Special Issue Article Damage detection in plates using two-dimensional directional Gaussian wavelets and laser scanned operating deflection shapes

Special Issue Article

Structural Health Monitoring

12(5-6) 457–468

� The Author(s) 2013

Reprints and permissions:

sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1475921713492365

shm.sagepub.com

Damage detection in plates usingtwo-dimensional directional Gaussianwavelets and laser scanned operatingdeflection shapes

Wei Xu1, Maciej Radzienski2, Wies1aw Ostachowicz2,3 and Maosen Cao1,2

AbstractMode shape analysis by wavelet transform has been used effectively for vibration-based damage detection in plates. Asan extension of previous studies, this study focuses on an improved method for damage detection in plates: scrutiny ofoperating deflection shapes by two-dimensional directional Gaussian wavelet transforms. With this method, the pro-posed two-dimensional directional Gaussian wavelet can characterize directional information about damage; moreover,the operating deflection shapes can be used to address the real-time dynamic characteristics of a plate. To identify dam-age, the local surface of the plate is scanned using a scanning laser vibrometer to generate the local operating deflectionshape, which is interrogated by two-dimensional directional Gaussian wavelets for damage. The feasibility of the methodis numerically demonstrated using a low-magnitude operating deflection shape of a two-sided clamped plate, incorporat-ing white noise with signal-to-noise ratio of 40 dB. The applicability of the method is then experimentally validated bydetecting a cross-like notch in a suspended aluminum plate with the operating deflection shapes measured by a scanninglaser vibrometer. Numerical and experimental results show that the method is capable of revealing directional featuresof small damage with high precision and strong robustness against noise. It appears that this damage detection method isrelated only to the spatially distributed measurement of vibrational responses in local critical regions of the plate. Withthis local property, the method requires no numerical or physical benchmark models for the entire structure in questionnor any prior knowledge of either the material properties or the boundary conditions of the structure. (The Matlabcode performing directional Gaussian wavelet transform can be provided by the corresponding author as per request.)

KeywordsDamage detection, mode shape, operating deflection shape, two-dimensional wavelet transform, plate, vibration, scan-ning laser vibrometer

Introduction

Vibration-based damage identification of beam-typestructures using one-dimensional (1D) wavelet trans-form has been widely investigated during the lastdecade.1–5 In contrast, vibration-based damage diagno-sis of plate-type structures by means of two-dimen-sional (2D) wavelet transform is a relatively new areaof research. In addition, the increasing applications ofplates and the rapid prevalence of wavelet transformssupport the development of wavelet transform–baseddamage detection of plate-type structures.

Existing wavelet transform–based methods fordetecting damage in a plate can be categorized into twogroups: (1) 1D wavelet transform is carried out onmode shape lines of a plate to identify damage.6–10 The

mode shape lines are obtained by degrading the 2Dmode shape of a plate in the length (x) and width (y)directions and (2) a 2D mode shape of a plate is directlyanalyzed by 2D wavelets to characterize damage in athree-dimensional (3D) scale-space domain.11–14

1Department of Engineering Mechanics, Hohai University, Nanjing,

People’s Republic of China2Mechanics of Intelligent Structures Department, Institute of Fluid-Flow

Machinery—Polish Academy of Sciences, Gdansk, Poland3Faculty of Automotive and Construction Machinery, Warsaw University

of Technology, Warsaw, Poland

Corresponding author:

Maosen Cao, Department of Engineering Mechanics, Hohai University,

Nanjing, 210098, People’s Republic of China.

Email: [email protected]

For the first group of methods, representative inves-tigations are as follows. Wang and Deng6 decomposeda 2D displacement surface of a plate into two sets ofdisplacement lines in the x and y directions, each with a1D Haar wavelet implemented to identify a through-thickness crack. The results showed that the 1D Haarwavelet was effective for locating the crack. Chang andChen7 examined local damage in a plate using a 1DGabor wavelet8 to deal with the mode shape lines ofthe plate. The results indicated that the damage couldbe effectively localized by the abrupt change in waveletcoefficients. Douka et al.9 used a 1D Symlet wavelet tocope with the y-directional mode shape lines of a rec-tangular plate with a crack parallel to one edge. Thelocation of the crack was identified by the abnormalityof wavelet coefficients. Moreover, the depth of thecrack was estimated with the Holder exponent.10

Essentially, this group of methods follows the regimeof wavelet-based damage detection of a beam, wherethe 1D wavelet is utilized to handle the 1D mode shape.It is noteworthy that degrading a 2D mode shape intotwo sets of x- and y-directional mode shape lines cancause loss of modal information, such as the disappear-ance of diagonal modal features, possibly impairing theaccuracy of damage detection.

For the second group of methods, typical studies areas follows. Loutridis et al.11 constructed a separable-product 2D wavelet based on the 1D Symlet waveletand applied this 2D wavelet to mode shapes of a platefor crack identification. They successfully locatedcracks of various depths. Rucka and Wilde12 utilized aseparable-product 2D wavelet arising from a 2D reversebiorthogonal wavelet to process the mode shapes of acracked plate. The location and size of the crack wereidentified by steep changes in the modulus and theangle of wavelet coefficients. Kim et al.13 employed the2D Haar wavelet to solve damage index equations in amulti-resolution wavelet domain and proposed anindex of damage via inverse wavelet transform. Thenumerical results showed the effectiveness of the indexin characterizing damage in a plate. Fan and Qiao14

proposed a 2D continuous wavelet-based algorithmusing a family of wavelets derived from a 2D Gaussianfunction for damage detection in plates. This algorithmwas shown to be effective in revealing damage frommode shapes of numerical and experimental cases. Ingeneral, this group of methods features direct charac-terization of damage using 2D wavelets to process 2Dmode shapes. The consistency in dimensions in thisgroup of methods is more advantageous than the meth-ods of the first group for identifying damage in plates.

Most 2D wavelet transform–based methods fordamage detection in plates have a common trait: modeshapes acquired by conventional contact measurementare processed by a 2D wavelet transform to depict

damage. This trait exposes several limitations primarilyresulting from the method of measurement: (1) the con-tact measurement easily imposes noise on the measuredmode shapes, attributable to the complex linkagecables, additional masses due to sensors, sensitivity tovariations in environmental factors such as tempera-ture, humidity, and so on; (2) because of the difficultyin deploying dense sampling points, the spatial resolu-tion of contact measurement is commonly low, incap-able of satisfying the requirements for slight damagedetection; and (3) to measure mode shapes of a plateusing contact measurement, interrupting its normalrunning status is usually a precondition for setting upsensing devices. These limitations provide motivationfor the use of an advanced measurement technique toacquire dynamical responses of a structure underinspection.

Currently, the scanning laser vibrometer (SLV) isbeing increasingly used in vibration response measure-ment for structural damage detection applications.Briefly, the merits of a SLV in acquiring dynamicresponses are fourfold:15–21 (1) noncontact measure-ment greatly diminishes the adverse influence of mea-surement noise; (2) optical scanning affords theopportunity of real-time acquisition of dynamicresponses; (3) a SLV facilitates high-resolution spatialmeasurement due to its facility of optically scanning avibration surface; and (4) arbitrary harmonics scopedin a wide-frequency band can be acceptable excitationsfor a SLV. These features render a SLV suitable formeasuring the operating deflection shapes (ODSs) ofan in-service structure, greatly supporting online healthmonitoring.16 In this study, an ODS represents a gener-alized dynamic deflection of a structure under harmo-nic vibration, possibly caused by active, ambient, orself-excitation. An ODS is dominated by a modeshape only when the vibration frequency is equal toone of modal frequencies, in which case a mode shapecan be viewed as a particular ODS. Recently, ODSshave been used in structural damage detection by sev-eral researchers.17–19

In this study, a 2D directional Gaussian wavelettransform is expressly elaborated, differing from theconventional wavelet transform in the inclusion of arotational parameter that allows description of thedirectional features of dynamic responses; moreover,processing ODSs by 2D directional Gaussian wavelettransform is explored with the aim of providing asophisticated method for damage detection in plates.

This study is organized as follows: Section ‘‘2Ddirectional Gaussian wavelet transform’’ formulatesthe 2D directional Gaussian wavelet transform, withparticular emphasis on its characteristics for damagedetection in plates. Section ‘‘Method demonstration innumerical simulations’’ presents numerically simulated

458 Structural Health Monitoring 12(5-6)

cases to demonstrate the feasibility of applying 2Ddirectional Gaussian wavelets to ODSs for damagedetection. Section ‘‘Experimental validation’’ validatesthe applicability of the proposed method through anexperimental program with the ODSs acquired by aSLV. Several conclusions are drawn in section‘‘Conclusion.’’

2D directional Gaussian wavelet transform

2D directional Gaussian wavelets

A 1D wavelet is defined as22

cu;s xð Þ¼ 1ffiffisp c

x� u

s

� �ð1Þ

which satisfies the wavelet admissibility condition

�

c vð Þ�� ��2

vj j dv\‘ ð2Þ

where s and u are the scale (or dilation) and translation(or space) parameters, respectively; cðxÞ is the motherwavelet with the Fourier counterpart, cðvÞ. The energyof cðxÞ is normalized by

�

c2 xð Þdx ¼ 1 ð3Þ

Analogous to a 1D wavelet, a 2D wavelet isdefined as

cu;v;sðx; yÞ ¼1

sc

x� u

s;y� v

s

� �ð4Þ

where u and v are the translation (or space) parametersin the x and y directions, respectively; cðx; yÞ is themother wavelet.

To increase flexibility, a rotational parameter u isintroduced into cðx; yÞ, yielding a 2D directionalwavelet

cu;v;s;uðx; yÞ ¼1

sc

x0 � u

s;y0 � v

s

� �ð5:1Þ

with

x0

y0

� ¼ cos u sin u

�sin u cos u

� �x

y

� ð5:2Þ

where u specifies the orientation of the wavelet.For a 2D signal f ðx; yÞ, for example, an ODS of a

plate, the associated 2D directional wavelet transformcan be expressed as

Wf u; v; s; uð Þ¼ f ;cu;v;s;u

�¼ 1

s

ðþ‘

�‘

ðþ‘

�‘

f x; yð Þc

x cos uþ y sin u� u

s;�x sin uþ y cos u� v

s

� �dxdy

ð6Þ

where Wf ðu; v; s; uÞ denotes wavelet coefficients. Forprovided values of u; v; s, and u, Wf ðu; v; s; uÞ measuresthe degree of similarity between f ðx; yÞ and cu;v;s;u at thespecific location ðu; vÞ, scale s and angle u.

In numerical implementation of the convolution inequation (6), border distortion may occur adjacent tothe edges of the analyzed signal, signified by abruptchanges of wavelet coefficients.23 This distortion isattributed to the finite size of the analysis signal andwavelets involved in numerical convolution. To presentthe wavelet analysis results clearly, a simple method ofignoring the subregions containing border distortions isadopted in this study.14

The classical Gaussian function, g0ðx; yÞ ¼ðffiffiffiffiffiffiffiffiffip=2

p�1

e�ðx2þy2Þ, is used to construct a (m, n) order

Gaussian mother wavelet as24

gm;nðx; yÞ ¼Cm;nð�1Þmþn ∂mþng0ðx; yÞ∂xm∂yn

ð7Þ

where Cm;n is a normalization constant such thatgmþnðx; yÞ has unit energy.

From gm;nðx; yÞ, a family of 2D Gaussian waveletscan be derived as

gm;nu;v;sðx; yÞ ¼

1

sgm;n x� u

s;y� v

s

� �ð8Þ

Similar to equation (5), a family of 2D directionalGaussian wavelets can be expressed as

gm;nu;v;s;uðx; yÞ ¼

1

sgm;n

x cos uþy sin u� u

s;�x sin uþy cos u� v

s

� �ð9Þ

It is known that conventional Gaussian waveletshave attractive properties for damage detection, suchas symmetry, smoothness, differentiability, localizabil-ity, and explicit mathematical expressions.25–27 Besidesthese properties, the 2D directional Gaussian waveletshave the merit of characterizing directional features ofdamage. In what follows, a specific 2D directionalGaussian wavelet is selected for damage detection inplates.

Xu et al. 459

Specific wavelets used for damage detection

In equation (7), m ¼ 2; n¼ 2 specifies a (2, 2) order 2Dmother Gaussian wavelet

g2;2ðx; yÞ ¼ 1

3ffiffiffiffiffiffiffiffiffip=2

p ð�2þ 4x2Þð�2þ 4y2Þe�ðx2þy2Þ ð10Þ

This mother wavelet is illustrated in Figure 1. Fromg2;2ðx; yÞ, a family of 2D directional Gaussian waveletscan be derived as

g2;2u;v;s;uðx; yÞ ¼

1

3ffiffiffiffiffiffiffiffiffip=2

p �2þ 4x cos uþ y sin u� u

s

� �2 !

�2þ 4�x sin uþ y cos u� v

s

� �2 !

e�ðððx cos uþy sin u�uÞ=sÞ2þðð�x sin uþy cos u�vÞ=sÞ2Þ

ð11Þ

Four directional Gaussian wavelets with u being0;p=8;p=4; 3p=4, respectively, are derived from equa-tion (11), with their planforms illustrated in Figure 2(a)to (d), respectively.

The wavelet g2;2u;v;s;uðx; yÞ has some significant features

for detecting damage in plates: (1) Directionality: theorientation parameter renders this wavelet appropriateto clarifying damage in arbitrary directions; (2)Symmetry: the mother wavelet is symmetrical in the xand y directions, making this family of wavelets rota-tionally symmetrical, which is suited to reveal underly-ing damage-caused jump singularity in ODSs of a plate;(3) Smoothness: as a derivative of the Gaussian func-tion, this wavelet is mathematically a smooth function,with strong capability of denoising; (4) Multiscale: as amultiscale operator, g

2;2u;v;s;uðx; yÞ characterizes damage

using a series of scale signatures ranging from macro-

profiles of the damage at coarse scales to micro-detailsof the damage at fine scales; and (5) Differentiation: asa differentiation operator, g

2;2u;v;s;uðx; yÞ is appropriate for

the protrusion of weak singularity related to slightdamage from dynamic responses.

Method demonstration in numericalsimulations

The feasibility of using the 2D directional Gaussianwavelet to detect damage in plates is examined using 3Delastic finite element simulations. An aluminum plate,bearing a cross-like patch damage, clamped at its lowerand right edges, is considered, as shown in Figure 3. Theplate has the dimensions 450 mm length, 410 mm width,and 3 mm depth in the x, y, and z directions, respectively.Its Young’s modulus, Poisson’s ratio, and mass densityare taken as 70 GPa, 0.33, and 2700 kg/m3, respectively.The cross-like patch damage is centered at x = 282 mmand y = 257.5 mm, with each branch containing five10 mm 3 10 mm 3 1 mm pseudo (removed) cubes.A local surface of the plate, whose opposite counterpartcovers the cross-like patch damage, spanning from 85 to400 mm in the x direction and from 115 to 430 mm inthe y direction, is taken as the measurement zone to gen-erate the local ODS for damage identification.

The numerical model of the plate is built with20-node 3D structural solid elements (SOLID 95) usingthe commercial software ANSYS�. The measurementzone covers 63 3 63 elements, from which the ODS isacquired when the plate is subjected to a harmonicexcitation located at the point of x = 120 mm and y =80 mm (outside the measurement zone), perpendicularto the plate. Figure 3 illustrates the finite element meshalong with the zoomed-in cross-like patch damage inthe plate.

Figure 1. (a) 2D mother Gaussian wavelet and (b) its planform.2D: two-dimensional.

460 Structural Health Monitoring 12(5-6)

ODSs

A preliminary analysis of the displacement frequencyresponse function (FRF) using the sweep frequencymethod is performed to determine the basic dynamic

property of the plate, with a particular subset of theobtained FRF, as illustrated in Figure 4. From thisFRF profile, a lower valued frequency point of 1510Hz is chosen as excitation frequency for generating a

Figure 2. Planforms of 2D directional Gaussian wavelets: (a) s = 2, u = 0; (b) s = 3, u = p/8; (c) s = 4, u = p/4; and (d) s = 6, u = 3p/8.2D: two-dimensional.

Figure 3. Finite element mesh of plate along with zoomed-in cross-like patch damage.

Xu et al. 461

lower magnitude ODS, with the consideration that thesmaller magnitude of ODS is helpful to justify the cap-abilities of the damage detection algorithm. Vibrationof the plate subjected to this excitation is numericallysimulated by the finite element model built above. Thezoomed-in local ODS of the measurement zone is dis-played in Figure 5(a) that is extracted from the globalODS of the plate, as shown in Figure 5(b). As a refer-ence, the 29th and 30th mode shapes adjacent to theODS are presented in Figure 6. Clearly, the globalODS (Figure 5(b)) has lower magnitude than the 29thand 30th mode shapes (Figure 6). This lower magnitudeODS at a non-modal frequency is of greater generality

for reflecting the real-time dynamic characteristics ofthe plate.

Damage detection

The local ODS of the measurement zone is analyzedusing the 2D directional Gaussian wavelets, g

2;2u;v;s;uðx; yÞ,

(equation (11)). Six wavelets specified withu ¼ 08; 158; 308; 458; 608; 758 can evenly sweep the ODSwhile caring about its principal directional information.After wavelet transform depending on each wavelet,the planforms of wavelet coefficients are illustrated inFigure 7 for s = 10. Clearly, each directional waveletnot only pinpoints the location but also depicts thedirectional feature of the damage in its favorable direc-tion. The collective diagrams of wavelet coefficient forevery wavelet comprehensively characterize the config-uration of the cross-like patch damage.

Noise robustness

The 2D directional Gaussian wavelet can reveal damageunder noisy environments due to its intrinsic multiscaleproperty of suppressing noise and strengthening dam-age features.22,25 To demonstrate this trait, Gaussiannoise with the relatively low signal-to-noise ratio (SNR)of 40 dB (the ratio of the root mean square (RMS)amplitude of noise to the RMS amplitude of signal is1%) is added to the local ODS of the measurementzone (Figure 5(a)) to yield a noise-contaminated ODS.The ODS is processed by g

2;2u;v;s;uðx; yÞ with u = 0� to

illustrate robustness to noise. The planforms of waveletcoefficients for s ¼ 10; 12:5; 15; 17:5 are shown inFigure 8(a) to (d), respectively. Clearly, the cross-likepatch damage is progressively identified with the

Figure 4. A subset of displacement FRF of plate.FRF: frequency response function.

Figure 5. (a) Zoomed-in local ODS of measurement zone and(b) extracted from global ODS of plate.ODS: operating deflection shape.

Figure 6. (a) The 29th and (b) 30th mode shapes of the plate.

462 Structural Health Monitoring 12(5-6)

increase of scale from 10 to 17.5; simultaneously, theinterference of background noise is gradually sup-pressed. At the smallest scale 10, intensive noise domi-nates the planform, severely obscuring the features ofthe damage; at the greatest scale 17.5, the configurationof the cross-like patch damage can be basically

differentiated from the background noise. By virtue ofthe multiscale property of 2D directional Gaussianwavelets, a proper interval of scale exists over whichnoise can be largely suppressed, while the effective signalis mostly strengthened, demonstrating the strong robust-ness of 2D directional Gaussian wavelets to noise.

Figure 7. Effect of u of g2;2u;v;s;uðx;yÞ (s ¼ 10) on damage characterization: (a) u = 0�, (b) u = 15�, (c) u = 30�, (d) u = 45�, (e) u = 60�,

and (f) u = 75�.

Xu et al. 463

Experimental validation

The proposed method of applying 2D directionalGaussian directional wavelets to ODSs for damagedetection in a plate is experimentally validated using aSLV to measure the ODSs of the plate.

Experimental setup

An aluminum plate of elastic modulus 70.5 GPa andmaterial density 2680 kg/m3, with dimensions of 1000mm 3 1000 mm 3 4 mm in the x, y, and z directions,respectively, is used as experimental specimen, as shownin Figure 9(a). The gray square in Figure 9(a) lying onthe intact surface (the damage exists on the oppositesurface) is taken as the measurement zone, with an areaof 920 mm 3 920 mm, that is, spanning from 40 to 960mm in the x and y directions, respectively. This mea-surement zone contains cross-like notch damage shapedin an ‘‘X,’’ with each inclined branch being 40 mm long,1 mm wide and 2 mm deep (2 mm reduction in depthaway from the original surface). The cross-like notch ismanually manufactured by gradually carving the plate

using a hard, sharp tool. Figure 9(b) shows the magni-fied cross-like notch. A preliminary modal test showsthat this notch causes insignificant change in naturalfrequencies.

Vibration of the plate is induced by a harmonic exci-tation exerted by a circular 10-mm-diameter smartpiezoelectric lead zirconate titanate (PZT) actuator,28

which is located at 90 and 65 mm from the lower andright edges of the plate, respectively. As the platevibrates steadily, the measurement zone, as shown inFigure 9(a), is scanned by a SLV (Ploytec PSV-400) togenerate the local ODS comprising 451 3 449 mea-surement points. The experimental setup is shown inFigure 10. The obtained local ODS of the measurementzone is processed by the 2D directional Gaussian wave-lets to interrogate the damage in the plate.

Results

When a harmonic excitation of 830 Hz is arbitrarilychosen, the local ODS of the measurement zone and itsplanform are shown in Figure 11(a) and (b), respec-tively. In Figure 11(b), the layout of the cross-like

Figure 8. Illustration of noise robustness of g2;2u;v;s;uðx;yÞ with u ¼ 0 for damage detection: (a) s = 10, (b) s = 12.5, (c) s = 15, and

(d) s = 17.5.

464 Structural Health Monitoring 12(5-6)

notch is indicated by a dashed circle. This local ODS isdealt with by the 2D directional Gaussian waveletg

2;2u;v;s;uðx; yÞ at s = 3. A group of wavelets specified by

u ¼ 08; 258; 508, and 758 are separately employed toscrutinize the ODS for damage interrogation. Thewavelet coefficients obtained and their respective plan-forms are illustrated in Figure 12. Clearly, the peak ofwavelet coefficients in each diagram can reveal thecross-like notch. Furthermore, distinct directionalinformation can be observed from the planforms foru ¼ 08; 258; and 758, suggesting the strong directionalfeatures of the notch in those directions, whereas noobvious directional information can be distinguishedfrom the planforms for u ¼ 508, implying weak direc-tional features in those directions. From the joint plan-forms, four subbranches of the cross-like notch can bereadily identified.

For performance comparison, the conventional 2DMexican hat wavelet, c2;2ðx; yÞ, that is, the second deri-vative of the 2D Gaussian function, g0ðx; yÞ, is adoptedto analyze the ODS just processed by the 2D directional

Figure 9. An aluminum plate with a cross-like notch:(a) measurement zone containing a cross-like notch and(b) zoomed-in cross-like notch.PZT: lead zirconate titanate.

Figure 10. Experimental setup.

Figure 11. Local ODS at 830 Hz and its planform for the measurement zone: (a) local ODS at 830 Hz and (b) layout of cross-like notch.ODS: operating deflection shape.

Xu et al. 465

Figure 12. Identified cross-like notch using g2;2u;v;s;uðx;yÞ (s = 3) with (a) u = 0�, (b) u = 25�, (c) u = 50� and (d) u = 75�.

466 Structural Health Monitoring 12(5-6)

Gaussian wavelet, g2;2ðx; yÞ, producing the wavelet coef-ficients, as shown in Figure 13. Comparing the resultsin Figures 12 and 13, it can be seen that g2;2ðx; yÞ identi-fies the cross-like notch with higher accuracy and relia-bility than c2;2ðx; yÞ.

Conclusion

An efficient method of detecting damage in plates using2D directional Gaussian wavelet transforms is pre-sented in this study. With this method, a SLV-basednoncontact measurement method is used to acquire theODSs of a plate. The obtained ODSs are processed by2D directional Gaussian wavelets for damage interro-gation. The performance of the method is numericallydemonstrated using the ODSs of a two-sided clampedplate, contaminated by noise; the applicability of themethod is experimentally validated using an aluminumplate with a cross-like notch. From the study, the fol-lowing are observed:

1. SLV-based noncontact measurement provides anadvanced approach to acquiring vibrationresponses of a plate, with predominant featuressuch as high spatial measurement resolution,greatly diminishing noise interference, suited toharmonic excitations within a wide-frequencyscope, and without the need to interrupt the nor-mal running of the structure under investigation.

2. Compared with mode shapes, the ODS obtained bythe SLV can serve as a more practical quantity todepict the real-time vibration characteristics of anin-service plate-type structure.

3. The proposed 2D directional Gaussian waveletshave advantages over conventional Gaussian wave-lets in characterizing the directional features ofdamage in a plate.

4. Processing the ODS of a plate using 2D directionalwavelets is sophisticated in revealing the detailedconfiguration of the damage under a high-noise orpractical measurement condition.

5. The proposed damage detection method requiresno numerical and physical benchmark models forthe entire structure under inspection nor any priorknowledge of either the material properties or theboundary conditions of the structure.

6. In practical applications, data fusion of waveletcoefficients for a set of ODSs at different excita-tion frequencies can be considered as an supple-mentary means to reduce the adverse effectof the ODS’ node lines on damage depictionand alleviate the interference of measurementnoise.

These distinctive features make this proposedmethod a promising prototype for developing onlinehealth monitoring systems for plate-type structures.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This study was supported by the Marie Curie IndustryAcademia Partnership and Pathways Grant (grant no. 251309STA-DY-WI-CO) within the 7th European CommunityFramework Programme) (to M.C. and W.O.) and by theNational Natural Science Foundations of China for Grants(no. 50978084 and 11172091) (to W.X.).

References

1. Cao MS and Qiao PZ. Integrated wavelet transform and

its application to vibration mode shapes for the damage

Figure 13. Identified cross-like notch using the Mexican hat Gaussian wavelet.

Xu et al. 467

detection of beam-type structures. Smart Mater Struct

2008; 17(5): 17.2. Taha MMR, Noureldin A, Lucero JL, et al. Wavelet

transform for structural health monitoring, a compen-dium of uses and features. Struct Health Monit 2006; 5:267–295.

3. Filho JV, Baptista FG and Inman DJ. Time-domainanalysis of piezoelectric impedance-based structuralhealth monitoring using multilevel wavelet decomposi-tion. Mech Syst Signal Process 2011; 25(5): 1550–1558.

4. Cao MS, Chen L, Su ZQ, et al. A multi-scale pseudo-force model in wavelet domain for identification of dam-age in structural components. Mech Syst Signal Process

2012; 28: 638–659.5. Wu N and Wang Q. Experimental studies on damage

detection of beam structures with wavelet transform. PhilTrans Math Phys Eng Sci 2007; 365: 303–315.

6. Wang Q and Deng XM. Damage detection with spatialwavelets. Int J Solid Struct 1999; 36: 3443–3468.

7. Chang CC and Chen LW. Damage detection of a rectan-gular plate by spatial wavelet based approach. Appl

Acoust 2004; 65: 819–832.8. Kishimoto K. Wavelet analysis of dispersive stress waves.

JSME Int J A: Mech M 1995; 38: 416–424.9. Douka E, Loutridis S and Trochidis A. Crack identifica-

tion in plates using wavelet analysis. J Sound Vib 2004;270: 279–295.

10. Hong JC, Kim YY, Lee HC, et al. Damage detectionusing the Lipschitz exponent estimated by the wavelettransform: application to vibration modes of a beam. IntJ Solid Struct 2002; 39: 1803–1816.

11. Loutridis S, Douka E, Hadjilentiadis LJ, et al. A two-dimensional wavelet transform for detection of cracks inplates. Eng Struct 2005; 27: 1327–1388.

12. Rucka M and Wilde K. Application of continuous wave-let transform in vibration based damage detection methodfor beams and plates. J Sound Vib 2006; 297: 536–550.

13. Kim BH, Kim H and Park T. Nondestructive damage eva-luation of plates using the multi-resolution analysis of two-dimensional Haar wavelet. J Sound Vib 2006; 292: 82–104.

14. Fan W and Qiao PZ. A 2-D continuous wavelet trans-form of mode shape data for damage detection of platestructures. Int J Solid Struct 2009; 46: 4379–4395.

15. Pai PF and Young LG. Damage detection of beams usingoperational deflection shapes. Int J Solid Struct 2001; 38:3161–3192.

16. Keller E and Ray A. Real-time health monitoring ofmechanical structures. Struct Health Monit 2003; 2:191–203.

17. Sriram P. Scanning laser Doppler technique for velocityprofile sensing on a moving surface. Appl Optic 1990; 29:2409–2417.

18. Stanbridge AB, Martarelli M and Ewins DJ. Measuringarea vibration mode shapes with a continuous-scan LDV.Measurement 2004; 35: 181–189.

19. Khan AZ, Stanbridge AB and Ewins DJ. Detecting dam-age in vibrating structures with a scanning LDV. Optic

Laser Eng 2000; 32: 583–592.20. Waldron K, Ghoshal A, Schulz MJ, et al. Damage detec-

tion using finite element and laser operational deflection

shapes. Finite Elem Anal Des 2002; 38: 193–226.21. Pai PF, Oh YJ and Lee SY. Detection of defects in circu-

lar plates using a scanning laser vibrometer. Struct Health

Monit 2002; 1: 63–88.22. Mallat S. A wavelet tour of signal processing. 3rd ed. San

Diego, CA: Academic Press, 2008.23. Rucka M and Wilde K. Crack identification using wave-

lets on experimental static deflection profiles. Eng Struct

2006; 28: 279–288.24. Antoine JP. Two-dimensional wavelets and their relatives.

Cambridge: Cambridge University Press, 2004.25. Jiang X, John Ma ZG and Ren WX. Crack detection

from the slope of the mode shape using complex continu-ous wavelet transform. Comput Aided Civ Infrastruct Eng

2012; 27(3): 187–201.26. Wu N and Wang Q. Experimental studies on damage

detection of beam structures with wavelet transform. IntJ Eng Sci 2011; 49(3): 253–261.

27. Zhong S and Oyadiji SO. Detection of cracks in simply-supported beams by continuous wavelet transform ofreconstructed modal data. Comput Struct 2011; 89:127–148.

28. Park G, Sohn H, Farrar CR, et al. Overview of piezoelec-tric impedance-based health monitoring and path for-ward. Shock Vib Digest 2003; 35: 451–463.

468 Structural Health Monitoring 12(5-6)