Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 49 (2014) 24–38

0888-32http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/ymssp

Localization of damage with speckle shearography and higherorder spatial derivatives

H. Lopes a, F. Ferreira a, J.V. Araújo dos Santos b,n, P. Moreno-García c

a DEM/ISEP, Instituto Politécnico do Porto, Rua Dr. António Bernardino de Almeida, 431, 4200-072 Porto, Portugalb IDMEC/IST, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugalc INEGI, Instituto de Engenharia Mecânica e Gestão Industrial, Campus da FEUP, Rua Dr. Roberto Frias, 400, 4200-465 Porto, Portugal

a r t i c l e i n f o

Article history:Received 14 August 2012Received in revised form20 December 2013Accepted 27 December 2013Available online 18 January 2014

Keywords:Speckle shearographyStroboscopic laser illuminationPulsed laser illuminationDamage localizationModal rotationsHigher order spatial derivatives

70/$ - see front matter & 2014 Elsevier Ltd.x.doi.org/10.1016/j.ymssp.2013.12.016

esponding author. Tel.: þ351 218419463; faail address: viriato@ist.utl.pt (J.V. Araújo dos

a b s t r a c t

Two speckle shearography systems are described in this paper. The first is based onstroboscopic laser illumination and temporal phase modulation, whereas the secondsystem relies on double pulse laser illumination and spatial phase modulation. Thesesystems are applied to measure the phase maps of modal rotation fields of a damagedlaminated composite plate. In order to decrease the propagation of noise, a newdifferentiation methodology is presented. It relies on the differentiation of the measuredphase maps before they are post-processed. This leads to an improvement in thelocalization of damage. It was found that the fourth order spatial derivative of modeshapes also presents better damage localizations, in particular with the phase mapsmeasured by the first shearography system.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The use of composite materials in lightweight structural applications, such as those developed in the automotive andaeronautical industries, has seen a huge increase in recent decades. These materials present types of defects and damagemechanisms different from those of metals. The lack of effective and global non-destructive inspection techniquesmotivated the development of new methodologies based on the vibrational characteristics of structures. Due to thecomplexity of the problem and the difficulty in finding a robust solution, different approaches have been proposed [1–14].This large number of approaches indicates that a universal method applicable to all kind of structures and damage types isnot available [15]. Nevertheless, methods of damage localization based on the analysis of perturbations or discontinuities inmodal curvatures or strain fields are the most well established and applied. Pandey et al. [1] proposed the use of differencesbetween curvature mode shapes of damaged and undamaged beams. Ratcliffe [3] proposed the use of polynomial functions,fitted with the data of the Laplacian operator, which is similar to beam curvatures, and applied it to the damaged modes,instead of the undamaged ones. Sampaio et al. [4] expanded the use of curvatures of mode shapes to a desired frequencyspectrum by using frequency response functions (FRF). The FRF curvatures have also been used by Maia et al. [14], but takinginto account the number of times that each sensor has a maximum. All these works are applied to the analysis of isotropicbeams. Lestari et al. [9] expanded the method of curvature mode shape differences to carbon/epoxy beams. Guan andKarbhari [10] presented an improvement to the computation of curvatures for sparse measurements, based on the use of a

All rights reserved.

x: þ351 218417915.Santos).

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–38 25

polynomial depending on vertical displacements and rotations. This polynomial can be differentiated twice to obtain thecurvature. Methods based on higher order derivatives have also been developed in recent years. Ismail and Abdul Razak [16]proposed the use of the ratio between the fourth order derivative of a mode shape and the mode shape itself as a damageindicator for Euler–Bernoulli beams. A similar method along with statistical treatment was proposed by Gauthier et al. [17].Whalen [18] compared the results for second, third and fourth derivatives of the mode shapes using an analytical model ofdamage. Santos et al. [19] used the Timoshenko beam model and defined several damage localization indicators based onhigher order derivatives of modal displacements and rotation fields. Abdo [20] extended some of these damage indicators byapplying them to plates, using the summation of fourth derivatives in the x and y directions. The derivatives needed in allthese methods can only be obtained by numerically differentiating experimental modal displacements or rotations. In orderto minimize the amplification and propagation of experimental noise, due to the numerical differentiation process, accuratemodal full-fields measurements are required [21,22].

In conventional experimental modal analysis, the use of accelerometers or other kind of contact sensors, the gluingmaterials and the need for connecting cables result in the addition of mass to the system. Depending on the masses ratioand its location relatively to the modal amplitude, a significant change in the dynamic behavior of the structure can takeplace. Moreover, the number of measured points is usually small, leading to a set of sparse measurements. On the otherhand, speckle interferometry techniques, such as electronic speckle pattern interferometry (ESPI) and speckle shearography,allow full-field, non-contact and high sensitivity resolution measurements of the modal displacement and modal rotationfields of the structures surface, respectively. The main limitation of the ESPI technique comes from the high density offringes obtained from measurements of displacements fields, including the rigid-body displacements, which makes difficultthe interpretation of the fringe patterns [23–25]. However, speckle shearography provides a way to measure displacementgradients, being therefore practically insensitive to rigid-body motions. In addition, it requires a simpler opticalinterferometer setup and a laser with low coherence length. Thus, more compact systems can be built, which are alsomore robust to external perturbations. Shearography is based on the principle of the speckle interference between twowavefronts reflected by the surface of the object. These two wavefronts are laterally shifted, i.e. sheared. This shift can becreated using a glass-shaped wedge placed in the front half of the lens, a rotation of two glass plates, a Wollaston prism or aMichelson optical interferometer setup with a slight rotation of one of the mirrors [26]. The last option is preferred, since itallows an easy adjustment of the shearing value. Another alternative, which does not require moving parts, is proposed inRef. [27]. A comprehensive description of shearography and its applications can be found in Refs. [28–38]. Recent reviews ofthis technique can also be found in Refs. [39–41].

A damage localization method based on the analysis of second and third order spatial derivatives of measured modalrotation fields is proposed in this paper. Since the rotation field corresponds to the first spatial derivative of thedisplacement field, the second and third derivatives of the rotations correspond to the third and fourth derivatives of thedisplacements, respectively. Therefore, the direct measurement of rotation fields has the advantage of reducing the order ofthe numerical differentiation by one. The modal rotations of a multi damaged laminated composite plate are measuredusing speckle shearography with stroboscopic laser illumination and temporal phase modulation. The experimentalmeasurements thus obtained present a higher signal-to-noise ratio when compared with previous ones, obtained usingspeckle shearography with pulsed laser [22]. Besides these improvements in the quality of the experimental measurements,a new differentiation methodology is proposed. Unlike previous works (see e.g. [22]), the differentiation is performed beforethe phase maps are post-processed, thus leading to a decrease in the propagation of noise. The higher order spatialderivatives of the modal rotations of a laminated composite plate are computed by applying central finite differences. Thesederivatives are also filtered, using low pass filters. The damages are directly localized by analyzing the perturbations in thesecond, third and fourth spatial derivatives of the out-of-plane modal displacements. Thus, there is no need for previousknowledge of the undamaged structure behavior.

2. Methods

2.1. Speckle shearography

Speckle shearography has been mainly applied to the measurement of static rotation fields, because of its simpleexperimental arrangement. However, the measurement of dynamic responses requires the use of more complexillumination and synchronization systems. This leads to an experimental setup which is more difficult to adjust. Therefore,reports on efficient and accurate measurements of vibration responses using speckle shearography are relatively recent[24,42,43]. Before these works, modal rotation fields, which can be viewed as gradients of mode shapes, were approximatelymeasured using the time-average method [26]. This method has the advantages of using the same optical interferometersetup used in the static measurements and allows the observation of the vibration contour fringes at video rate. The methodis based on the subtraction of speckle interference patterns produced by stationary harmonic motion of objects duringseveral cycles of vibration. In this case, the recording time is very long compared to the period of vibration. Black intensityfringes are observed as contours of equal amplitude of vibration, being the fringe intensity modulated by the Bessel functionJ0, where the contrast decreases with the increase of the fringes order associated with the amplitude of vibration. Onlyrecently, the development and application of spatial phase modulation and temporal phase modulation to speckle

Fig. 1. Schematic diagram of the speckle shearography system with double pulse laser illumination used for the measurement of modal rotation fields.

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–3826

shearography made possible a quantitative evaluation of the phase distribution or phase map and the accuratemeasurement of the corresponding modal rotation field [42,43].

In order to measure modal rotation fields, a double pulse laser illumination and a spatial phase modulation technique arecombined with speckle shearography. The double pulse laser and the vibration amplitude are controlled by an externalsynchronization signal. To subsequently extract the phase map, a spatial carrier is introduced in the interference pattern by asmall rotation between the two wavefronts. The introduction of this spatial carrier is accomplished by using a Mach–Zehnder interferometer (Fig. 1). In this kind of interferometer, the speckle pattern created by the rough and diffuse surface ofthe plate is divided in two optical paths by the first beam splitter (BS1), reflects in Mirror 1 and Mirror 2 and is recombinedin the second beam splitter (BS2). The shearing value is set by translating Mirror 1 and the spatial carrier is introduced byrotating Mirror 2.

The recording of the spatial carrier requires the use of small optical apertures, limiting the frequency of themeasurements to 1/6 of the number of pixels of the CCD array [44]. Also, with this interferometer we cannot obtainuniform distributions of the spatial carrier, leading to errors in the measurement. The determination of the interferencephase involves the isolation of the spectral information around the spatial carrier and can be more easily performed throughthe application of forward and inverse fast Fourier transforms [45]. The intensity of the interference in the wave domainnumber Iðu; vÞ is expressed by [26]

Iðu; vÞ ¼ Aðu; vÞþ C ðu; vÞþ Cnðu; vÞ ð1Þ

where Aðu; vÞ represents the background intensity, C ðu; vÞ is the intensity of the phase interference modeled by the carrierphase, n denotes the complex conjugate, being u and v the order of the wave number in the horizontal and verticaldirections, respectively. The intensity of the interference can be separated from the background intensity in the frequencydomain by applying a simple window filter. The spatial carrier can be adjusted by controlling the rotation of Mirror 2 andthe window filter is adjusted by controlling the optical aperture of the system. Therefore, Aðu; vÞ, C ðu; vÞ and C

nðu; vÞ can beeasily separated in the frequency domain. After demodulation of the spatial carrier, the interference phase can be calculatedby [26]

Φ x; yð Þ ¼ arctanIm½cðx; yÞ�Re½cðx; yÞ� ð2Þ

where x and y are coordinates in the spatial domain and cðx; yÞ is given by the inverse Fourier transform of C ðu; vÞ. The twocaptured modal rotation amplitudes correspond to a reference state and a deformed state of the structure. By subtractingthe deformed interference phase ΦDðx; yÞ from the reference interference phase ΦRðx; yÞ, it is possible to extract the phasemap Δϕðx; yÞ:

Δϕðx; yÞ ¼ΦDðx; yÞ�ΦRðx; yÞ if ΦDðx; yÞZΦRðx; yÞΦDðx; yÞ�ΦRðx; yÞþ2π if ΦDðx; yÞoΦRðx; yÞ

(ð3Þ

When the sensitivity vector is perpendicular to the measurement surface, a relation between the gradient of the out-of-plane displacement field wðx; yÞ and the phase map can be established [26]:

Δϕ x; yð Þ � 2πΔxλ

∂wðx; yÞ∂x

ð4Þ

where Δx is the shearing value in the x direction, λ is the wavelength of the laser and ∂wðx; yÞ=∂x is the first spatial derivativeof the out-of-plane displacement field in the x direction. This gradient can be taken as a good approximation to the rotationfield of small deformations. If one considers a structure vibrating at a frequency corresponding to the i-th mode shape

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–38 27

wiðx; yÞ, the derivative in Eq. (4) defines the modal rotation θixðx; yÞ in the x direction at this frequency:

θix x; yð Þ ¼ ∂wiðx; yÞ∂x

ð5Þ

An alternative to the previously described technique relies on the combination of speckle shearography with strobo-scopic laser illumination and temporal phase modulation. The use of stroboscopic laser illumination, which is synchronizedwith the vibration excitation, allows freezing in time the speckle pattern [25]. Thus, the quantitative evaluation of the phasemap can be performed by applying temporal phase modulation. Furthermore, with this technique, the same kind ofMichelson interferometer used in static measurements can also be used to measure dynamic motions. The stroboscopicillumination can be generated from a continuous-wave laser either by using an electro-optic modulator or an acousto-opticmodulator. When an electro-optic modulator is used, short stroboscopic illumination pulses are created by switching thepolarization of a Pocket cell crystal by π/2, being the duration of the pulses controlled by a high voltage electric signal.This produces a more efficient illumination than the one obtained with the use of an acousto-optic modulator. However, thissystem is more expensive and, therefore, an acousto-optic modulator is more often used. In this case, the continuous-wavelaser beam passes through a crystal where traveling sound waves are generated by a piezoelectric actuator. Thisphenomenon produces periodic variations in the refractive index of the crystal. The laser light beam is laterally deflectedby selecting the grating first order diffraction. This is done by adjusting the incident angle θ of the laser beam, as shown inFig. 2(a). With the purpose of isolating the stroboscopic beam pulses from the zero order diffraction beam, a spatial filter ismounted in front of the acousto-optic modulator. In order to freeze the interference pattern, these pulses should be narrowenough. However, they should also be wide enough to illuminate the surface of the object. The stroboscopic illuminationis synchronized with the harmonic vibration excitation by modulating the piezoelectric excitation signal, as depicted inFig. 2(b).

Because the speckle pattern is frozen in time, a Michelson optical interferometer for static measurements can be used.This setup, shown in Fig. 3, allows the use of the temporal phase modulation technique for the quantitative determination ofthe interference pattern. The speckle pattern generated on the surface of the object is split into two by the beam splitter, andthe slight rotation of one of the mirrors is used to laterally shift the two intensity paths and create the speckle interference.By translating another mirror, using a piezoelectric actuator, a temporal phase modulation technique, also known as phaseshifting or phase stepping, can be applied. The most usual method of temporal phase modulation is based on four intensity

Fig. 2. (a) Schematic diagram of the acousto-optical modulator with spatial filter and (b) synchronization of the vibration excitation and stroboscopicillumination signals.

Fig. 3. Schematic diagram of the speckle shearography system with stroboscopic laser illumination used for the measurement of modal rotation fields.

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–3828

distributions with a constant phase step between them of π=2. In this method, the phases of the speckle patterns for thereference state, ΦRðx; yÞ, and deformed state, ΦDðx; yÞ, are, respectively, a function of the intensity distributions, IR;1ðx; yÞ,IR;2ðx; yÞ, IR;3ðx; yÞ, IR;4ðx; yÞ, and ID;1ðx; yÞ, ID;2ðx; yÞ, ID;3ðx; yÞ, ID;4ðx; yÞ:

ΦR x; yð Þ ¼ arctanIR;4ðx; yÞ� IR;2ðx; yÞIR;1ðx; yÞ� IR;3ðx; yÞ

� �

ΦD x; yð Þ ¼ arctanID;4ðx; yÞ� ID;2ðx; yÞID;1ðx; yÞ� ID;3ðx; yÞ

� �ð6Þ

By applying Eq. (3), the phase map of the modal rotation field is obtained by subtracting the two interference phasesabove. Finally, the relation between the phase map and the modal rotation field is given by Eq. (4).

2.2. Damage localization

As described in the previous subsection, the modal rotation fields are directly measured using speckle shearography withstroboscopic laser illumination. Because rotations correspond to first order spatial derivatives of out-of-plane displacements,the application of numerical differentiation techniques, needed in many damage localization methods, is reduced in oneorder. This is particularly important because the present damage localization method is based on the analysis ofperturbations and discontinuities in second, third and fourth order spatial derivatives of modal displacement fields. It isknown that, in practice, the differentiation of experimental data leads to amplification and propagation of experimentalnoise, namely in higher frequencies. It is possible to mitigate some of these effects by combining differentiation andlow-pass filters techniques [22]. However, this can lead to the elimination of signal components with higher frequencies,which are essential for a correct representation of higher order derivatives of the modal fields. Furthermore, and since thedamage influence is associated with this kind of signal components, their elimination can lead to unsuccessful damagelocalizations. The separation of the true signal from the high frequency noise can be more easily accomplished by increasingthe measurements spatial resolution and by improving the signal-to-noise ratio. Also, by decreasing the number ofnumerical operations applied to the experimental data, it is possible to reduce the noise amplification and propagation. Inorder to cope with these problems, in the present approach the numerical differentiation is performed in the phase mapsand not in the rotational fields, as in previous works [22,46]. Indeed, to obtain these fields, the phase maps must beunwrapped and in this process the noise can be amplified, as described more thoroughly below.

Any phase map presents phase discontinuities that must be correctly removed in order to obtain a continuousdescription of the rotational field. Normally, this can be a very challenging task using real data, due to the presence ofexperimental noise. To overcome this problem, several unwrapping algorithms have been proposed [47]. According to theirstrategy, these algorithms can be grouped into two different categories [47]: the path-following methods and the minimum-norm methods. The strategy of the first category is based on defining a unique integration path of the phase map. This isachieved by placing restriction lines to the integration path or by defining the integration path using the phase map qualityinformation. In either case, this leads to singular regions that are represented by local perturbations. Normally, theapplication of low-pass filters to remove these perturbations will introduce errors in the signals. On the other hand, theminimum-norm methods are based on the strategy of global minimization of phase transitions. By definition, thesemethods do not necessarily preserve the phase map information, mainly in noisy areas, where the data is smoothed and thesignal is changed. Therefore, the unwrapping methods always produce signal errors in the rotational field, which are furtherpropagated and amplified by the numerical differentiation process.

In view of the above, in the present work, the second, third and fourth order spatial derivatives of the displacement fieldsare successively computed by applying first-order central finite differences and low-pass filters to the phase maps of themodal rotations. Because these phase maps and respective derivatives are defined between �π and π, they havediscontinuities, which can be removed by applying unwrapping algorithms. This leads to a better visualization ofperturbations in the derivatives, and therefore a better analysis of the damage localization, since after unwrapping oneobtains a continuous description of the rotational field.

The spatial derivatives of order n of the phase map, defined between �π and π, relative to the i-th mode shape areapproximated by

∂nΔϕiðx; yÞ∂xn

� arctansin

∂n�1Δϕiðxþhx=2; yÞ∂xn�1 �∂n�1Δϕiðx�hx=2; yÞ

∂xn�1

� �

cos∂n�1Δϕiðxþhx=2; yÞ

∂xn�1 �∂n�1Δϕiðx�hx=2; yÞ∂xn�1

� �8>>><>>>:

9>>>=>>>;

hx= ð7Þ

where hx is the lateral shift size in the x direction. Eq. (7) is a mathematical description of the process of laterally shifting themaps by hx and subtracting them. The high frequency noise is removed by applying the average filtering technique.However, to apply this technique it is first necessary to transform the maps into continuous ones. This is accomplished byshifting them into the complex domain. The applied filtering technique uses image convolution:

∂nΔ ~ϕiðx; yÞ

∂xn¼ ∂nΔ ~ϕiðx; yÞ

∂xn� h m;nð Þ ð8Þ

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–38 29

where � is the convolution operator, hðm;nÞ is the filter array, and m and n represent the horizontal and vertical dimensionsof the filtering window, respectively. In this case, the average filter window was applied. After applying Eq. (8), the filteredspatial derivative of order n of the phase map is described in the spatial domain by

∂nΔϕiðx; yÞ∂xn

¼ arctan∂nΔ ~ϕiðx; yÞ

∂xn

!ð9Þ

Finally, by applying the Goldstein unwrapping algorithm [47] the derivatives of the modal fields can be constructed.Indeed, by taking into consideration Eqs. (4) and (5), a relation between the unwrapped filtered spatial derivative of order nof the phase map, ∂nΔϕ

_

iðx; yÞ=∂xn, the spatial derivative of order nþ1 of the displacement field, ∂nþ1wiðx; yÞ=∂xnþ1, and thespatial derivative of order n of the rotational field ∂nθixðx; yÞ=∂xn can be established:

∂nΔϕ_

iðx; yÞ∂xn

¼ 2πΔxλ

∂nþ1wiðx; yÞ∂xnþ1 with

∂nþ1wiðx; yÞ∂xnþ1 ¼ ∂nθixðx; yÞ

∂xnð10Þ

2.3. Structure and equipment

The structure analyzed is a multidamaged laminated composite plate with in-plane dimensions 276.5�198.0 mm2 andthickness 1.825 mm. The stacking sequence of the layers is [0/90/þ45/�45/0/90]s [22]. Two internal damages wereproduced by dropping a steel sphere into the in-plane surface at two distinct points, as shown in Fig. 4. The first and secondimpacts correspond to energies of 13.5 and 26.2 J, respectively. It should be noted that no damage was observed on theimpacted surface.

Fig. 4. Impacted surface of the plate and localization of the two impacts.

Fig. 5. View of the experimental setup and equipment.

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–3830

Flexible rubber bands were used to suspend the plate, thus creating almost free boundary conditions. In order touniformly illuminate the measured surface, a very small layer of white powder was applied to it, as shown in Fig. 4.A Coherents Verdi laser with a wavelength of 532 nm, was used as the illumination source. A harmonic signal is generated

1st 3rd

5th 8th

Fig. 6. Filtered phase maps of the first, third, fifth and eighth modal rotation fields obtained by speckle shearography with stroboscopic laser illuminationand temporal phase modulation.

1st 3rd

5th 8th

Fig. 7. Filtered phase maps of the first, third, fifth and eighth modal rotation fields obtained by speckle shearography with double pulse illumination andspatial phase modulation.

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–38 31

with an amplifier and loudspeaker to excite the plate at its natural frequencies, as in [22], whereas a pulse signal is appliedto an acousto-optic modulator to generate the stroboscopic illumination. These two synchronized signals are created using aTektronixs AFG320 dual signal generator. The intensity distribution is captured by a Dalsas Falcon 4M30, digital camera,with a frame rate of 31 fps and a CMOS sensor array of 4 megapixels.

In order to quantify the interference phases by applying temporal phase modulation, four intensity distributions arerecorded in the reference and deformation states, with a constant phase shift of π=2. This shifting is accomplished bytranslating one of the mirrors of the interferometer through the actuation of a piezoelectric (see Figs. 3 and 5). Thispiezoelectric is controlled by a National Instrumentss PCI 6722 card and a Burleighs PZ70 amplifier. No excitation is appliedto the plate in the reference state. The information of the deformation state is taken after the vibration excitation is appliedand the plate response becomes stationary. By adjusting the amplitude of the harmonic signal or the phase of the pulsesignal, it is possible to observe in real time the raw fringes of the phase map, thus controlling the amplitude of the modalrotation field.

3. Results and discussion

The filtered phase maps of the first, third, fifth and eighth modal rotation fields obtained using the proposedshearography technique are shown in Fig. 6, being the shearing value Δx of 10 mm. As a result of the high spatialresolution, a high density of fringes is accomplished in all the maps. The analysis of these maps already reveals smallperturbations in the phase map fringes pattern near the regions where the second impact was produced, i.e. the lower rightedge of the plate. Fig. 7 shows the filtered phase maps of the same modal rotation fields relative to the horizontal direction,

1stmode (1stder.) 1stmode (2ndder.) 1stmode (3rdder.)

3rdmode (1stder.) 3rdmode (2ndder.) 3rdmode (3rdder.)

5thmode (1stder.) 5thmode (2ndder.) 5thmode (3rdder.)

8thmode (1stder.) 8thmode (2ndder.) 8thmode (3rdder.)

Fig. 8. First, second and third order derivatives of the phase maps obtained with stroboscopic laser illumination and temporal phase modulation.

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–3832

obtained by speckle shearography, with a shearing value Δx of 10 mm, but pulsed laser illumination and spatial phasemodulation. These maps have been used to localize the damage in a previous study [22]. By comparing the phase maps inFigs. 6 and 7, we see that a significant improvement in the quality of the measurements is accomplished by usingshearography with stroboscopic laser illumination and temporal phase modulation.

Although we can observe some perturbations in the phase maps of the modal rotation fields, which are due to thedamage, it is possible to see a more clear localization of the damage by differentiating these phase maps successively, asdescribed in Section 2.2. This new numerical differentiation process was applied to the phase maps obtained using the twoshearography systems. The differentiation is performed in the horizontal direction, i.e. the differentiation is in order to the xaxis, with a lateral shift size hx of 6 mm for the phase maps in Fig. 6 and 12 mm for the phase maps in Fig. 7, as defined byEq. (7). According to this equation, the numerical differentiation of the phase maps leads to a reduction of the number offringes, which can be defined by the lateral shift. Thus, a higher lateral shift was set for the phase maps of Fig. 7, in order toobserve the fringes perturbations. Figs. 8 and 9 show the first, second and third spatial derivatives of the phase maps inFigs. 6 and 7, respectively. It can be seen in Fig. 8 that the images present noticeable local perturbations in a region near thesecond impact. However, the same is not true for all the images in Fig. 9, which were obtained by speckle shearography withpulsed illumination and spatial phase modulation. Most important is the fact that the analysis of fringe perturbation in thederivatives of the phase maps already and clearly show the localization of the highest damage. This represents a meaningfulimprovement relatively to the numerical differentiation method proposed in previous studies [22]. Furthermore, in [22], theresponse of the undamaged structure is required.

1stmode (1stder.) 1stmode (2ndder.) 1stmode (3rdder.)

3rdmode (1stder.) 3rdmode (2ndder.) 3rdmode (3rdder.)

5thmode (1stder.) 5thmode (2ndder.) 5thmode (3rdder.)

8thmode (1stder.) 8thmode (2ndder.) 8thmode (3rdder.)

Fig. 9. First, second and third order derivatives of the phase maps obtained with double pulse illumination and spatial phase modulation.

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–38 33

By unwrapping the derivatives of the phase maps, the damage can be more easily and clearly identified. The unwrappedfirst, second and third order derivatives of the phase maps of the modal rotation fields are presented in Figs. 10–15. Notethat, according to Eq. (10), these images correspond to the second, third and fourth derivatives of the modal displacement

Fig. 10. Second order spatial derivative of the modal displacement fields obtained by stroboscopic laser illumination and temporal phase modulation.

Fig. 11. Second order spatial derivative of the modal displacement fields obtained by double pulse illumination and spatial phase modulation.

Fig. 12. Third order spatial derivative of the modal displacement fields obtained by stroboscopic laser illumination and temporal phase modulation.

Fig. 13. Third order spatial derivative of the modal displacement fields obtained by double pulse illumination and spatial phase modulation.

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–3834

Fig. 14. Fourth order spatial derivative of the modal displacement fields obtained by stroboscopic laser illumination and temporal phase modulation.

Fig. 15. Fourth order spatial derivative of the modal displacement fields obtained by double pulse illumination and spatial phase modulation.

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–38 35

Fig. 16. Second order spatial derivative of the modal displacement fields obtained by differentiating the unwrapped phase maps, according to [22].

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–3836

field or mode shape, respectively. For derivatives obtained using the stroboscopic laser illumination technique, the localperturbations clearly show the localization of the damage created by the second impact (Figs. 10, 12 and 14). The localperturbations due to the damage are not so easily identified in the unwrapped derivatives of the phase maps measuredusing the double pulse laser illumination, due to the presence of fluctuations in the signal, which are associated to the lowsignal-to-noise ratio of the measurements (Figs. 11, 13 and 15). These fluctuations are amplified and tend to spread toneighboring regions by the numerical differentiation process.

The fourth order derivatives of the modal displacement fields, in Fig. 14, present a smooth and plane surface inundamaged areas. Furthermore, they also present the highest magnitude near the damaged region, comparatively to theother order derivatives. The present results confirm a better damage localization relatively to the one reported in a previousstudy [22]. Indeed, by comparing the plots in Fig. 10 with the ones in Fig. 16, which are obtained by differentiating theunwrapped phase maps as described in [22], one sees that now we have a more clear peak at the damage location. This is aresult of the combination of the new differentiation methodology, relying on differentiation of raw phase maps, and the useof stroboscopic laser illumination. Furthermore, in Ref. [22] only the second order derivative of the modal displacement fieldis considered, whereas now also the third and fourth order derivatives are computed.

4. Conclusions

Two speckle shearography systems used for the measurement of modal rotation fields are described in this paper.The first system combines double pulse illumination with spatial phase modulation. In the second, stroboscopic laserillumination and temporal phase modulation are used. A comparative analysis between the measured phase maps of adamaged laminated composite plate, and the ones obtained in previous studies, using the first system, is presented. Theresults show the superior quality of the experimental measurements obtained with the second speckle shearographysystem. A new differentiation methodology is also proposed in order to decrease the experimental noise propagation.Contrary to previous works, the differentiation is applied to the phase maps, therefore avoiding the propagation of noisecaused by their post-processing. The results show that the proposed numerical differentiation process is much better thanthe previous one. The combination of modal rotation fields, measured by speckle shearography with stroboscopic laserillumination, and the new differentiation methodology, allows the computation of the second, third and fourth orderderivatives of the mode shapes. The comparative analysis between the phase maps, obtained using the two experimentalmethodologies, show the superior resolution and the higher signal-to-noise ratio of the speckle shearography withstroboscopic laser illumination. Based on the analysis of the spatial derivatives of these mode shapes, it was possible to

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–38 37

localize one of the damages in the plate, in particular with the fourth order derivative. Finally, the results presented provethat the proposed methodology is more effective for damage localization relatively to the one proposed in previous studies.

Acknowledgments

The authors greatly appreciate the financial support of FCOMP-01-0124-FEDER-10236, through Project Ref. FCT PTDC/EME-PME/102095/2008.

References

[1] A.K. Pandey, M. Biswas, M.M. Samman, Damage detection from changes in curvature mode shapes, J. Sound Vib. 145 (2) (1991) 321–332.[2] A.K. Pandey, M. Biswas, Damage detection in structures using changes in flexibility, J. Sound Vib. 169 (1) (1994) 3–17.[3] C.P. Ratcliffe, Damage detection using a modified Laplacian operator on mode shape data, J. Sound Vib. 204 (4) (1997) 505–517.[4] R.P.C. Sampaio, N.M.M. Maia, J.M.M. Silva, Damage detection using the frequency-response-function curvature method, J. Sound Vib. 226 (5) (1999)

1029–1042.[5] M.A. Abdo, M. Hori, A numerical study of structural damage detection using changes in the rotation of mode shapes, J. Sound Vib. 251 (2) (2002)

227–239.[6] A. Messina, Detecting damage in beams through digital differentiator filters and continuous wavelet transforms, J. Sound Vib. 272 (2004) 385–412.[7] W. Lestari, P. Qiao, Damage detection of fiber-reinforced polymer honeycomb sandwich beams, Compos. Struct. 67 (3) (2005) 365–373.[8] M. Rucka, K. Wilde, Crack identification using wavelets on experimental static deflection profiles, Eng. Struct. 28 (2006) 279–288.[9] W. Lestari, P. Qiao, S. Hanagud, Curvature mode shape-based damage assessment of carbon/epoxy composite beams, J. Intell. Mater. Syst. Struct. 18 (3)

(2007) 189–208.[10] H. Guan, V. Karbhari, Improved damage detection method based on element modal strain damage index using sparse measurement, J. Sound Vib. 309

(3–5) (2008) 465–494.[11] S. Park, H.-H. Shin, C.-B. Yun, Wireless impedance sensor nodes for functions of structural damage identification and sensor self-diagnosis, Smart

Mater. Struct. 18 (5) (2009) 055001.[12] M.-K. Yoon, D. Heider, J. Gillespie, C. Ratcliffe, R. Crane, Local damage detection with the global fitting method using mode shape data in notched

beams, J. Nondestruct. Eval. 28 (2009) 63–74.[13] M. Yoon, D. Heider, J. Gillespie, C. Ratcliffe, R. Crane, Local damage detection with the global fitting method using operating deflection shape data,

J. Nondestruct. Eval. 29 (2010) 25–37.[14] N.M.M. Maia, J.V. Araújo dos Santos, R.P.C. Sampaio, C.M. Mota Soares, Damage identification using curvatures and sensitivities of frequency-response-

functions, in: A. Güemes (Ed.), Proceedings of Third European Workshop on Structural Health Monitoring, 2006, Granada, Spain, 2006, pp. 547–554.[15] J.V. Araújo dos Santos, N.M.M. Maia, C.M. Mota Soares, C.A. Mota Soares, Structural damage identification: a survey, in: Trends in Computational

Structures Technology, Saxe-Coburg Publications, Stirlingshire, UK, 2008, pp. 1–24.[16] A.A.R.Z. Ismail, H. Abdul Razak, Determination of damage location in RC beams using mode shape derivatives, Eng. Struct. 28 (2006) 1566–1573.[17] J.F. Gauthier, T.M. Whalen, J. Liu, Experimental validation of the higher-order derivative discontinuity method for damage identification, Struct. Control

Health Monit. 15 (2) (2008) 143–161.[18] T.M. Whalen, The behavior of higher order mode shape derivatives in damaged beam-like structures, J. Sound Vib. 309 (3–5) (2008) 426–464.[19] J.V. Araújo dos Santos, H.M. Reis Lopes, N.M.M. Maia, A damage localisation method based on higher order spatial derivatives of displacement and

rotation fields, J. Phys.: Conf. Ser. 305 (2011) 012008.[20] M.A.-B. Abdo, Damage detection in plate-like structures using high-order mode shape derivatives, Int. J. Civil Struct. Eng. 2 (3) (2012) 801–816.[21] H.M.R. Lopes, R.M. Guedes, M.A. Vaz, An improved mixed numerical-experimental method for stress field calculation, Opt. Laser Technol. 39 (5) (2007)

1066–1073.[22] H.M.R. Lopes, J.V. Araújo dos Santos, C.M.M. Soares, R.J.M. Guedes, M.A.P. Vaz, A numerical-experimental method for damage location based on

rotation fields spatial differentiation, Comput. Struct. 89 (19–20) (2011) 1754–1770.[23] M. Conrad, M. Sayir, Composite ceramic-metal plates tested with flexural waves and holography, Exp. Mech. 41 (2001) 412–420.[24] F. Santos, M. Vaz, J. Monteiro, A new set-up for pulsed digital shearography applied to defect detection in composite structures, Opt. Lasers Eng. 42 (2)

(2004) 131–140.[25] W. Steinchen, L. Yang, Digital Shearography: Theory and Application of Digital Speckle Pattern Shearing, SPIE Press, Bellingham, Washington, 2003.[26] T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2005.[27] C. Falldorf, S. Osten, C.v. Kopylow, W. Jüptner, Shearing interferometer based on the birefringent properties of a spatial light modulator, Opt. Lett. 34

(18) (2009) 2727–2729.[28] M.Y.Y. Hung, C.E. Taylor, Speckle-shearing interferometric camera—a tool for measurement of derivatives of surface displacements, in: Proceedings of

SPIE, vol. 41, 1974, pp. 169–175.[29] Y.Y. Hung, A speckle-shearing interferometer: a tool for measuring derivatives of surface displacements, Opt. Commun. 11 (2) (1974) 132–135.[30] Y.Y. Hung, R.E. Rowlands, I.M. Daniel, Speckle-shearing interferometric technique: a full-field strain gauge, Appl. Opt. 14 (3) (1975) 618–622.[31] Y.Y. Hung, C.Y. Liang, Image-shearing camera for direct measurement of surface strains, Appl. Opt. 18 (7) (1979) 1046–1051.[32] Y.Y. Hung, Shearography: a new optical method for strain measurement and nondestructive testing, Opt. Eng. 21 (3) (1982) 391–395.[33] Y.Y. Hung, Applications of digital shearography for testing of composite structures, Compos. Part B—Eng. 30 (7) (1999) 765–773.[34] W. Steinchen, L. Yang, G. Kupfer, P. Mäckel, Non-destructive testing of aerospace composite materials using digital shearography, Proc. Inst. Mech.

Eng.: Part G: J. Aerosp. Eng. 212 (1) (1998) 21–30.[35] D. Paoletti, G.S. Spagnolo, M. Facchini, P. Zanetta, Artwork diagnostics with fiber-optic digital speckle pattern interferometry, Appl. Opt. 32 (31) (1993)

6236–6241.[36] P.K. Rastogi, Measurement of curvature and twist of a deformed object by electronic speckle-shearing pattern interferometry, Opt. Lett. 21 (12) (1996)

905–907.[37] W. Osten, M.K. Kalms, W.P.O. Jueptner, Some ways to improve the recognition of imperfections in large-scale components using shearography, Proc.

SPIE 3745, Interferometry 099: Applications (1999) 244–256.[38] W. Osten, M.K. Kalms, W.P.O. Jueptner, G. Tober, W. Bisle, D. Scherling, Shearography system for the testing of large-scale aircraft components taking

into account noncooperative surfaces, in: Proceedings of SPIE, vol. 4101, 2000, pp. 425–431.[39] G.S. Spagnolo, A. Martocchia, D. Papalillo, L. Cozzella, Simple educational tool for digital speckle shearography, Eur. J. Phys. 33 (4) (2012) 733–750.[40] H.J. Feng, J. Zhang, X.K. Liu, Studies on digital shearography for testing of aircraft composite structures and honeycomb-based specimen, Appl. Mech.

Mater. 121–126 (2011) 1264–1268.[41] J.A. dos Santos, H. Lopes, Application of speckle interferometry to damage identification, in: Computational Methods for Engineering Science,

Saxe-Coburg Publications, Stirlingshire, UK, 2012, pp. 299–330.

H. Lopes et al. / Mechanical Systems and Signal Processing 49 (2014) 24–3838

[42] G. Pedrini, Y.-L. Zou, H.J. Tiziani, Quantitative evaluation of digital shearing interferogram using the spatial carrier method, Pure Appl. Opt. 5 (3) (1996)313–321.

[43] L. Yang, W. Steinchen, G. Kupfer, P. Mäckel, F. Vössing, Vibration analysis by means of digital shearography, Opt. Lasers Eng. 30 (2) (1998) 199–212.[44] G. Pedrini, B. Pfister, H. Tiziani, Double pulse-electronic speckle interferometry, J. Mod. Opt. 40 (1) (1993) 89–96.[45] M. Takeda, H. Ina, S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, J. Opt. Soc.

Am. 72 (1) (1982) 156–160.[46] J.V. Araújo dos Santos, H.M.R. Lopes, M. Vaz, C.M. Mota Soares, C.A. Mota Soares, M.J.M. de Freitas, Damage localization in laminated composite plates

using mode shapes measured by pulsed TV holography, Compos. Struct. 76 (6) (2006) 272–281.[47] D.C. Ghiglia, M.D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software, Wiley, New York, 1998, p. 493, Chapter XIV.