A Numerical-experimental Method for Dama

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A numerical-experimental method for damage location based on rotation fieldsspatial differentiation

H.M. Reis Lopes a, J.V. Araújo dos Santos b,⇑, C.M. Mota Soares b, R.J. Miranda Guedes c, M.A. Pires Vaz c

a ESTIG, Instituto Politécnico de Bragança, Campus de Sta. Apolónia, Apartado 134, 5301-857 Bragança, Portugalb IDMEC/IST, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugalc DEMEGI, Faculdade de Engenharia do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

a r t i c l e i n f o

Article history:Received 27 November 2008Accepted 5 September 2010Available online 16 April 2011

Keywords:Multiple damage locationLaminated plateCurvature fieldShear interferometryGaussian function derivativeFiltering techniques

a b s t r a c t

This paper presents a structural damage location method that decreases the number of spatial differen-tiations needed to compute modal curvature fields. The method is numerically and experimentallyapplied to isotropic and laminated rectangular plates, respectively. A speckle shear interferometer is usedto measure the rotation fields of the laminated plate, while the isotropic plate is analysed by finite ele-ments. It was found that the Gaussian differentiation is the most suited technique to compute the curva-ture fields. It is also demonstrated the superior performance of the curvature method using measuredrotation fields instead of measured displacement fields.

� 2010 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.

1. Introduction

The interest in predictive structural monitoring, aiming at dam-age early detection, has been motivated by, amongst other factors,the risk of human lives loses, due to unpredictable structural fail-ures, such as airplane crashes and bridges or buildings collapses.Damage detection and localisation aims at preventing the struc-tures failure by repairing or replacing the damaged component.The need of global methods for damage detection has been essen-tially motivated by aeronautical and aerospace applications. Inthese applications, the less critical components have been progres-sively replaced by components manufactured with compositematerials. Although their high specific stiffness and strength, com-pared to those of metals, composite materials are more sensitive tocertain type of damages and present different kinds of defects ordamage mechanisms. The main damage types in laminated com-posites are interlaminar debonding, micro-cracks and micro-buck-ling, besides inclusions. These internal damages usually result frommanufacturing processes or in service loadings. Delamination orinterlaminar debonding is undetected by visual means and, there-fore, one of the most critical type of damage, being also the mostcommon in aeronautical components.

There are several types of methods which use dynamic charac-teristics, namely those based on modal frequencies, displacements,

rotations and curvatures [1–14]. Among these, the global methodsbased on curvature changes can be given some prominence. Pan-dey et al. [3] introduced the mode shapes curvature method. In or-der to localise damage in beams, the proposed method uses thedifference in the curvature fields associated with mode shapes inthe undamaged and damaged states. The curvatures are extractedby applying second order central finite differences to the modeshapes. Contrary to the mode shapes, one can prove that the higherchanges in mode shapes curvatures are coincident with the dam-age region and their magnitudes are proportional to its severity.An extension of the curvature differences method was proposedby Sampaio et al. [7], who also used second order central finite dif-ferences to compute the frequency response functions curvatures.Recently, Guan and Karbhari [12] stated that the computation ofthe curvatures by numerical differentiation is the main cause forthe poor performance of the curvature method, in particular undersparse and noisy measurements. In order to improve the numericaldifferentiation results they proposed the use of a fourth order cen-tral difference method, instead of applying the second order cen-tral difference. The damage index method formulation presentedby Stubbs et al. [4] is also based on mode shapes curvatures infor-mation. This damage index relates the undamaged and damagedcurvature fields in each segment of the structure. Its deviation fromthe indices normal distribution average is used as an indicator forthe identification of the probable damaged region. Also in the con-text of curvature methods, Ratcliffe [5] developed a procedure thatdoes not require undamaged structure information. The curvature

0045-7949/$ - see front matter � 2010 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruc.2011.03.014

⇑ Corresponding author. Tel.: +44 958 8846.E-mail address: [email protected] (J.V. Araújo dos Santos).

Computers and Structures 89 (2011) 1754–1770

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computation is carried out by applying a Laplacian operator to themode shapes in order to locate damage experimentally. The dam-age position is identified from discontinuities in the curvaturefield. Data extracted from finite elements allows the damage iden-tification of local thickness reduction higher than 10%. Subse-quently and in order to show graphical discontinuities of damagelower than 10%, a modification on the Laplacian was presented.This method was named gapped smoothing damage detectionmethod. A smoothed curvature profile, representative of theundamaged structure, is subtracted to the curvature profile. Theperformance of this method was experimentally tested in a steelbeam with localised damage, characterised by a half thicknesstransversal cut. The damage was successfully located using the firsttwo mode shapes. However, the results low accuracy lead theseresearchers to suggest the use of strain gauges as an alternativetechnique to measure the curvatures. The gapped smoothing dam-age detection method was applied to a composite beam in order tolocate a delamination [6]. The extension of this method to twodimensions and the subsequent application to plates was per-formed to localise delaminations [10]. A finite model update algo-rithm based on experimental modal data was proposed for damagedetection [13]. A quantitative assessment of damage detectionusing different modal selection criteria is presented. The near-real-time composite structures health monitoring based on wave-let signal processing of lamb wave propagation was presented bySohn et al. [14]. A set of piezoelectric patches works as an activeactuators-sensors pair to generate a specific wavelet and to mea-sure the propagation of lamb wave signals. In this study, the pro-posed damage detection strategy was proved to correctly detectthe simulated delamination. A more complete description of theseand other damage identification methods can be found in a re-cently published survey [15].

The presence of material properties discontinuities results indiscontinuities or perturbations in the modal curvature field. This

Fig. 1. Computation of curvature field from measured: (a) displacement field and (b) rotation field.

Fig. 3. Damaged plate fourth mode: (a) x and (b) y rotation fields obtained usingANSYS�.

Fig. 2. Rectangular plate finite element mesh and simulated damaged areas.

H.M. Reis Lopes et al. / Computers and Structures 89 (2011) 1754–1770 1755


fact was investigated by Araújo dos Santos et al. [11], whomeasured mode shape displacements of a laminated compositeplate using double pulsed TV holography. The numerical differen-tiation method proposed by Lopes et al. [16] was used to computethe second order spatial derivatives of the mode shapes displace-ment fields. In order to minimise the propagation and amplifica-tion of experimental high frequency noise, this method uses bothdifferentiation and filtering techniques. It allowed the localisationof a delamination in the plate centre due to low velocity impact.Small perturbations or discontinuities in the curvature field, asso-ciated with the existence of delaminations, are in general mixed upwith high frequency noise. Furthermore, these signal components,essential for damage localisation, are also attenuated or removedby filtering techniques, which are usually applied in the computa-tion of high order derivatives. In order to partially overcome theseproblems, a new numerical-experimental method is proposed inthis paper. This method is based on the computation of the curva-ture field from the experimental rotation field, and not, as in [11],from the experimental displacement field. In the present work, therotation fields are directly measured using a speckle shear interfer-ometric technique with a double pulse laser. Therefore, the num-ber of spatial derivative computations is reduced to one, since

the curvature field is obtained by differentiating only once therotation field. The present technique yields an improvement onthe sensitivity of the curvature difference method in structuraldamage localisation.

2. Methodology

Curvature fields can only be obtained by numerical spatial dif-ferentiation of measured displacement or rotation fields. Theseexperimental fields are contaminated with high frequency noisewhich is propagated and amplified by the differentiation process[17]. This difficulty can be partially overcome by using low-pass fil-ters. Nevertheless, the signal higher order spatial terms, essentialfor the correct representation of derivatives, in particular their dis-continuities can also be eliminated in this process [18]. Therefore, arobust methodology is needed to assure that this information isnot eliminated by applying filters. The solution lies on a progres-sive filtering in the course of the successive spatial derivativescomputation. This approach aims at preserving the signal higherorder components and simultaneously avoids the propagation ofexperimental high frequency noise.

y [m

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y [m

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0 0.069 0.138 0.207 0.2765x [m]

(a)

Fig. 4. Differences between the first mode: (a) x and (b) y rotation fields magnitude.

1756 H.M. Reis Lopes et al. / Computers and Structures 89 (2011) 1754–1770


y [m

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x [m]0 0.069 0.138 0.207 0.2765

(a)

x [m]0 0.069 0.138 0.207 0.2765

(b)

Fig. 5. Differences between the first mode: (a) xx, (b) xy and (c) yy curvature fields magnitude.



2.1. Numerical differentiation and filtering techniques

Central finite differences techniques are commonly used for thedifferentiation of sparse numerical data. When such techniques areapplied to experimental measurements an amplification of highfrequency noise occurs. Therefore, a more robust differentiationtechnique should be used in these cases. For full-field measure-ments the high order signal and high frequency noise can be par-tially split [17]. This allows the application of more advanceddifferentiation techniques. The differentiation and smoothing tech-niques are combined to obtain the spatial derivatives. In this work,the Gaussian, Fourier and spline differentiation techniques wereapplied.

The first differentiation and smoothing technique used is basedon the one-dimensional Gaussian function G(x) defined by [19]:

GðxÞ ¼ 1ffiffiffiffiffiffiffi2pp exp � x2

2

� �: ð1Þ

The spatial differentiation can be obtained from this Gaussian func-tion, since the one-dimension n-th order Gaussian derivative is[19]:

dnGðxÞdxn ¼ 1ffiffiffiffiffiffiffi

2pp� �n

Hnxffiffiffiffiffiffiffi2pp� �

GðxÞ; ð2Þ

where Hn(x) are the Hermite polynomials:

H0ðxÞ ¼ 1;H1ðxÞ ¼ 2x;

Hnþ1ðxÞ ¼ 2xHnðxÞ � 2ðn� 1ÞHn�1ðxÞ:ð3Þ

The application of such a differentiation scheme to a measured fieldmakes it necessary that the Gaussian function derivatives be de-fined in a discrete domain. In a discrete domain the x representsthe kernel points of the one-dimensional Gaussian derivative. Theuse of discrete Gaussian derivatives has the advantage of spatialvariables separation, allowing a straightforward computation oftwo-dimensional field derivatives.

Spatial derivatives can also be obtained using a spline smooth-ing technique, which is based on third order coefficients of B-splines functions, computed by fitting these functions to a set ofpoints of a measured field. For a uniformly spaced set of one-dimensional points xi, a cubic B-spline function is written as [20]:

SðxÞ ¼Xm

j¼1

Bj;3ðxiÞaj for i ¼ 1; . . . ;m; ð4Þ

where the coefficient Bj,3 are obtained by the de Boor’s recurrencerelation [20], m stands for the number of points and aj are non-zero

y [m

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x [m]0 0.069 0.138 0.207 0.2765

(a)

Fig. 6. Differences between the first mode: (a) x and (b) y rotation fields magnitude (rotation fields with ±1% signal amplitude of pseudorandom noise).



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x [m]0 0.069 0.138 0.207 0.2765

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x [m]0 0.069 0.138 0.207 0.2765

(b)

Fig. 7. Differences between the first mode: (a) xx, (b) xy and (c) yy curvature fields magnitude (differentiation of rotation fields with ±1% signal amplitude of pseudorandomnoise using Gaussian technique).



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x [m]0 0.069 0.138 0.207 0.2765

(a)

x [m]0 0.069 0.138 0.207 0.2765

(b)

Fig. 8. Differences between the first mode: (a) xx, (b) xy and (c) yy curvature fields magnitude (differentiation of rotation fields with ±1% signal amplitude of pseudorandomnoise using FFT technique).



y [m

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x [m]0 0.069 0.138 0.207 0.2765

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x [m]0 0.069 0.138 0.207 0.2765

(b)

Fig. 9. Differences between the first mode: (a) xx, (b) xy and (c) yy curvature fields magnitude (differentiation of rotation fields with ±1% signal amplitude of pseudorandomnoise using spline technique).



coefficients of the function. These coefficients are computed byminimising the least square estimator:

Xm

j¼1

jSðxiÞ � f ðxiÞj2; ð5Þ

where f(xi) is the value of the measured field at point xi. The nth or-der derivative, up to the fourth order, of the cubic B-spline can nowbe directly computed according to the equation:

dnSðxÞdxn ¼

Xm

j¼1

Bj;3ðxiÞajþn with ajþn ¼ 0 for jþ n > m;

n ¼ 1;2;3;4: ð6Þ

Finally, a two-dimensional field spatial derivative and smoothingtechnique was also applied. It relies on the direct fast Fouriertransform (FFT) and its inverse (FFT�1). The computations are per-formed in the wave number domain, where the high frequency

components, usually associated with experimental noise, are sup-pressed and the remaining components multiplied by a wave num-ber vector. The m and n order spatial derivative of the field f(x, y) inthe directions x and y, respectively, are defined by [21]:

onþmf ðx; yÞoxnoym

¼ FFT�1 ð�juÞnð�jvÞm � FFT½f ðx; yÞ� � �hðu;vÞ� �

; ð7Þ

Table 1Localisations and energies of impacts.

Position (x, y) (mm) Impact energy (J)

Impact 1 (74, 51) 13.5Impact 2 (215, 160) 26.2

Fig. 10. (a) Experimental set-up to perform impact and (b) location of the two impacts.

Fig. 11. General view of experimental set-up for the measurements of the mobility frequency response functions (FRF).

Table 2Natural frequencies for the three states of the plate.

Frequencynumber

Undamaged Impact 1 Impact 1 plus 2

Frequency(Hz)

Frequency(Hz)

Difference(%)

Frequency(Hz)

Difference(%)

1 97.7 97.8 0.16 98.1 0.332 158.6 158.6 0.00 158.3 �0.193 251.6 251.6 0.00 250.3 �0.524 267.3 267.2 �0.04 266.7 �0.195 329.1 328.9 0.06 328.4 �0.156 440.2 440.2 0.00 438.8 �0.327 492.7 492.5 �0.04 492.0 �0.108 532.5 532.8 0.06 526.8 �1.139 730.9 730.9 0.00 728.3 �0.36

10 776.1 776.1 0.00 771.1 �0.6411 787.8 788.0 0.03 780.8 �0.9112 858.9 858.8 �0.01 858.8 0.0013 970.6 970.5 �0.01 965.5 �0.52



where �hðx; yÞ is the filtering function, where u and v are the wavenumber vector in the direction x and y, respectively and j ¼

ffiffiffiffiffiffiffi�1p

.The filtering technique used in this work is the average tech-

nique, which is usually applied for the removal of high frequencyspatial distribution noise, e.g. speckle noise, to improve the qualityof images [22]. This filter performs spatial filtering by applying thearithmetic mean to the level values in a square or rectangular win-dow surrounding a point, and can be computed by using an imageconvolution technique, according to the equation:

f ðx; yÞ ¼ f ðx; yÞ � hðm;nÞ; ð8Þ

where the symbol � is the convolution operator, f(x, y) is a mea-sured full-field, h(m, n) the window filter, such that m and n arethe x and y axes window sizes, respectively. This technique provesto be more computational efficient than others, such as the Gauss-ian, median, Wiener filter, cubic splines, FFT, windowed FFT, shorttime FFT or wavelet transform techniques [23].

Instead of applying a unique filter to the measured field, it ispresently proposed a progressive filtering of the information inthe course of the process of the successive spatial derivatives com-putation. A single filter is normally used for the elimination of highfrequency noise. In this process, the high order signal components,important for the localisation of damages, are also eliminated. Onthe other hand, the high frequency noise is propagated and ampli-fied by the numerical differentiation, becoming preponderant tothe signal components. The proposed methodology tries to dealsimultaneously with these two opposite problems by attenuatingboth high frequency signal and noise, without removing the most

important high order signal components. For experimental mea-surements with high spatial sample resolution, an effective imple-mentation of this method can be performed by relaxing the filterand applying progressive low-pass filtering technique trough thenumerical differentiation process. Fig. 1(a) shows a scheme of thexx curvature field kxx(x, y) computation from the displacement fieldw(x, y) as performed in [11]. In the present work, and since themeasured field are the rotation field hx(x, y) or hy(x, y), the curva-ture field computation is highly simplified, as can be seen inFig. 1(b). In this case, only one spatial differentiation and two filter-ing processes are needed to obtain the xx curvature field. The yyand xy curvatures fields, kyy(x, y) and kxy(x, y), can be computedin a similar way.

The filtering and differentiation techniques parameters arefound by the minimisation of the root mean square error calculatebetween the post-processed experimental field and the numericalsimulation field. The optimal parameters strongly dependent of thecase study and can only are found by an iterative and heuristic pro-cess. It was seen that, when using experimental data, these param-eters need few adjustments and with few iterations one achievesthe optimal solution [23].

2.2. Damage localisation method

Each mode shape l of an undamaged plate can be characterisedby two rotation fields, hx(l)(x, y), and hy(l)(x, y), and three curvaturefields, kxx(l)(x, y), kxy(l)(x, y), and kyy(l)(x, y). If there is some kind ofdamage, the plate will present a different dynamic behaviour.

Fig. 12. Experimental equipment for rotation fields measurements: (a) shear interferometric system, (b) suspended plate and loudspeaker, (c) LUMONICS� double pulse Rubylaser.

Fig. 13. Post-processing of mode shape phase maps measurements: (a) phase map obtained with interferometric system, (b) phase map after applying the phase filtertechnique, (c) mode shape rotation field.



Hence, for a damaged plate the two rotation fields and the threecurvature fields will be ~hxðlÞðx; yÞ; ~hyðlÞðx; yÞ; ~kxxðlÞðx; yÞ; ~kxyðlÞðx; yÞ, and~kyyðlÞðx; yÞ. Based on this, the differences between undamaged anddamaged rotation fields and undamaged and damaged curvaturefields can be defined as damage location indicators. For instance,if one takes the lth mode shape, the difference between the magni-tude of the undamaged and the damaged x rotation in the pointwith coordinates (xi, yi) is given by ~hxðlÞðxi � yiÞ � hxðiÞðxi; yiÞ. The dif-ference between the magnitude of the undamaged and the dam-aged y rotations, xx curvatures, xy curvatures, and yy curvaturesin the same point and for the same mode are obtained in a similarway.

3. Applications

The damage localisation method was applied to: (i) a rectangu-lar isotropic plate numerically studied and (ii) a rectangular lami-nated plate subjected to an experimental analysis. In the firstapplication, the undamaged and damaged curvature fields of theisotropic plate were computed by differentiating numerical rota-tion fields. For the analysis of the laminated plate, its undamaged

curvature fields were computed relying on numerical rotationfields and the damaged curvature fields were computed by differ-entiating experimental rotation fields.

3.1. Numerical application

In order to validate the damage localisation method in the pres-ence of two different size damages and chose the best differentia-tion technique, a thin rectangular isotropic plate was numericallystudied using the finite element method. The plate was discretisedand a modal analysis was performed in ANSYS� using 60 � 80SHELL93 finite elements [24]. In order to analyse the interactionof two different sized damages in the first twelve natural frequen-cies, their position and dimension were selected according to theanti-nodal regions distribution of the twelve mode shapes transla-tion fields (Fig. 2). The two damages are numerically simulated byreducing in 50% the stiffness of some of the elements.

A visual inspection of mode shapes translation fields does notreveal the damages presence. However, some of the mode shapesrotation fields show local changes that can be correlated to thedamages, in particular the larger one. It was also found that thesuccess of damage location depends on the rotation field consid-

Fig. 14. Undamaged plate modes x rotation fields.



ered. As an example, the fourth mode y rotation field hy(4)(x, y) hasvisible perturbations in the damaged areas, while the correspond-ing x rotation field hx(4)(x, y) has almost no irregularities, as can beseen in Fig. 3.

3.1.1. Damage localisation with clean rotation fieldsThe rotation fields that came directly from ANSYS� were sub-

tracted in order to compute their differences. There are largespread areas with high differences in the rotation fields magnitude,as shown in Fig. 4. These areas contain the damaged elements.Fig. 5 shows the differences between the first mode curvaturefields magnitude. Their high values are now restricted to smallerareas, which contain the damaged elements. This is more notice-able in the differences between the xx curvature field magnitude(Fig. 5(a)). Since in this damage analysis the numerical rotationfields present no noise, the curvature fields were computed by dif-ferentiating the rotation fields using finite differences. By compar-ing Figs. 4 and 5 one sees that the differences in curvature fields areone order of magnitude larger than the differences in the rotationfields, which shows that curvature differences are a better indica-tor of damage.

3.1.2. Damage localisation with noisy rotation fieldsTo simulate the noise that is always present in experimental

data, the damage method was also applied to the rotation fieldspreviously contaminated with ±1% signal amplitude of pseudoran-dom noise. It can be seen in Fig. 6 that the differences in rotationfields fail to locate the damages. However, the curvature differ-ences perform better, although only the larger damage is located.The best damage localisation is obtained when the Gaussian tech-nique is used to compute the curvature fields (Fig. 7). The worstdamage localisations are obtained with the FFT technique(Fig. 8), because of the border effect. This technique, contrary tothe Gaussian one, is not able to deal with the edge discontinuities,even if the field is replicated beyond the plate edges. Its poor per-formance is due to the difficulty in obtaining continuous deriva-tives of non-periodic fields, such as the present rotation fields. Itis also possible to locate the larger damage using the spline tech-nique, which does not require a mirrored field (Fig. 9). The Gauss-ian technique shows a better performance, because it deals withhigh frequencies at a local level, i.e. at pixel level. For global anal-ysis techniques, such as the FFT and the spline techniques, the highfrequencies signals are superimposed to high frequency noise,being removed in the smoothing process.

Fig. 15. Damaged plate modes x rotation fields (impacts 1 plus 2).



3.2. Experimental application

A rectangular carbon fibre reinforced epoxy plate was experi-mentally analysed. The plate is 0.2765 m long, 0.198 m wide,1.825 � 10�3 m thick, and the layers elastic properties, specifiedby the manufacture, are: E1 = 126.103 GPa, E2 = 15.0 GPa,E3 = 15.746 GPa, m12 = 0.217, m23 = m13 = 0.208, G12 = 5.678 GPa,G23 = 5.547 GPa, G13 = 3.603 GPa. The layers stacking sequence is[0/90/+45/�45/0/90]s and specific mass is q = 1690 kg/m3.

3.2.1. Damage characteristicsThe carbon fibre reinforced epoxy plate was subjected to delam-

inations produced by two low velocity impacts, using a steel spher-ical impactor of 0.988 kg. The impact energy was computed basedon the height of the sphere fall. The localisations and energies ofthe two impacts are presented in Table 1. The plate was fixedagainst a support structure along the two shorter edges(Fig. 10(a)). The sphere was guided trough an acrylic tube withsmall cuts, in order to reduce air resistance. The plate surfacewas painted with a thin coat of removable white paint so thatthe impact zone can be registered and the supporting structure isslightly tilted relatively to the tube axis in order to project thesphere, after the impact and, hence, avoid a second impact in the

plate. The two impact locations are indicated in Fig. 10(b) and nodamage can be visually detected.

3.2.2. Natural frequencies and rotation fieldsThe experimental modal response tests performed, considering

the plate free in space, consisted in the measurement of its naturalfrequencies and respective natural modes rotation fields beforedamage, after impact 1, and impacts 1 plus 2. The plate was sus-pended using elastic rubber bands, creating nearly free boundaryconditions (Fig. 11). The natural frequencies were obtained bymeasuring the mobility frequency response functions (FRF). Thetransient excitation was applied using a small impact hammerPCB� model 084A17 and the velocity response was measured with-out contact, using a laser vibrometer Polytec� model OFV-3001. Areflective tape was fixed in the plate surface to increase the laserbeam reflectivity and, therefore, improve the signal quality. Bothsignals, i.e. the excitation and response signals, were amplifiedand conditioned in the Bruel & Kjaer 2525 amplifier and analysedin frequency with the dynamic signal analyser Oros�-OR35 system.

For each one of the plate states, i.e. undamaged case, impact 1,and impact 1 plus 2, the mobility function H1 type was measuredin the upper right corner of the plate in the same surface as theone where the mode shapes are measured. These functions charac-

Fig. 16. Undamaged plate modes xx curvature fields.



teristics lead to the conclusion that, in general, damage produces adecrease in the natural frequencies values. This decrease is morepronounced for higher order natural frequencies.

Table 2 shows the natural frequencies for the three states stud-ied and their respective differences. One can see that after impact 1the plate natural frequencies present a negligible variation rela-tively to those in the undamaged state, being of the same orderas the frequency resolution (156.25 mHz). However, the damagecase induced by impact 1 plus 2 produce, in general, a noticeablereduction in the plates natural frequencies, being the maximumvalue of 1.13% registered in the eighth natural frequency. For thisparticular state, there is an increase in the first natural frequency.Similar results have been reported in the literature [9,11].

Speckle shear interferometry is a full-field technique that allowsnon-contact measurements of displacements spatial gradients witha resolution of a few tenth of micrometers [25]. Since it measuresthe displacement spatial gradient in a predetermined direction, thistechnique is almost insensitive to rigid body motions. It belongs tothe optical techniques class that also include other full-field tech-niques, such as holographic interferometry, projections Moiré, dig-ital holography, and point-wise techniques, such as Dopplervibrometry [26]. The shear interferometric technique was firstdeveloped by two independent research groups [27,28].

A Mach–Zehnder shear interferometer (Fig. 12(a)) with a spatialcarrier in the primary fringes [29,30] was built to measure the nat-ural modes rotation fields of the plate in free-free condition(Fig. 12(b)). The measurement system also uses a LUMONICS� dou-ble pulse Ruby laser to generate a pair of impulses with 690 nmwave length, separated by 400 ls (Fig. 12(c)). The holograms ob-tained from this double exposure were recorded using a1608 � 1208 pixels CCD camera JAI� model CV-M2 and then areprocessed to obtain the wrapped phase maps, which representsthe spatial variation of rotation fields. In order to obtain the fullrotation fields, the high frequency phase noise and phase disconti-nuities need to be removed by dedicated image processing tech-niques [23]. The phase filter technique proposed by Kemao et al.[31] and phase unwrapping technique proposed by Volkov et al.[32] were applied to obtain the mode shapes rotation fields.Fig. 13 presents the results of the filtering and unwrapping pro-cesses of the phase map of the plate tenth mode shape.

The plate was suspended by elastic rubber bands and excitedusing a loudspeaker placed behind it (Fig. 12(b)). The plate frontalsurface was painted with a thin coat of removable white paint toassure a uniform irradiation intensity of the measured surface.

A Tektronix� model AFG320 signal generator along with theloudspeaker was used to excite the plate at its natural frequencies.

Fig. 17. Damaged plate modes xx curvature fields (Impacts 1 plus 2).



For each natural frequency, the signal amplitude was adjusted toframe the vibration amplitude within the systems measurementrange. This same signal was used to synchronize the laser triggerand the plate harmonic motion.

3.2.3. Damage localisationThe modes of the undamaged plate rotation fields were numer-

ically obtained using a model of 254 � 404 ANSYS� SHELL93 finiteelements [24], while the modes of the damaged plate rotationfields were measured using the pulsed laser and the shear interfer-ometer, with a spatial phase modulation [29]. The image that de-scribes each rotation field has a resolution of 996 � 1295 pixels.

In Figs. 14 and 15 are presented the x axis rotation fields of thefirst twelve modes of the undamaged plate and the correspondingones after impacts 1 and 2, respectively. A global comparative anal-ysis between the several rotation fields pairs show that, with theexception of the fourth mode, they present similar configurations.

As seen in the numerical application, small perturbations or dis-continuities in the rotation fields, due to the presence of damage,are more pronounced in the curvature fields. The xx curvaturefields are obtained by applying the Gaussian differentiation tech-nique described in Section 2.1. These curvature fields of theundamaged plate and the corresponding ones after impacts 1and 2 can be seen in Figs. 16 and 17, respectively.

The visual cross-examination of the curvature fields distribu-tion reveals perturbations in the configurations relative to the

undamaged structure, therefore exposing the presence of adelamination in the region where the impact 2 took place. Thisresult confirms that the curvature fields are more sensitive todamage than the rotation fields. With the technique now pro-posed, where the rotation fields are experimentally measured,the numerical differentiation to obtain the curvature fields is per-formed only once and not twice, decreasing the computationalcost. Furthermore, in each differentiation is applied a low-pass fil-ter that is responsible for eliminating the experimental noise andthe higher order terms that characterise the perturbations in thecurvatures associated with the damage. Hence, by decreasing thenumber of differentiations the perturbations due to damage arepreserved.

In order to corroborate the localisation results obtained by theanalysis of the damage plate curvature fields, the differences be-tween them and the undamaged plate curvature fields were alsocomputed. It should be noted that the maximum amplitude, thespatial resolution and the phase of the undamaged and damagedcurvature fields had to be previously adjusted. The curvature fieldsmagnitude differences of the undamaged plate and the damagedplate, after impacts 1 plus 2, are shown in Fig. 18. Its analysis re-veals, in several natural modes, the presence of an area that con-tains the location of impact 2. There is a lack of perturbations inthe curvature fields differences near the low energy impact area,i.e. impact 1. One should note that the small impact area is repre-sented mainly by high signal components. This means that the per-

Fig. 18. Differences between the xx curvature fields magnitude (impacts 1 plus 2).



turbations due to small damages are mixed with high frequencynoise, being both eliminated in the differentiation and filteringprocesses.

Fig. 19 shows a plate image and the delamination area lo-cated using the xx curvature fields differences of the eighth nat-ural mode shape, which is also the mode that presents thehigher difference between the undamaged and the damagedplate natural frequencies (Table 2). The delamination area is de-fined by the contour that inscribes the two maximum juxta-posed amplitudes, presenting a disposition near the regionwhere impact 2 occurred.

4. Conclusion

This paper presents a numerical-experimental technique thatdecreases the number of spatial differentiations needed tonumerically compute modal curvature fields, by means of mea-suring modal rotation fields. This process leads to a significantimprovement in structural damage localisation. This improve-ment is due to the preservation of higher order terms that char-acterise the perturbations in the curvatures associated with thedamage. If two differentiations instead of one are performed,these perturbations, if not sufficiently high, are lost in thesmoothing and filtering processes. The use of different spatial dif-ferentiation techniques to compute the curvature fields, such asthe Gaussian function derivative, the FFT based differentiationand the spline differentiation, was discussed. It was found thatthe Gaussian differentiation is the one which present better re-sults, since it deals with high frequencies at local level. This tech-nique also deals well with the discontinuities at the edges. Acomparison between modal rotation and modal curvature basedmethods for damage location was also presented. The numericaland experimental applications of both methods to an isotropicand a laminated plate show that the differences in curvaturesare more efficient than the differences in rotations. The proposedmethodology could be very useful in the development of a struc-tural health monitoring tool for the inspection of laminate struc-tures in industrial environment. Another major improvement inthe characterisation of damages will be the development experi-mental/numerical tool for quantification of their severity, whichwill be performed by integrating the experimental mode, shapesrotation fields into a finite elements model.

Acknowledgements

The authors greatly appreciate the financial support of FCT/POCTI (2010)/FEDER through Projects POCTI/EME/56616/2004,PPCDT/EME/56316/2004, POCI/EME /63236/2004, the EU throughFP6-STREP Project contract No. 013517-NMP3-CT-2005-0135717.

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x [m]

y [m

]

0 0.075 0.14 0.205 0.2765

0

0.05

0.1

0.15

0.2

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