Moreover, a damage threshold was defined based on the wavelet coefficient values to identify the damage locations by ignoring the possible noises induced in the process stage. The measured data and post-processing algorithm clearly demonstrate the effectiveness of shearography and the application of wavelet transform for damage identification. This effectiveness is also a result of the proposed selection of modal rotations that should be post-processed.

It should be mentioned that any mother wavelet has individual geometrical configurations and characteristics, which can be changed by changing the scale value. On the other hand, any damage scenario in a structural element causes a specific change in the inherent features of the structure, such as its natural frequency and mode shape. Therefore, no helpful suggestion can be provided for the users to select the best mother wavelets before conducting the trial and error process.

3 Material and Methods

This section presents the characteristics of the beams and their damage. The shearography system used to measure the modal rotations is also described. The influence of damage is measured by parameters to evaluate the changes in natural frequencies and modes. Finally, two damage indicators based on wavelet transform and respective mother wavelets are also defined.

3.1 Material, Geometric, and Damage Characteristics of Aluminium Beams

The structures analysed are three aluminium beams with 400 mm length, 40 mm width, and 2.85 mm thickness. The first beam (B1) is undamaged, i.e., the cross-sections have a constant area of 40 × 2.85 mm2, whereas the second (B2) and third (B3) beams have two slots created by a milling machine. Table 1 lists the location, depth, width, and thickness reduction of each slot, which can be observed in Fig. 1.

Table 1 Location, depth, width, and thickness reduction of slot 1 (S1) and slot 2 (S2)
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Fig. 1
figure 1

Location and width of slot 1 (S1) and slot 2 (S2)

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3.2 Experimental Set-up for the Measurement of Modal Rotations

The measurement of modal rotations of the three beams is performed by shearography. Shearography is a non-contact, full-field, and high-resolution optical method based on the speckle phenomenon (dos Santos and Lopes 2018). The first step to obtaining the modal rotations is the measurement of natural frequencies by exciting the beams with an impact hammer and acquiring the response with a microphone. The use of the impact hammer and the microphone has the advantage of not adding mass to the beams, which is the case when one uses force transducers and accelerometers. After the input and output signals are sampled, Frequency Response Functions (FRFs) are evaluated. Finally, the natural frequencies are identified from these FRFs by peak analysis.

The experimental set-up for the measurement of the modal rotations of the beams is shown in Fig. 2. In this figure, we can see the speckle effect on the surface of the beam, which is produced by the laser light reflection on the surface. The beams were connected to a frame by two flexible wires glued to each one of the extremities. The damaged beams were placed such that the two slots are facing the opposite side of the inspection surface. To avoid rigid body motions and stabilize the beams, soft foams were placed on the opposite side of the measurement surfaces. The excitation of the beams is accomplished by two loudspeakers, which are placed close to the surfaces. The laser used to illuminate the beams has 1.3 W of power with a wavelength of 532 nm. To spread the well-collimated light source and obtain a homogeneous illumination, the laser passes through a calibrated lens. An in-house built shearography system using a 4-megapixel camera, Dalsa-Falcon 4M30, was used to record the intensity of the interference pattern produced. This image has a spatial resolution of 2300 pixels in the horizontal direction and 225 pixels in the vertical direction, yielding a sampling interval of 0.17 mm. The high spatial resolution of the measurement makes it possible to describe the modal rotation field with greater precision.

Fig. 2
figure 2

Experimental set-up for the measurement of modal rotations

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The shearography system uses stroboscopic laser illumination and a temporal phase modulation technique (dos Santos and Lopes 2012). To freeze the speckle pattern, the stroboscopic laser illumination, produced by an acoustic-optic modulation, and the loudspeaker excitation of the beams are synchronized, thus allowing the measurement of the light phase distributions before and after the acoustic excitation and subsequent the light phase change, i.e., the phase map, by phase subtraction. This phase map is afterward filtered and unwrapped by applying the sine/cosine average filter (Aebischer and Waldner 1999) and the Goldstein unwrapping method (Ghiglia and Pritt 1998), respectively. In the present work, the two-dimensional modal rotations \({\theta }_{i}\left(x,y\right)\) are reduced to one-dimensional profiles \({\theta }_{i}\left(x\right)\), by considering only the measurement at a line with an \(y\) coordinate equal to half the width of the beam. The modal curvatures \({\kappa }_{i}\left(x\right)\) are computed by applying a combined smoothing and differentiation process (Lopes et al. 2007), leading to a less noisy signal, which is the case if we use finite differences to approximate the derivative of the modal rotations (dos Santos and Lopes 2018). Therefore, the phase map of the \(i\)-th mode \(\Delta {\phi }_{i}\left(x,y\right)\) thus obtained is related to the \(i\)-th mode first derivative of the out-of-plane displacement, i.e., the rotation \({\theta }_{i}\left(x,y\right)=\partial {w}_{i}(x,y)/\partial x\) by:

$$\Delta {\phi }_{i}\left(x,y\right)=\frac{4\pi\Delta x}{\lambda }{\theta }_{i}(x,y)$$
(1)

where \(\lambda\) is the wavelength of the laser (532 nm) and \(\Delta x\) is the shearing amount (5 mm). A more comprehensive description of this type of set-ups and post-processing of captured images for measurement of modal rotations can be found in dos Santos and Lopes (2018) and Francis et al. (2010).

3.3 Evaluation of Changes in Natural Frequencies and Modal Rotations

The changes in the natural frequencies are evaluated by computing the relative percentage difference (RPD) between the \(i\)-th natural frequencies of the undamaged beam (B1), \({f}_{i}\), and the two damaged beams (B2 and B3), \(\widetilde{{f}_{i}}\):

$$RPD({f}_{i},\widetilde{{f}_{i}})=\left|\frac{{f}_{i}-\widetilde{{f}_{i}}}{{f}_{i}}\right|\times 100$$
(2)

It should be mentioned that in this study, the bold marks were designated to vectors as the ordinary parameters are single arrays and parameters. On this basis, the Modal Assurance Criterion (MAC) (Maia and Silva 1997) between the vectors of the undamaged modal rotations \({{\varvec{\theta}}}_{i}\) and the damaged modal rotations \({\widetilde{{\varvec{\theta}}}}_{i}\) is defined as:

$$MAC\left({{\varvec{\theta}}}_{i},{\widetilde{{\varvec{\theta}}}}_{i}\right)=\frac{{\left|{{\varvec{\theta}}}_{i}^{{\text{T}}}\boldsymbol{ }{\widetilde{{\varvec{\theta}}}}_{i}\right|}^{2}}{\left|{{\varvec{\theta}}}_{i}^{{\text{T}}}\boldsymbol{ }{{\varvec{\theta}}}_{i}\right|\left|{\widetilde{{\varvec{\theta}}}}_{i}^{{\text{T}}}\boldsymbol{ }{\widetilde{{\varvec{\theta}}}}_{i}\right|} \,{\text{with}} \,0<MAC\left({{\varvec{\theta}}}_{i},{\widetilde{{\varvec{\theta}}}}_{i}\right)\le 1$$
(3)

However, the \(MAC\left({{\varvec{\theta}}}_{i},{\widetilde{{\varvec{\theta}}}}_{i}\right)\) has values that do not differ very much from damage case to damage case. A more sensitive parameter to the changes in modes is the Normalized Modal Difference (NMD) (Maia and Silva 1997; dos Santos et al. 2006):

$$NMD\left({{\varvec{\theta}}}_{i},{\widetilde{{\varvec{\theta}}}}_{i}\right)=\sqrt{\frac{1-MAC\left({{\varvec{\theta}}}_{i},{\widetilde{{\varvec{\theta}}}}_{i}\right)}{MAC\left({{\varvec{\theta}}}_{i},{\widetilde{{\varvec{\theta}}}}_{i}\right)}}\, {\text{with}}\, 0<NMD\left({{\varvec{\theta}}}_{i},{\widetilde{{\varvec{\theta}}}}_{i}\right)<\infty$$
(4)

Since the RPD is a global parameter, which is related to natural frequencies, whereas the NMD is related to modal characteristics and thus can be seen as a parameter describing local changes, a way to describe both the global and local changes (GLC) due to damage can be defined by:

$$GLC\left(({f}_{i},\widetilde{{f}_{i}},{{\varvec{\theta}}}_{i},{\widetilde{{\varvec{\theta}}}}_{i})\right)=RPD({f}_{i},\widetilde{{f}_{i}})\times NMD\left({{\varvec{\theta}}}_{i},{\widetilde{{\varvec{\theta}}}}_{i}\right)$$
(5)

3.4 Damage Indicators and Mother Wavelets

The proposed damage identification process requires the measurement of modal rotations of an undamaged beam (B1), to be used as reference data, and the equivalent modal rotations of the two damaged beams (B2 and B3). This leads to the definition of damage indicators based on differences in modal rotations and differences in modal curvatures of the undamaged and damaged beams. A wavelet transform \(WT\), using distinct mother wavelets, is applied to these differences, such that

$${\varvec{M}}{\varvec{R}}\&{\varvec{D}}{\varvec{U}}(i)=WT\left({\widetilde{{\varvec{\theta}}}}_{i}-{{\varvec{\theta}}}_{i}\right)$$
(6)
$${\varvec{M}}{\varvec{C}}\&{\varvec{D}}{\varvec{U}}(i)=WT\left({{\widetilde{{\varvec{\kappa}}}}_{i}-{\varvec{\kappa}}}_{i}\right)$$
(7)

for the damage indicators based on modal rotations \({\varvec{M}}{\varvec{R}}\&{\varvec{D}}{\varvec{U}}(i)\) and modal curvatures \({\varvec{M}}{\varvec{C}}\&{\varvec{D}}{\varvec{U}}(i)\), respectively, of the \(i\)-th mode.

Time–frequency analysis has been mentioned in different historical contexts. In 1807, Joseph Fourier proposed that every static function could be represented as an infinite number of sine waves and cosine waves of different frequencies (Hanteh et al. 2021). The proposed transform, nowadays called the Fourier transform, was the first to transfer time domain information to the frequency domain and find out the prominent frequencies in the signal. This idea was expanded to non-static functions with a changing statistical distribution over time. It was gradually understood that information could be centred on time and frequency when signals are broken up into separate components that are not pure sine waves. There are various families that can generate and modify signals over both time and frequency domains, such that any signal can be converted into a set of local basic functions, also called mother wavelets (Hanteh and Rezaifar 2021). An example of a wavelet transform application would be an analysis that shows how the frequencies change within a short time period. Basically, this transform produces the frequency components for different resolutions by varying the mother wavelets. These mother wavelets can be used to analyse each local property of a signal (Jahangir and Esfahani 2013). Wavelet transforms are sensitive to singularities in signals, and thus they are often able to detect abrupt changes in mode shapes due to the presence of damage.

In the present work, the post-processing of differences in modal rotations and differences in modal curvatures is carried out by applying a continuous wavelet analysis, which is defined below. A signal \(x\left(t\right)\in \left[-\infty ,+\infty \right]\) can be analysed by 1D continuous wavelet according to:

$$CWT\left(s,b\right)=\frac{1}{\sqrt{\left|s\right|}}{\int }_{-\infty }^{+\infty }x(t){\Psi }^{*}\left(\frac{t-b}{s}\right)dt$$
(8)

The translation and scaling parameters are \(b\) and \(s\), \(t\) is the time change defined for each mother wavelet function and \(CWT\left(s,b\right)\) are the wavelet coefficients. \({\Psi }^{*}\left(s, b,t\right)\) is the mother wavelet, being its complex conjugate given by:

$${\Psi }^{*}\left(s,b, t\right)=\frac{1}{\sqrt{\left|s\right|}}\Psi \left(\frac{t-b}{s}\right)$$
(9)

The differences in modal rotations and the differences in modal curvatures were post-processed, according to Eqs. (6) and (7), respectively, with 14 distinct wavelet families and a total of 84 mother wavelets. The families and respective mother wavelets are described in Matlab software (2014). Some of the mother wavelets are depicted in Fig. 3.

Fig. 3
figure 3

Plots of some mother wavelets and respective scaling functions

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4 Results and Discussion

This section contains a preliminary analysis of the differences in natural frequencies and the changes in modal rotations. Damage identifications obtained by post-processing differences in modal curvatures and modal rotations are also presented and discussed. In view of the results, the adequacy and ability of each mother wavelet, each wavelet family, and each mode to successfully identify the damage is verified. Finally, a study on the boundary effect and noise on the identifications as a function of the mother wavelet applied is also presented.

4.1 Differences in Natural Frequencies and Changes in Modal Rotations

Table 2 lists the natural frequencies of the three beams, the relative percentage difference (RPD) between the natural frequencies of the undamaged beam B1 and the damaged beams B2 and B3, and the normalized modal difference (NMD) of the modal rotations. It can be seen that beam B2 has lower values of these parameters than beam B3, thus indicating that the reduction in thickness at the slots of beam B2 is smaller than the one of beam B3.

Table 2 Measured natural frequencies of beams 1, 2, and 3 (B1, B2, and B3), RPD and NMD of B2 and B3
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4.2 Damage Identification Results

4.2.1 Wavelet Families, Mother Wavelets, Modes, and Modal Characteristic

The ability of the distinct mother wavelets to identify both slots in beams B2 and B3 with the post-processing of differences in modal curvatures, as defined by Eq. (7), can be seen in Table 5 of Appendix A, where the peak values (PV) of the wavelet coefficients in both slots of the two beams are listed as a function of the wavelet mothers and modes. This Table is the result of an automated procedure that looks for the two highest PV and verifies if they are located at the positions of the slots. If not, the identification of a slot is taken as unsuccessful, and an em dash is assigned to it. It is clear that Shannon, frequency B-spline, and complex Morlet wavelets are more efficient in identifying the slots. Indeed, as reported in Figs. 4, 5, and 6, the PV of wavelet coefficients for shan1-0.1, fbsp2-1-0.1, and cmor1-0.1 are located at the site of the slots and have larger values than the ones given by other families, and no boundary effect is observed. It is worth noting that with the same wavelet and mode, the PV is higher in beam B3 than in beam B2. For example, with the shan1-0.1 wavelet and mode M3 we obtain a maximum of 2.1040 in beam B2 for slot 2 and 11.317 for this same slot.

Fig. 4
figure 4

Absolute values of wavelet coefficients with shan1-0.1 wavelet applied to modal curvatures M1, M2, and M3

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Fig. 5
figure 5

Absolute values of wavelet coefficients with fbsp2-1-0.1 wavelet applied to modal curvatures M1, M2, and M3

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Fig. 6
figure 6

Absolute values of wavelet coefficients with cmor1-0.1 wavelet applied to modal curvatures M1, M2, and M3

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In this paper, 14 wavelet families including 84 mother wavelets were considered to detect the location of single and double slots in the tested aluminium beams. After applying each wavelet family and the predefined damage thresholds, the number of times that wavelet family was successful in identifying the exact location of the slot (s) was counted to determine the best wavelet family among others. Table 3 lists the total number of identified slots by applying all the mother wavelets within each family and the average number of slots identified by one wavelet of a given family. This average is evaluated by computing the ratio of the total number of identified slots within each family to the total number of mother wavelets in the family, which is presented in the second column of Table 3. It can be seen that the Shannon and frequency B-spline families are the ones with the largest average number of identified slots by one wavelet, followed by the complex Morlet family. Although the total number of identified slots with the biorthogonal and reverse biorthogonal families is high, i.e., 32 and 28, respectively, the average number of identified slots is around two since 15 mother wavelets were tested for each one of these families.

Table 3 Total number of identified slots by each family and average number of slots identified by one wavelet of a given family
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It is also noteworthy that with mode M3 both slots are identified, and the absolute values of the wavelet coefficients are high and approximately the same. In Table 4 we can see the values of the GLC parameter for the three modes and notice the large values of this parameter for mode M3, which indicates that the most changed mode is the third and, thus, more sensitive to damage. Indeed, in beam B2 we have a GLC of mode M3, which is one order of magnitude larger than the GLC of the other two modes, and the number of identified slots is 43 with M3 and 6 for modes M1 and M2, thus indicating a good correlation between GLC of a mode and its ability to identify damage.

Table 4 Values of GLC and total number of identified slots in B2 and B3 by all mother wavelets
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Another modal characteristic that can be post-processed besides the modal curvature is modal rotation. However, the post-processing of difference in modal rotations, based on Eq. (6), with shan1-0.1 and fbsp2-1–0.1 and cmor1-0.1 yields less satisfactory identifications of the slots than the post-processing of differences in modal curvatures, since, as seen in Fig. 7, the results exhibit boundary effects. We can also observe in Fig. 7 that for mode M3 the results show humps. It is thus advisable to read the results obtained with modal rotations data with care and as a complement to the ones given by post-processing the differences in modal curvatures.

Fig. 7
figure 7

Absolute values of wavelet coefficients with shan1-0.1 wavelet applied to modal rotations M1, M2, and M3

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4.2.2 Boundary Effect and Noise

A problem often found in the analysis of signals with wavelets, in particular in the application of wavelet transform to damage identification, is the existence of boundary effects in the results. Another issue is that some wavelets yield very noisy results, such that the noise masks the damage influence, and no clear peak at the damage locations can be observed. The existence of these undesirable characteristics is more noticeable when one tries to identify small damage and thus, only results with beam B2 are reported here.

The picking up of the wavelets with results where the boundary effect is more and less noticeable is carried out by evaluating the ratio of the maximum value at the left and right extremities to the maximum value in the other zones of the beam. The mother wavelet where we observe the largest boundary effect in the results is gauss4, as reported in Fig. 8a. It is perfectly clear that with this wavelet we are not able to identify the slots. Figure 8b presents the results with the shan1-0.5, which is the wavelet where we have no boundary effect, and thus it is possible to unmistakably localize the slots.

Fig. 8
figure 8

Absolute values of wavelet coefficients with, a gaus4 and b shan1-0.5 wavelets applied to modal curvatures M1, M2, and M3 of beam B2

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The mother wavelets with more and less noisy results were found by counting the number of fluctuations (peaks) that surpassed the threshold value (Sarmadi et al. 2021). In this search, the values of the coefficients in the extremities were not taken into account. For the modal curvature post-processing, the noisiest results are obtained with the coif4, mexh and cgau4 mother wavelets, whereas the sym6 and cmor1-0.1 mother wavelets yield less noisy identifications. It should be mentioned that the sensitivity of mother wavelets and the influence of noise can be related directly to their mathematical properties as the noisiest mother wavelets such as coif4, mexh and cgau4 mother wavelets resulted in more noisy coefficients and smoothest mother wavelets such as sym6 and cmor1-0.1 mother wavelets resulted in less noisy detections. Figure 9 shows the results of the application of mexh and cmor1-0.1 wavelets to modal curvatures. We can observe that the mexh wavelet does not allow the correct identification of the slots. The results with the post-processing of modal rotations with gaus4 present the highest level of noise, whereas the shan1-0.5 (fbsp1-1-0.5), shan1-0.1 and fbsp2-1-0.1 mother wavelets give the least noisy identifications. Indeed, as can be observed in Fig. 10, contrary to the shan1-0.5 wavelet, which has clear peaks at the positions of the slots, the gaus4 mother wavelet yields only noise, making it impossible to locate the slots.

Fig. 9
figure 9

Absolute values of wavelet coefficients with mexh and cmor1-0.1 wavelets applied to modal curvatures M1, M2, and M3 of beam B2

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Fig. 10
figure 10

Absolute values of wavelet coefficients with gaus4 and shan1-0.5 wavelets applied to modal rotations M1, M2, and M3 of beam B2

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5 Conclusions

The selection of the best mother wavelets and mode for successful multiple damage identification in aluminium beams is reported in this paper. The modal rotations are experimentally measured with shearography. The differences in modal rotations and differences in modal curvatures are post-processed by continuous wavelet transform with 14 wavelet families and 84 mother wavelets. A novel parameter for the selection of the best mode, based on the measurement of global and local modal changes due to damage, is proved to be effective.

In general, families with complex mother wavelets yield the best damage identification results, namely the Shannon, frequency B-spline, and complex Morlet families, since the average number of identified slots by each wavelet belonging to these families is the highest. Furthermore, this type of wavelets, contrary to Gaussian wavelets, present the lowest noise in the results and are not sensitive to boundary effects. Among the mother wavelets belonging to the aforementioned families, the highest values of the absolute values of coefficients of shan1-0.1, fbsp2-1–0.1, and cmor1-0.1 mother wavelets are observed at the slot locations. It is noteworthy that these three wavelets have the same bandwidth parameter and centre frequency, 1 and 0.1, respectively. In brief, the reported damage identification results clearly show that the combination of shearography measurements and a proper selection of mother wavelets can lead to successful multiple damage identifications in beams. This success can be further improved by selecting the most changed mode according to the proposed selection parameter.