Research ArticleReal-Time Structural Damage Assessment Using ArtificialNeural Networks and Antiresonant Frequencies

V Meruane and J Mahu

Department of Mechanical Engineering University of Chile Beauchef 850 8370448 Santiago Chile

Correspondence should be addressed to V Meruane vmeruaneinguchilecl

Received 21 April 2013 Accepted 2 September 2013 Published 12 February 2014

Academic Editor Gyuhae Park

Copyright copy 2014 V Meruane and J Mahu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The main problem in damage assessment is the determination of how to ascertain the presence location and severity of structuraldamage given the structurersquos dynamic characteristics The most successful applications of vibration-based damage assessment aremodel updating methods based on global optimization algorithms However these algorithms run quite slowly and the damageassessment process is achieved via a costly and time-consuming inverse process which presents an obstacle for real-time healthmonitoring applications Artificial neural networks (ANN) have recently been introduced as an alternative to model updatingmethods Once a neural network has been properly trained it can potentially detect locate and quantify structural damage in ashort period of time and can therefore be applied for real-time damage assessmentThe primary contribution of this research is thedevelopment of a real-time damage assessment algorithm using ANN and antiresonant frequencies Antiresonant frequencies canbe identified more easily and more accurately than mode shapes and they provide the same information This research addressesthe setup of the neural network parameters and provides guidelines for the selection of these parameters in similar damageassessment problems Two experimental cases validate this approach an 8-DOF mass-spring system and a beam with multipledamage scenarios

1 Introduction

Damage assessment must detect and characterize damageat the earliest possible stage and estimate how much timeremains before maintenance is required the structure under-goes failure or the structure is no longer usable Damageassessment offers tremendous potential for life and safetyandor economic benefits because it reduces themaintenancecost and increases the safety and reliability of the structureDamage assessment is a subject of great importance forseveral industry sectors as well as for the safety of citizensand can be applied to civil engineering structures (such asbuildings or bridges) transport vehicles (such as airplaneshelicopters trains ships or cars) and industrial equipment(such as mills turbines pumps and boilers among others)

The main problem of damage assessment is to ascertainthe presence location and severity of structural damagegiven the structurersquos dynamic characteristics The most

successful applications of vibration-based damage assess-ment are model updatingmethods based on global optimiza-tion algorithms [1ndash5] Model updating is an inverse methodthat identifies the uncertain parameters in a numerical modeland is commonly formulated as an inverse optimizationproblem In inverse damage detection the algorithm usesthe differences between the models of the structure that areupdated before and after the presence of damage to localizeand determine the extent of damage The basic assumptionis that the damage can be directly related to a decreaseof stiffness in the structure Nevertheless these algorithmsare exceedingly slow and the damage assessment process isachieved via a costly and time-consuming inverse processwhich presents an obstacle for real time health monitoringapplications Recent studies have introduced artificial neuralnetworks (ANN) as an alternative to model updating indamage assessment [6ndash8] A trained neural network canpotentially detect locate and quantify structural damage

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 653279 14 pageshttpdxdoiorg1011552014653279

2 Shock and Vibration

in a short period and hence it can be used for real timedamage assessment Damage detection by ANN has theadvantage that it is a general approach Unlike many otherdamage detection methods that are often developed forspecific quantities [9ndash11] in principle ANN can be appliedto any correlation coefficient that is sensitive to damageAdditionally ANN can be used with structures that exhibita nonlinear response [12]

One of the main challenges in structural damage assess-ment is the selection of an appropriate measure of the systemresponse that is sufficiently sensitive to small damage Thefact thatmanymeasures have been studied over the past years(and continue to be investigated) with no consensus as to theoptimumchoice is a testament to the difficulty of the problemThis measure can be constructed in the time frequency ormodal domains and the latter two are the most broadlyused

The idea of directly using the frequency response func-tions (FRFs) to train the neural networks has attracted manyresearchers Among all of the dynamic responses the FRFis one of the easiest to obtain in real time because the insitu measurement is straightforward However the numberof spatial response locations and spectral lines is overly largefor neural network applications The direct use of FRFs willlead to networks with a large number of input variablesand connections thus rendering them impractical Hence itbecomes necessary to extract features from the FRFs and usethese features as inputs to the neural networks Castellini andRevel [13] presented an algorithm to detect and locate struc-tural damage based on laser vibrometry measurements and aneural network for data processing Using features extractedfrom the frequency response functions as inputs to the neuralnetwork the authors were able to use the same networkto detect and locate damage in three different experimentalstructures To reduce the number of input variables Zang andImregun [14] applied a principal component analysis (PCA)technique to the measured FRFs The output of the neuralnetwork gives the actual state of the structure undamaged ordamaged The algorithm was able to distinguish between theundamaged and damaged cases with satisfactory accuracyFang et al [15] selected key spectral points near the resonancefrequencies in the FRF data These selected points were usedas the inputs of a neural network and the outputs werethe stiffness reduction factors The algorithm showed highaccuracy in identifying damage to a simulated cantileverbeam under different damage scenarios

The input data can be further reduced if modal analysisis performed first Thus the input variables are the modalparameters of the structure The natural frequencies andmode shapes are themost frequently used parameters a largebody of literature on the use of natural frequencies in damagedetection is available [16] However the natural frequenciesdo not always provide the spatial information necessary tolocate the damage and thus additional information is alsonecessary such as the mode shapes In general the naturalfrequencies detect and quantify the presence of damagewhereas the mode shapes provide the location [17] Zapicoet al [18] presented an algorithm designed to identify thedamage to an experimental two-floor structure they used

the natural frequencies as the input variables and traineda neural network with data from a numerical model Theneural network contained two outputs representing the levelof damage of each floorThe proposed methodology was ableto predict the damage with an error of less than 86 Yun etal [19] used the natural frequencies andmode shapes as inputdata for a neural network used to detect damage in the jointsof framed structures The neural network was trained usinga noise-injection learning algorithm to reduce the effects ofexperimental noise The authors found that the algorithmcould estimate damage with reasonable accuracy but theperformance strongly depended on the level of experimentalnoise A disadvantage of using mode shape displacementsis that the mode shape identification requires many mea-surement locations Furthermore mode shape sensitivity todamage is not significant [3] The mode shape curvaturesare more sensitive to small structural modifications thanthe mode shape displacements [20] Sahin and Shenoi [21]studied the effectiveness of different combinations of global(natural frequencies) and local (mode shape curvatures)vibration data as inputs for an artificial neural network Theyconcluded that the best performance was obtained when thenatural frequencies were used as the inputs to a first networkused to predict the severity of the damage and themode shapecurvatures were used as the inputs to a second network topredict the damage location

Recently researchers have proposed the use of anti-resonant frequencies as an alternative to mode shapes [22]Anti-resonant frequencies correspond to the zeros (dips) ofthe FRFs and offer an attractive alternative because theycan be determined more easily and with less error thanthe mode shapes while still providing the same informa-tion The antiresonances can be derived from the pointfrequency response functions (FRFs) in which the responsecoordinate is the same as the excitation coordinate orfrom transfer FRFs in which the response coordinate differsfrom the excitation coordinate Point FRFs are preferredbecause matching problems arise when the antiresonancesfrom transfer FRFs are used Moreover the distribution ofthe transfer antiresonances can be significantly modifiedwith small structural changes [22] However the procedureused to obtain point FRFs differs from common modaltesting that is the excitation degree of freedom (DOF)is shifted together with the response DOF which maybecome impractical or expensive Williams and Messina [23]introduced antiresonances from point FRFs to the MultipleDamage Location Assurance Criterion (MDLAC) algorithmThey concluded that the incorporation of anti-resonant dataimproves the accuracy of the damage predictions Dilenaand Morassi [24] studied the problem of crack detection inbeams using resonant and anti-resonant frequencies Theyfound that the use of antiresonances aids in avoiding thenonuniqueness of the damage location that occurs whenonly natural frequencies are used However they also foundthat the experimental noise and modeling errors are usuallyamplified when antiresonances are included Bamnios et al[25] proposed a scheme for crack location in beams using theshift in the first anti-resonant frequency versus themeasuringposition to detect and locate a crack They stated that this

Shock and Vibration 3

method could be used to roughly locate the crack andsubsequently other methods can be applied to determinethe crack characteristics more precisely The changes inthe resonant and anti-resonant frequencies were used byInada et al [26] to locate and quantify the delaminationof a composite beam This group implemented a two-stepprocedure first the delamination domain is identified fromthe anti-resonant changes and next the location and sizeare defined using natural frequency changes This methodwas effective in identifying the delamination locations andsizes Wang and Zhu [27] suggested a method for identifyingcracks in beams which makes use of the natural frequenciesand antiresonances from the point FRFsThemethodology issimilar to that proposed by Bamnios et al [25] in that the shiftin the first anti-resonant frequency versus the driving pointlocation is used to locate damage They validated the methodwith a numerical example of a simply supported beam withthree cracks Their method was able to accurately predict thelocations and qualitatively estimate the crack size Meruaneand Heylen [28] showed that antiresonances are a goodalternative to mode shapes in damage assessment Howeverthey stated that further research is required for identificationof the experimental antiresonances and for the matching ofexperimental and numerical antiresonances Thus antires-onances present an attractive alternative to mode shapes asthe input values of neural networks for the following reasonsthey require a lower number of inputs are less contaminatedby noise and provide the same information Neverthelessmethods that use antiresonances are still under developmentand the application of antiresonances to structural damagedetection has not been fully investigated primarily becausethe inverse optimization problem used with antiresonances isparticularly challenging and robust optimization algorithmsare needed However these limitations should not present aproblem for methods based on neural networks

11 Artificial Neural Networks An ANN is a data processingalgorithm that attempts to emulate the processing scheme ofthe human brain [29] An ANN is formed by ldquoneuronsrdquo thatare interconnected to build a complex network Knowledge isacquired by a learning process and stored in the interneuronconnections known as the ldquosynaptic weightsrdquo

Different types of network architectures exist and amongthem themultilayer perceptron (MLP) is themost frequentlyused An MLP network consists of an array of input neuronsknown as the input layer an array of output neurons known asthe output layer and a number of hidden layers Each neuronreceives a weighted sum from the neurons in the precedinglayer and provides an input to every neuron of the next layerThe activation of each neuron is governed by a functionknown as the transfer function Typical selections for thetransfer function are linear log-sigmoid and hyperbolictangent-sigmoid as shown in Figure 1

Figure 2 illustrates the principle of a damage assessmentalgorithm using ANN The vibration characteristics of thestructure act as the inputs to the neural network and theoutputs are the damage indices of each element in thestructure

The outputs of a three-layer MLP are given by

119910119894= 119891 (

NHsum

119895=1

(119908119894119895119891 (

NIsum

119896=1

V119895119896

119909119896

+ 119887V119896) + 119887119908119894

)) (1)

where 119909 = 1199091 1199092 119909

119899 and 119910 = 119910

1 1199102 119910

119899 are the

input and output vectors 119908119894119895and V119895119896are the interconnection

weights 119887 represents the bias (or threshold) terms 119891(sdot) is thetransfer function and NI and NH are the number of inputand hidden nodes respectively

The values of the weights are updated by training theneural network with data from the structure The trainingproblem consists of finding theweights that willminimize themean square error 119864

119904

119864119904

=1

NO

NOsum

119896=1

(119910119896

minus 119900119896)2 (2)

where 119900119896is the desired output at the 119896th output node and

NO is the number of output nodes Many optimizationtechniques are able to address the network-training problemThe simplest method is gradient descent also known as thesteepest descent This algorithm begins with an initial weightvector and iteratively updates it by moving in the direction ofthe greatest error decrease The gradient descent rule can bewritten as

Δ119908120591

119894= minus120578

120597119864119904

120597119908120591

119894

(3)

where Δ119908120591

119894is the change for the 119894th weight at step 120591 The

backpropagation algorithm calculates the gradient at eachstep The convergence of this algorithm strongly dependson the learning rate 120578 If 120578 is too large the algorithm mayovershoot leading to divergent oscillations However if 120578 istoo small the search might proceed quite slowly Fang etal [15] showed that an adaptive learning rate significantlyimproves the training performance Another improvementto the gradient descent algorithm involves ensuring thateach search direction is conjugate to all previous directionsthus avoiding unnecessary loops in the search process Thescale-conjugated gradient algorithm proposed byMoslashller [30]combines this concept and variable steps Many additionaloptimization algorithms have been proposed to train neuralnetworks but the Levenberg-Marquant algorithm has beenthe most efficient [31] This method is a gradient-basedalgorithm specifically designed to minimize the sum-of-squares error [32]

A disadvantage of ANN is the need for large trainingsets It is highly difficult and time-consuming to producesufficiently large training data sets from experiments Analternative to generating training samples is to use a numer-ical model of the structure Castellini and Revel [13] showedthat it is possible to produce correct damage predictions inan experimental structure using a neural network that wastrained with samples generated by a finite element modelNevertheless this approach is highly dependent on theaccuracy of the numerical model There are two approaches

4 Shock and Vibration

f(n)

+1

minus1

n

(a)

f(n)

+1

minus1

n

(b)

f(n)

+1

minus1

n

(c)

f(n)

+1

minus1

n

(d)

f(n)

+1

minus1

n

(e)

Figure 1 Transfer functions (a) hyperbolic tangent sigmoid (b) logarithm sigmoid (c) linear (d) saturating linear and (e) symmetricsaturating linear

Shock and Vibration 5

to overcome this problem The first is to update the numer-ical model using experimental data from the undamagedstructure However even after updating differences will stillremain between the numerical and experimental modelsThe second alternative is to define an input parameter thatconsiders the initial errors in the numerical model thusavoiding the need for an accurate numerical modelThis goalis achieved using the changes in the data instead of theirabsolute values The main assumption is that any changein the structural properties is caused by damage Thus anyerror in the undamaged model of the structure that is alsopresent in the damaged model will be removed [33] Leeet al [34] showed that natural frequency changes becausestructural damage in a system without modeling errors isapproximated the same as those in a system with modelingerror Hence changes in the natural frequencies are lesssensitive to modeling errors than the natural frequenciesthemselves This group demonstrated the applicability ofa neural network trained with mode shape changes Thedamage locations were estimated with reasonable accuracyalthough false alarms were detected at several locations

Simulated data derived from a numerical model arenoise-free whereas actual measurements are never free fromexperimental noise Noise in themeasurements will cause thenetwork to estimate parameters that are different from theactual properties of the structure A solution is to introduceartificial noise into the numerical data used to train thenetwork This process is known as data perturbation scheme[35] Yun et al [19] used a noise-injection learning algorithmand a data perturbation schemeThey implemented a neural-network-based damage assessment algorithm to detect thedamage in structural joints The joints are modeled assemirigid connections using rotational springs and accurateresults are obtained in the cases of moderated noise Sahinand Shenoi [21] trained a neural network with artificiallyadded noise to detect single damage in beam-like compositelaminates The network contains two outputs the locationand the amount of damage They studied different combina-tions of features extracted from the resonant frequencies andmode shape curvatures as inputs to the neural network andtheir results show that feature selection plays a crucial rolein the predictions accuracy Zang and Imregun [14] proposeda different approach to address experimental noise and usedprincipal component projection of the FRF data as an inputto a neural network with two outputs of healthy or damagedThe authors stated that reconstructing the response using thehighest principal components should not only achieve datacompression but also remove a proportion of the noise ThePCA compression acts as a noise filter which can be useful inmodal analysis as well

The successful application of a neural network dependson the representation and the learning algorithms Never-theless their selection is problem-dependent and is usuallydetermined by trial and error [36] Sahoo and Maity [7]used a Genetic Algorithm (GA) to automate the trial-and-error process The network parameters (number of neuronslearning rate etc) are set as variables in an optimizationproblem handled by a GA They used an MLP network withtwo hidden layers trained by a backpropagation algorithm

Structure

Vibration characteristics

Artificial neural networks

Input layer

Hidden layers

Output layer

Stiffness reduction factors

Figure 2 Damage assessment with artificial neural networks

The inputs are the natural frequencies and strains in certainlocations of the structure Fang et al [15] explored the use ofa tunable steepest descent (TSD) algorithm that dynamicallyadjusts the learning speed during the training process Theyshowed that this methodology significantly increases thetraining speed while maintaining the learning stabilityTheseauthors used FRF data measured at key spectral pointsaround the resonant frequencies

The primary contribution of this research is the devel-opment of a real time damage assessment algorithm usingANN and anti-resonant frequencies In literature similarlyto mode shapes anti-resonant frequencies are used as com-plementary information to natural frequencies NeverthelessMotthershead [37] demonstrated that the sensitivity of anti-resonant frequencies can be expressed as a combination of thesensitivities of mode shapes and natural frequencies Henceanti-resonant frequencies contain the same information asnatural frequencies and mode shapes together which is thehypothesis of the present studyThis study is restricted to anti-resonant frequencies obtained from point FRFs Althoughmeasurement of the point FRFs is time-consuming becausethe excitation point is moved together with the response

6 Shock and Vibration

location this method offers several advantages for damageassessment as follows

(i) All antiresonances contain independent informationbecause they correspond to the resonant frequenciesof the system grounded at different degrees of free-dom

(ii) For a given FRF the number of anti-resonant frequen-cies does not change from one damage scenario tothe next this is a valuable property if we require thesefrequencies to act as inputs to a neural network

(iii) An anti-resonant frequency always lies betweentwo resonant frequencies Hence there is no doubtwhether a dip in an FRF is an anti-resonant frequencyor a minimum

This study addresses the setup of the neural networkparameters and provides guidelines for their selection insimilar damage detection problems The proposed method-ology is evaluated with two experimental structures an 8-DOF mass-spring system and a beam with multiple damagescenarios

2 Neural Network for Damage Assessment

This study intends to train a neural network using anti-resonant frequencies and to determine its feasibility in assess-ing experimental damage Hence this study works with thesimplest neural network that has been able to detect locateand quantify structural damage the multilayer perceptron(MLP) with three layers (input hidden and output) aspresented by Fang et al [15] The outputs of the networkcorrespond to the damage indices of each element whereasthe inputs are the changes in anti-resonant frequencies Thenumber of hidden nodes is defined after a sensitivity analysisfor each application case

21 Inputs As proposed by Lee et al [34] the inputs tothe neural network are defined as the changes in the modalparameters rather than their absolute values With thisapproach the network is less sensitive to errors in the baselineFE model

Therefore the inputs correspond to the experimentalchange in the anti-resonant frequencies with respect to theintact case

119909119894times119899

=120596119863

119903119894119899minus 120596119880

119903119894119899

120596119880

119903119894119899

(4)

The superscripts 119863 and 119880 denote undamaged and damagedrespectively and120596

119903119894119899is the 119894th anti-resonant frequency of the

119899th FRFTo reduce the effects of experimental noise the simulated

data are polluted with random noise As proposed by Hjelm-stad and Shin [35] each set of perturbed data is created byadding uniformly distributed random noise to the numericaldata

120596119903119894119899

= 120596119903119894119899 (1 + 120585) (5)

where 120585 is a uniform random number with a specifiedamplitudeThe variance of the perturbing noise should be thesame as the variance of the measurement noise

22 Outputs The damage indices are represented by theelemental stiffness reduction factors 120573

119894defined as the ratios

between the stiffness reduction and the initial stiffness

119910119894= 120573119894 (6)

The stiffnessmatrix of the damaged structureK119889 is expressed

as a sum of element matrices multiplied by reduction factors

K119889

= sum

119894

(1 minus 120573119894)K119894 (7)

The value 120573119894= 0 indicates that the element is undamaged

whereas 0 lt 120573119894le 1 implies partial or complete damage

23 Training and Validation Patterns The distribution ofthe training patterns plays a crucial role in the success of aneural network The relationship between the anti-resonantfrequencies and the different damage levels is not linearand as a consequence the network might be unable tointerpolate Fang et al [15] recommend the use of trainingpatterns with evenly distributed damaged levels Combiningsimultaneous damages into the training increases the numberof training patterns and a large number of training patternscould overwhelm the training procedure In this study thetraining patterns were generated by considering up to twosimultaneous damage incidences with nine damage levelsevenly distributed between 0 and 80 Hence the totalnumber of training patterns depends on the number ofelements 119873 as

number of training patterns = 92

times 119862 (119873 2) (8)

where 119862(119873 2) is the number of combinations of 119873 elementstaken two at a time

The algorithm trains the network using the early stoppingtechnique [37] In this technique two sets of data are usedduring training a training set and a validation set Thetraining set is used for updating the network weights andbiases As defined in (2) the mean square error evaluated onthe validation set ismonitored during trainingWhen the val-idation error increases for six iterations the algorithm stopsthe training The weights of the network at the minimumvalidation error become the final weights

The validation set is a group of patterns that must bedifferent from the training patterns To ensure this conditionthe validation patterns were created with nine damage levelsevenly distributed between 5 and 85 and considered up totwo simultaneous damage incidences Note that the numberof training patterns is equal to the number of validationpatterns This set of validation patterns is also used to checkthe network performance after training

24 Measures of Network Performance The performance ofthe network is measured by three indicators the mean sizing

Shock and Vibration 7

error the damage missing error and the false alarm error asdefined by Yun et al [19] The mean sizing error (MSE) is theaverage quantification error

MSE =1

NOsum

119894

1003816100381610038161003816119910119894 minus 119900119894

1003816100381610038161003816 (9)

where119910119894and 119900119894are the estimated and desired outputs for node

119894 respectively and NO is the number of output nodesThe damage missing error (DME) is given by

DME =1

NTsum

119894

120576119868

119894 0 le DME le 1 (10)

where 120576119868

119894= 0 if the 119894th damaged element is correctly detected

and 120576119868

119894= 1 otherwise The NT corresponds to the number

of true damage locations If DME = 0 all damage locationsare correctly detected The value of 120576

119868

119894is calculated using the

following equation

120576119868

119894=

1 if 119900119894gt 0 119910

119894le 120572119888

0 otherwise(11)

It is assumed that an element is detected as damaged if theestimated damage 119910

119894is greater than a prescribed critical value

120572119888 The critical damage level 120572

119888is defined as being equal to

the average MSEThis value is the minimum damage that thenetwork can reasonably assess

The false alarm error (FAE) is defined as

FAE =1

NFsum

119894

120576119868119868

119894 0 le FAE le 1 (12)

where 120576119868119868

119894= 0 if the 119894th detected damage is truly damaged

and 120576119868119868

119894= 1 otherwise The NF is the number of predicted

damage locations If FAE = 0 all of the detected locations areactual damage locations The value of 120576

119868119868

119894is calculated using

the following equation

120576119868119868

119894=

1 if119910119894ge 120572119888 119900119894= 0

0 otherwise(13)

3 Identification and Matching ofAntiresonant Frequencies

31 Experimental Antiresonances The experimental antires-onances are identified from experimental point FRFs byldquodip-pickingrdquo [22] In this technique the antiresonances areselected by picking the dips from the magnitude plot ofa given FRF that have an associated change of +180∘ inthe phase plot This method is similar to the ldquopeak pick-ingrdquo technique for resonant frequencies Figure 3 shows anexample of the anti-resonant picking procedure When usingthis technique the errors in the estimated antiresonancesoriginate primarily from the frequency resolution of theFRFs If the dips in the FRFs are difficult to determine theantiresonances can be identified by a curve-fitting techniquebased on the rational fraction polynomial representation ofthe FRFs [38]

32 Numerical Antiresonances For a lightly damped struc-ture the anti-resonant frequencies are nearly unaffected bydamping and therefore they can be obtained from theundamped system using only the stiffness andmass matrices

By definition the FRFmatrix is the inverse of the dynamicstiffness matrix

H (120596) = (K minus 1205962M)minus1

=adj (K minus 120596

2M)

det (K minus 1205962M) (14)

The operators adj(sdot) and det(sdot) indicate the adjointand determinant respectively The anti-resonant frequenciescorrespond to the zeros of the FRFs The zeros of the 119894119896th FRF are the values of 120596 for which the numerator of119867119894119896

(120596)vanishes The numerator of 119867119894119896

(120596) is the 119894 119896th termof adj(Kminus 120596

2M) which is given by (minus1)119894+119896 det(K

119894119896minus 1205962M119894119896

)The subscripts 119894 and 119896 denote that the 119894th row and 119896th columnhave been deleted As a consequence the antiresonances ofthe 119894 119896th FRF are the frequency values that satisfy

det (K119894119896

minus 1205962M119894119896

) = 0 (15)

which is equivalent to solving the eigenvalue problem

(K119894119896

minus 1205962M119894119896

) u = 0 (16)

If 119894 = 119896 (16) represents a physical system obtained bygrounding the 119894th degree of freedom Therefore the anti-resonant frequencies obtained from point FRFs (119894 = 119896) areequivalent to the resonant frequencies of the structure withthe 119894th degree of freedom grounded

Because we are working with point FRFs matching theexperimental and numerical anti-resonant frequencies is aneasy task because each anti-resonant frequency lies betweentwo resonant frequencies

4 Eight-DOF Spring-Mass System

The structure shown in Figure 4 consists of an 8-DOF spring-mass system The Los Alamos National Laboratory (LANL)designed and constructed this system to study the effec-tiveness of various vibration-based damage identificationtechniques [39]

The system consists of eight translatingmasses connectedby springs and each mass consists of a disc of aluminumwith a diameter of 762mm and a thickness of 254mm Themasses slide on a highly polished steel rod and are fastenedtogether with coil springs The spring and mass locationsare designated sequentially with the first located the closestto the shaker attachment In the undamaged configurationall springs are identical and have linear stiffness coefficientsDamage is simulated by replacing the fifth spring with adifferent spring with a lower stiffness (55 stiffness reduc-tion)The acceleration is measured horizontally at eachmassgiving a total of eightmeasuredDOFsThe structure is excitedrandomly by an electrodynamic shaker The responses aremeasured in the eight DOFs in the undamaged and damagedcases

Seven anti-resonant frequencies are identified from theFRF corresponding to the first mass which corresponds to

8 Shock and Vibration

Mag

nitu

de (a

bs)

102

100

104

10minus2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

2855988

10261549

8125

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

Phas

e (de

g)

50

0

minus50

minus100

minus150

minus200

(b)

Figure 3 Identification of experimental anti-resonant frequencies

m8 m7 m6 m5 m4 m3 m2 m1 f(t)

k7 k6 k5 k4 k3 k2 k1x8 x7 x6 x5 x4 x3 x2 x1

Figure 4 Experimental system with eight degrees of freedom

a point FRF Given that the anti-resonant frequencies areindependent seven antiresonances should be sufficient toassess the stiffness of each of the seven springs

Table 1 lists the anti-resonant frequencies identified inthe undamaged and damaged cases The second and fourthantiresonances provide the larger variation due to the exper-imental damage

The numerical model is built in Matlab with springs andconcentrated masses and the initial parameters are set asfollows

(i) mass 1 5593 g (this mass is greater than the othersbecause of the hardware required to attach the shaker)

(ii) masses 2 to 8 4194 g(iii) spring constants 567 kNmminus1

Table 2 shows the numerical and experimental anti-resonant frequencies The maximum difference between theexperimental and numerical antiresonances is 07 As aconsequence the numerical model provides a faithful repre-sentation of the experimental structure and can be used totrain the neural network

41 Construction of the Neural Network The network con-tains seven inputs and seven outputsThe best parameters forthe neural network are defined after performing a sensitivity

Table 1 Experimental antiresonances

Antiresonant frequencies (Hz) Variation ()Undamaged Damaged1231 1166 minus5283694 3244 minus12185961 598 0327996 7174 minus10289612 9302 minus32310944 10686 minus23611572 11267 minus264

Table 2 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical123 123 043369 367 054596 592 070800 795 055961 956 0521094 1090 0401157 1155 019

analysis The network is trained under different configura-tions and the selected configuration is the one that providesthe lowest validation error

First the effects of five different transfer functions usedfor the output layer are studied and these transfer functionsare formulated as follows (a) hyperbolic tangent sigmoid(b) logarithm sigmoid (c) linear (d) saturating linear and(e) symmetric saturating linear Figure 2 illustrates thesefunctions During the experimental runs the remainingparameters of the network are characterized as follows thehidden transfer function is a logarithm sigmoid the numberof hidden neurons is 30 the training data are polluted with2 random noise and the Levenberg-Marquardt algorithmis used to train the network The network is trained 10times with each of the output transfer functions Table 3summarizes the average validation performance obtained

Shock and Vibration 9

Table 3 Network validation performance with different transfer functions

Output transfer function 119864119904

Hidden transfer function 119864119904

Hyperbolic tangent sigmoid 00067 Hyperbolic tangent sigmoid 00045Logarithm sigmoid 00134 Logarithm sigmoid 00044Linear 00060 Linear 00130Saturating linear 00127 Saturating linear 00050Symmetric saturating linear 00046 Symmetric saturating linear 00047

Table 4 Network validation performance with different trainingalgorithms

Training algorithms Time (s) 119864119904

Gradient descent 237 00467Gradient descent with adaptive learning 9 00289Scaled conjugate gradient 12 00158Levenberg-Marquardt 15 00046

for each function The symmetric saturating linear functionprovides the best performance

Second the performance of each of these five transferfunctions in the hidden layer is investigated The remainingnetwork parameters are the symmetric saturating linearoutput transfer function 30 hidden neurons 2 noise andthe Levenberg-Marquardt training algorithmThe network istrained 10 times with each of these output transfer functionsTable 3 summarizes the average validation performance Thelogarithm sigmoid function provides the best performance

Next the performance of four training algorithms isstudied Table 4 summarizes the average results for 10 runsThe gradient descent algorithm provides the worst perfor-mance in terms of training time and validation mean squareerror An adaptive learning rate substantially improves theresults the training time is 26 times lower and the meanerror is reduced by a factor of 16 The scale conjugate-gradient reaches a lower mean square error with a slightlyhigher training time Nevertheless the Levenberg-Marquardtalgorithm provides the lowest mean square validation errorby far and its training time is nearly the same as that with thescale conjugate gradient algorithm

Figure 5 shows the average validation error for differentnumbers of hidden nodes Seventy nodes in the hidden layergive the lower validation error 119864

119904 Less than 70 neurons are

not sufficient to fit the data whereas greater than 70 neuronsoverfit the neural network

The last parameter corresponds to the level of noise inthe training data The selection of this parameter is relatedto the level of experimental noise Hence this parameterwas defined by studying the performance of the trainednetwork with the experimental data and different levels ofperturbation noise Figure 6 shows the mean sizing error(MSE) obtained when the network is tested with experimen-tal dataThis case corresponds to a 55 stiffness reduction inelement 5 The MSE is the difference between the predicteddamage level on each element and the actual damage level asdefined in (9) A data perturbation noise of 15 provides theminimum error in the predicted damage

20 40 60 80 100 120 1403638

442444648

5

Number of hidden nodes

Es

times10minus3

Figure 5 Network validation performance as a function of thenumber of hidden nodes

0 05 1 15 2 25 3 35 40

00501

01502

02503

035

Noise level ()

MSE

Figure 6 Experimental mean sizing error for different levels ofinjected noise

42 Network Validation The selected network parametersare

(i) network three layers of multilayer perceptron(ii) number of input nodes 7(iii) number of hidden nodes 70(iv) number of output nodes 7(v) transfer function in the hidden layer Logarithm

sigmoid(vi) transfer function in the output layer Symmetric

saturating linear(vii) training method Levenberg-Marquardt

The validation patterns verify the performance of the net-work and these patterns are polluted with 15 noise Theresulting mean sizing error is MSE = 337 and thus thenetwork quantification accuracy is 97 Figure 7 shows thedamage missing error (DME) and false alarm error (FAE)divided by the damage levelsThe results of the DME indicate

10 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

Damage level ()

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(b)

Figure 7 Damage missing error (DME) and false alarm error (FAE) for different damage levels

1 2 3 4 5 6 70

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Figure 8 Assessment of experimental damage (55 stiffness reduc-tion in element 5)

that the detection of damage with low severities is poor Infact 80 of the damage with severities lower than 10 and40of the damagewith severities of 10ndash20 are not detectedThe algorithm can detect damage with severities larger than30with confidence it correctly detects 996 of the damagewith severities over 30The false alarm results indicate thatmost of the damage detected with levels lower than 10are false damage (near 75) The amount of false damagedetection is drastically reduced with an increment in the levelof damage In fact 9924 of the damage detected with levelsover 30 is true damage

Figure 8 shows the results of the damage detected usingthe experimental case The damaged element is correctlyassessed the location is correct and the quantification erroris 5

5 Experimental Beam

The structure consists of a steel beam with a rectangularcross-section The dimensions of the beam are length = 1mand section = 25 times 10mm2 As shown in Figure 9 soft springssuspend the structure to simulate a ldquofree-freerdquo boundarycondition

A hammer excites the beam at four points distributedalong the beam and an accelerometer measures the responseat each excitation location Both the excitation force and themeasured responses lie in the horizontal direction In thisdirection the anti-resonant frequencies are more sensitive to

Table 5 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical

FRF1

903 89 1102934 289 1356063 604 04010328 1033 00015528 1576 151

FRF2

1288 128 0692609 258 1005831 586 05611128 1127 12616219 1650 170

FRF3

1288 128 069260 258 0665844 586 03411138 1127 11716175 1650 198

FRF4

903 89 1102929 289 1356061 604 04010328 1033 00015534 1576 151

Figure 9 Experimental beam

the experimental damage Five anti-resonant frequencies areidentified at each of the four experimental FRFs

The numerical model is built in Matlab with 2D beamelements The model contains 20 beam elements and 40degrees of freedom as shown in Figure 10 The shadowedelements indicate possible locations of damage Table 5 showsthe experimental and numerical frequencies and their differ-ences and the maximum difference is 198

The structure is subjected to four different damagescenarios containing single and double cracks Cracks are

Shock and Vibration 11

Table 6 Damage cases introduced to the beam

Case Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

1 315 7 5 mdash mdash mdash2 635 13 10 mdash mdash mdash3 303 7 5 640 13 104 360 8 5 810 17 15

1 2 3 4 5 6 7 8 9 10 1211 13 14 15 16 17 18 19 20

Figure 10 Numerical model and element numbering

C

Figure 11 Saw cut introduced to the beam

introduced to the structure by saw cuts of length 119897119888 as

illustrated in Figure 11 Table 6 summarizes the differentdamage scenarios and indicates the distance from the left endto the cut the corresponding element in the numericalmodeland the cut length

51 Network Validation The network uses the same param-eters selected for the first structure except for the number ofinput hidden and output nodes The number of input nodescorresponds to the number of anti-resonant frequencieswhich is 20 Elements 2 to 19 were defined as possiblelocations of damage thus giving 18 output nodes (damageindices) The number of hidden nodes was defined afterthe sensitivity analysis shown in Figure 12 The minimumvalidation error is obtained with 80 hidden nodes

Hence the network parameters are

(i) network three layers of multilayer perceptron(ii) number of input nodes 20(iii) number of hidden nodes 80(iv) number of output nodes 18(v) transfer function in the hidden layer logarithm sig-

moid(vi) transfer function in the output layer symmetric

saturating linear(vii) training method Levenberg-Marquardt

The training and validation patterns were polluted with15 random noise The mean sizing error obtained for thevalidation patterns is MSE = 153 and thus the networkquantification accuracy is 98 Figure 13 shows the damagemissing error (DME) and false alarm error (FAE) separatedby damage levels The results of the DME indicate that thedetection of damage with low severities is poor In fact

20 40 60 80 100 120 140 160 180 200

28

3

32

34

36

Number of hidden nodes

times10minus3

Es

Figure 12 Validation performance as a function of the number ofhidden nodes

near 90 of the damage with severities lower than 10 and70 of the damage with severities between 10 and 20 isnot detected The algorithm is able to detect damage withseverities larger than 40with confidence it correctly detects995 of the damage with severities over 40The false alarmresults indicate that most of the damage detected with levelslower than 10 is false damage (near 80) The amount offalse damage detection decreases with an increment of thelevel of damage 99 of the damage detected with levels over40 is true damage

Note that these results are slightly worse than the resultsobtained from the 8-DOF system This result is explained bythe fact that the anti-resonant frequencies are less sensitiveto damage in the free beam than the 8-DOF structure thusmaking it more difficult to detect small damage

Figure 14 shows the results of the damage detected in thefour experimental scenarios An arrow indicates the actualdamage location In the first three cases the damage iscorrectly located although false damage appears next to theactual locations Nevertheless the assumption that a crackonly affects the stiffness of the corresponding element mightnot always be true For instance in the fourth case the effectof the larger cut is represented more accurately by a stiffnessreduction of two elements rather than one In the last case thenetwork does not detect the small crack in element 8 becausethe effect of the larger cut hides the effect of the smaller cut

6 Conclusions

This paper presents a new methodology designed to assessexperimental damage using neural networks and anti-resonant frequencies A multilayer perceptron network wastrained with data obtained from a numerical model and

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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DistributedSensor Networks

International Journal of

2 Shock and Vibration

in a short period and hence it can be used for real timedamage assessment Damage detection by ANN has theadvantage that it is a general approach Unlike many otherdamage detection methods that are often developed forspecific quantities [9ndash11] in principle ANN can be appliedto any correlation coefficient that is sensitive to damageAdditionally ANN can be used with structures that exhibita nonlinear response [12]

One of the main challenges in structural damage assess-ment is the selection of an appropriate measure of the systemresponse that is sufficiently sensitive to small damage Thefact thatmanymeasures have been studied over the past years(and continue to be investigated) with no consensus as to theoptimumchoice is a testament to the difficulty of the problemThis measure can be constructed in the time frequency ormodal domains and the latter two are the most broadlyused

The idea of directly using the frequency response func-tions (FRFs) to train the neural networks has attracted manyresearchers Among all of the dynamic responses the FRFis one of the easiest to obtain in real time because the insitu measurement is straightforward However the numberof spatial response locations and spectral lines is overly largefor neural network applications The direct use of FRFs willlead to networks with a large number of input variablesand connections thus rendering them impractical Hence itbecomes necessary to extract features from the FRFs and usethese features as inputs to the neural networks Castellini andRevel [13] presented an algorithm to detect and locate struc-tural damage based on laser vibrometry measurements and aneural network for data processing Using features extractedfrom the frequency response functions as inputs to the neuralnetwork the authors were able to use the same networkto detect and locate damage in three different experimentalstructures To reduce the number of input variables Zang andImregun [14] applied a principal component analysis (PCA)technique to the measured FRFs The output of the neuralnetwork gives the actual state of the structure undamaged ordamaged The algorithm was able to distinguish between theundamaged and damaged cases with satisfactory accuracyFang et al [15] selected key spectral points near the resonancefrequencies in the FRF data These selected points were usedas the inputs of a neural network and the outputs werethe stiffness reduction factors The algorithm showed highaccuracy in identifying damage to a simulated cantileverbeam under different damage scenarios

The input data can be further reduced if modal analysisis performed first Thus the input variables are the modalparameters of the structure The natural frequencies andmode shapes are themost frequently used parameters a largebody of literature on the use of natural frequencies in damagedetection is available [16] However the natural frequenciesdo not always provide the spatial information necessary tolocate the damage and thus additional information is alsonecessary such as the mode shapes In general the naturalfrequencies detect and quantify the presence of damagewhereas the mode shapes provide the location [17] Zapicoet al [18] presented an algorithm designed to identify thedamage to an experimental two-floor structure they used

the natural frequencies as the input variables and traineda neural network with data from a numerical model Theneural network contained two outputs representing the levelof damage of each floorThe proposed methodology was ableto predict the damage with an error of less than 86 Yun etal [19] used the natural frequencies andmode shapes as inputdata for a neural network used to detect damage in the jointsof framed structures The neural network was trained usinga noise-injection learning algorithm to reduce the effects ofexperimental noise The authors found that the algorithmcould estimate damage with reasonable accuracy but theperformance strongly depended on the level of experimentalnoise A disadvantage of using mode shape displacementsis that the mode shape identification requires many mea-surement locations Furthermore mode shape sensitivity todamage is not significant [3] The mode shape curvaturesare more sensitive to small structural modifications thanthe mode shape displacements [20] Sahin and Shenoi [21]studied the effectiveness of different combinations of global(natural frequencies) and local (mode shape curvatures)vibration data as inputs for an artificial neural network Theyconcluded that the best performance was obtained when thenatural frequencies were used as the inputs to a first networkused to predict the severity of the damage and themode shapecurvatures were used as the inputs to a second network topredict the damage location

Recently researchers have proposed the use of anti-resonant frequencies as an alternative to mode shapes [22]Anti-resonant frequencies correspond to the zeros (dips) ofthe FRFs and offer an attractive alternative because theycan be determined more easily and with less error thanthe mode shapes while still providing the same informa-tion The antiresonances can be derived from the pointfrequency response functions (FRFs) in which the responsecoordinate is the same as the excitation coordinate orfrom transfer FRFs in which the response coordinate differsfrom the excitation coordinate Point FRFs are preferredbecause matching problems arise when the antiresonancesfrom transfer FRFs are used Moreover the distribution ofthe transfer antiresonances can be significantly modifiedwith small structural changes [22] However the procedureused to obtain point FRFs differs from common modaltesting that is the excitation degree of freedom (DOF)is shifted together with the response DOF which maybecome impractical or expensive Williams and Messina [23]introduced antiresonances from point FRFs to the MultipleDamage Location Assurance Criterion (MDLAC) algorithmThey concluded that the incorporation of anti-resonant dataimproves the accuracy of the damage predictions Dilenaand Morassi [24] studied the problem of crack detection inbeams using resonant and anti-resonant frequencies Theyfound that the use of antiresonances aids in avoiding thenonuniqueness of the damage location that occurs whenonly natural frequencies are used However they also foundthat the experimental noise and modeling errors are usuallyamplified when antiresonances are included Bamnios et al[25] proposed a scheme for crack location in beams using theshift in the first anti-resonant frequency versus themeasuringposition to detect and locate a crack They stated that this

Shock and Vibration 3

method could be used to roughly locate the crack andsubsequently other methods can be applied to determinethe crack characteristics more precisely The changes inthe resonant and anti-resonant frequencies were used byInada et al [26] to locate and quantify the delaminationof a composite beam This group implemented a two-stepprocedure first the delamination domain is identified fromthe anti-resonant changes and next the location and sizeare defined using natural frequency changes This methodwas effective in identifying the delamination locations andsizes Wang and Zhu [27] suggested a method for identifyingcracks in beams which makes use of the natural frequenciesand antiresonances from the point FRFsThemethodology issimilar to that proposed by Bamnios et al [25] in that the shiftin the first anti-resonant frequency versus the driving pointlocation is used to locate damage They validated the methodwith a numerical example of a simply supported beam withthree cracks Their method was able to accurately predict thelocations and qualitatively estimate the crack size Meruaneand Heylen [28] showed that antiresonances are a goodalternative to mode shapes in damage assessment Howeverthey stated that further research is required for identificationof the experimental antiresonances and for the matching ofexperimental and numerical antiresonances Thus antires-onances present an attractive alternative to mode shapes asthe input values of neural networks for the following reasonsthey require a lower number of inputs are less contaminatedby noise and provide the same information Neverthelessmethods that use antiresonances are still under developmentand the application of antiresonances to structural damagedetection has not been fully investigated primarily becausethe inverse optimization problem used with antiresonances isparticularly challenging and robust optimization algorithmsare needed However these limitations should not present aproblem for methods based on neural networks

11 Artificial Neural Networks An ANN is a data processingalgorithm that attempts to emulate the processing scheme ofthe human brain [29] An ANN is formed by ldquoneuronsrdquo thatare interconnected to build a complex network Knowledge isacquired by a learning process and stored in the interneuronconnections known as the ldquosynaptic weightsrdquo

Different types of network architectures exist and amongthem themultilayer perceptron (MLP) is themost frequentlyused An MLP network consists of an array of input neuronsknown as the input layer an array of output neurons known asthe output layer and a number of hidden layers Each neuronreceives a weighted sum from the neurons in the precedinglayer and provides an input to every neuron of the next layerThe activation of each neuron is governed by a functionknown as the transfer function Typical selections for thetransfer function are linear log-sigmoid and hyperbolictangent-sigmoid as shown in Figure 1

Figure 2 illustrates the principle of a damage assessmentalgorithm using ANN The vibration characteristics of thestructure act as the inputs to the neural network and theoutputs are the damage indices of each element in thestructure

The outputs of a three-layer MLP are given by

119910119894= 119891 (

NHsum

119895=1

(119908119894119895119891 (

NIsum

119896=1

V119895119896

119909119896

+ 119887V119896) + 119887119908119894

)) (1)

where 119909 = 1199091 1199092 119909

119899 and 119910 = 119910

1 1199102 119910

119899 are the

input and output vectors 119908119894119895and V119895119896are the interconnection

weights 119887 represents the bias (or threshold) terms 119891(sdot) is thetransfer function and NI and NH are the number of inputand hidden nodes respectively

The values of the weights are updated by training theneural network with data from the structure The trainingproblem consists of finding theweights that willminimize themean square error 119864

119904

119864119904

=1

NO

NOsum

119896=1

(119910119896

minus 119900119896)2 (2)

where 119900119896is the desired output at the 119896th output node and

NO is the number of output nodes Many optimizationtechniques are able to address the network-training problemThe simplest method is gradient descent also known as thesteepest descent This algorithm begins with an initial weightvector and iteratively updates it by moving in the direction ofthe greatest error decrease The gradient descent rule can bewritten as

Δ119908120591

119894= minus120578

120597119864119904

120597119908120591

119894

(3)

where Δ119908120591

119894is the change for the 119894th weight at step 120591 The

backpropagation algorithm calculates the gradient at eachstep The convergence of this algorithm strongly dependson the learning rate 120578 If 120578 is too large the algorithm mayovershoot leading to divergent oscillations However if 120578 istoo small the search might proceed quite slowly Fang etal [15] showed that an adaptive learning rate significantlyimproves the training performance Another improvementto the gradient descent algorithm involves ensuring thateach search direction is conjugate to all previous directionsthus avoiding unnecessary loops in the search process Thescale-conjugated gradient algorithm proposed byMoslashller [30]combines this concept and variable steps Many additionaloptimization algorithms have been proposed to train neuralnetworks but the Levenberg-Marquant algorithm has beenthe most efficient [31] This method is a gradient-basedalgorithm specifically designed to minimize the sum-of-squares error [32]

A disadvantage of ANN is the need for large trainingsets It is highly difficult and time-consuming to producesufficiently large training data sets from experiments Analternative to generating training samples is to use a numer-ical model of the structure Castellini and Revel [13] showedthat it is possible to produce correct damage predictions inan experimental structure using a neural network that wastrained with samples generated by a finite element modelNevertheless this approach is highly dependent on theaccuracy of the numerical model There are two approaches

4 Shock and Vibration

f(n)

+1

minus1

n

(a)

f(n)

+1

minus1

n

(b)

f(n)

+1

minus1

n

(c)

f(n)

+1

minus1

n

(d)

f(n)

+1

minus1

n

(e)

Figure 1 Transfer functions (a) hyperbolic tangent sigmoid (b) logarithm sigmoid (c) linear (d) saturating linear and (e) symmetricsaturating linear

Shock and Vibration 5

to overcome this problem The first is to update the numer-ical model using experimental data from the undamagedstructure However even after updating differences will stillremain between the numerical and experimental modelsThe second alternative is to define an input parameter thatconsiders the initial errors in the numerical model thusavoiding the need for an accurate numerical modelThis goalis achieved using the changes in the data instead of theirabsolute values The main assumption is that any changein the structural properties is caused by damage Thus anyerror in the undamaged model of the structure that is alsopresent in the damaged model will be removed [33] Leeet al [34] showed that natural frequency changes becausestructural damage in a system without modeling errors isapproximated the same as those in a system with modelingerror Hence changes in the natural frequencies are lesssensitive to modeling errors than the natural frequenciesthemselves This group demonstrated the applicability ofa neural network trained with mode shape changes Thedamage locations were estimated with reasonable accuracyalthough false alarms were detected at several locations

Simulated data derived from a numerical model arenoise-free whereas actual measurements are never free fromexperimental noise Noise in themeasurements will cause thenetwork to estimate parameters that are different from theactual properties of the structure A solution is to introduceartificial noise into the numerical data used to train thenetwork This process is known as data perturbation scheme[35] Yun et al [19] used a noise-injection learning algorithmand a data perturbation schemeThey implemented a neural-network-based damage assessment algorithm to detect thedamage in structural joints The joints are modeled assemirigid connections using rotational springs and accurateresults are obtained in the cases of moderated noise Sahinand Shenoi [21] trained a neural network with artificiallyadded noise to detect single damage in beam-like compositelaminates The network contains two outputs the locationand the amount of damage They studied different combina-tions of features extracted from the resonant frequencies andmode shape curvatures as inputs to the neural network andtheir results show that feature selection plays a crucial rolein the predictions accuracy Zang and Imregun [14] proposeda different approach to address experimental noise and usedprincipal component projection of the FRF data as an inputto a neural network with two outputs of healthy or damagedThe authors stated that reconstructing the response using thehighest principal components should not only achieve datacompression but also remove a proportion of the noise ThePCA compression acts as a noise filter which can be useful inmodal analysis as well

The successful application of a neural network dependson the representation and the learning algorithms Never-theless their selection is problem-dependent and is usuallydetermined by trial and error [36] Sahoo and Maity [7]used a Genetic Algorithm (GA) to automate the trial-and-error process The network parameters (number of neuronslearning rate etc) are set as variables in an optimizationproblem handled by a GA They used an MLP network withtwo hidden layers trained by a backpropagation algorithm

Structure

Vibration characteristics

Artificial neural networks

Input layer

Hidden layers

Output layer

Stiffness reduction factors

Figure 2 Damage assessment with artificial neural networks

The inputs are the natural frequencies and strains in certainlocations of the structure Fang et al [15] explored the use ofa tunable steepest descent (TSD) algorithm that dynamicallyadjusts the learning speed during the training process Theyshowed that this methodology significantly increases thetraining speed while maintaining the learning stabilityTheseauthors used FRF data measured at key spectral pointsaround the resonant frequencies

The primary contribution of this research is the devel-opment of a real time damage assessment algorithm usingANN and anti-resonant frequencies In literature similarlyto mode shapes anti-resonant frequencies are used as com-plementary information to natural frequencies NeverthelessMotthershead [37] demonstrated that the sensitivity of anti-resonant frequencies can be expressed as a combination of thesensitivities of mode shapes and natural frequencies Henceanti-resonant frequencies contain the same information asnatural frequencies and mode shapes together which is thehypothesis of the present studyThis study is restricted to anti-resonant frequencies obtained from point FRFs Althoughmeasurement of the point FRFs is time-consuming becausethe excitation point is moved together with the response

6 Shock and Vibration

location this method offers several advantages for damageassessment as follows

(i) All antiresonances contain independent informationbecause they correspond to the resonant frequenciesof the system grounded at different degrees of free-dom

(ii) For a given FRF the number of anti-resonant frequen-cies does not change from one damage scenario tothe next this is a valuable property if we require thesefrequencies to act as inputs to a neural network

(iii) An anti-resonant frequency always lies betweentwo resonant frequencies Hence there is no doubtwhether a dip in an FRF is an anti-resonant frequencyor a minimum

This study addresses the setup of the neural networkparameters and provides guidelines for their selection insimilar damage detection problems The proposed method-ology is evaluated with two experimental structures an 8-DOF mass-spring system and a beam with multiple damagescenarios

2 Neural Network for Damage Assessment

This study intends to train a neural network using anti-resonant frequencies and to determine its feasibility in assess-ing experimental damage Hence this study works with thesimplest neural network that has been able to detect locateand quantify structural damage the multilayer perceptron(MLP) with three layers (input hidden and output) aspresented by Fang et al [15] The outputs of the networkcorrespond to the damage indices of each element whereasthe inputs are the changes in anti-resonant frequencies Thenumber of hidden nodes is defined after a sensitivity analysisfor each application case

21 Inputs As proposed by Lee et al [34] the inputs tothe neural network are defined as the changes in the modalparameters rather than their absolute values With thisapproach the network is less sensitive to errors in the baselineFE model

Therefore the inputs correspond to the experimentalchange in the anti-resonant frequencies with respect to theintact case

119909119894times119899

=120596119863

119903119894119899minus 120596119880

119903119894119899

120596119880

119903119894119899

(4)

The superscripts 119863 and 119880 denote undamaged and damagedrespectively and120596

119903119894119899is the 119894th anti-resonant frequency of the

119899th FRFTo reduce the effects of experimental noise the simulated

data are polluted with random noise As proposed by Hjelm-stad and Shin [35] each set of perturbed data is created byadding uniformly distributed random noise to the numericaldata

120596119903119894119899

= 120596119903119894119899 (1 + 120585) (5)

where 120585 is a uniform random number with a specifiedamplitudeThe variance of the perturbing noise should be thesame as the variance of the measurement noise

22 Outputs The damage indices are represented by theelemental stiffness reduction factors 120573

119894defined as the ratios

between the stiffness reduction and the initial stiffness

119910119894= 120573119894 (6)

The stiffnessmatrix of the damaged structureK119889 is expressed

as a sum of element matrices multiplied by reduction factors

K119889

= sum

119894

(1 minus 120573119894)K119894 (7)

The value 120573119894= 0 indicates that the element is undamaged

whereas 0 lt 120573119894le 1 implies partial or complete damage

23 Training and Validation Patterns The distribution ofthe training patterns plays a crucial role in the success of aneural network The relationship between the anti-resonantfrequencies and the different damage levels is not linearand as a consequence the network might be unable tointerpolate Fang et al [15] recommend the use of trainingpatterns with evenly distributed damaged levels Combiningsimultaneous damages into the training increases the numberof training patterns and a large number of training patternscould overwhelm the training procedure In this study thetraining patterns were generated by considering up to twosimultaneous damage incidences with nine damage levelsevenly distributed between 0 and 80 Hence the totalnumber of training patterns depends on the number ofelements 119873 as

number of training patterns = 92

times 119862 (119873 2) (8)

where 119862(119873 2) is the number of combinations of 119873 elementstaken two at a time

The algorithm trains the network using the early stoppingtechnique [37] In this technique two sets of data are usedduring training a training set and a validation set Thetraining set is used for updating the network weights andbiases As defined in (2) the mean square error evaluated onthe validation set ismonitored during trainingWhen the val-idation error increases for six iterations the algorithm stopsthe training The weights of the network at the minimumvalidation error become the final weights

The validation set is a group of patterns that must bedifferent from the training patterns To ensure this conditionthe validation patterns were created with nine damage levelsevenly distributed between 5 and 85 and considered up totwo simultaneous damage incidences Note that the numberof training patterns is equal to the number of validationpatterns This set of validation patterns is also used to checkthe network performance after training

24 Measures of Network Performance The performance ofthe network is measured by three indicators the mean sizing

Shock and Vibration 7

error the damage missing error and the false alarm error asdefined by Yun et al [19] The mean sizing error (MSE) is theaverage quantification error

MSE =1

NOsum

119894

1003816100381610038161003816119910119894 minus 119900119894

1003816100381610038161003816 (9)

where119910119894and 119900119894are the estimated and desired outputs for node

119894 respectively and NO is the number of output nodesThe damage missing error (DME) is given by

DME =1

NTsum

119894

120576119868

119894 0 le DME le 1 (10)

where 120576119868

119894= 0 if the 119894th damaged element is correctly detected

and 120576119868

119894= 1 otherwise The NT corresponds to the number

of true damage locations If DME = 0 all damage locationsare correctly detected The value of 120576

119868

119894is calculated using the

following equation

120576119868

119894=

1 if 119900119894gt 0 119910

119894le 120572119888

0 otherwise(11)

It is assumed that an element is detected as damaged if theestimated damage 119910

119894is greater than a prescribed critical value

120572119888 The critical damage level 120572

119888is defined as being equal to

the average MSEThis value is the minimum damage that thenetwork can reasonably assess

The false alarm error (FAE) is defined as

FAE =1

NFsum

119894

120576119868119868

119894 0 le FAE le 1 (12)

where 120576119868119868

119894= 0 if the 119894th detected damage is truly damaged

and 120576119868119868

119894= 1 otherwise The NF is the number of predicted

damage locations If FAE = 0 all of the detected locations areactual damage locations The value of 120576

119868119868

119894is calculated using

the following equation

120576119868119868

119894=

1 if119910119894ge 120572119888 119900119894= 0

0 otherwise(13)

3 Identification and Matching ofAntiresonant Frequencies

31 Experimental Antiresonances The experimental antires-onances are identified from experimental point FRFs byldquodip-pickingrdquo [22] In this technique the antiresonances areselected by picking the dips from the magnitude plot ofa given FRF that have an associated change of +180∘ inthe phase plot This method is similar to the ldquopeak pick-ingrdquo technique for resonant frequencies Figure 3 shows anexample of the anti-resonant picking procedure When usingthis technique the errors in the estimated antiresonancesoriginate primarily from the frequency resolution of theFRFs If the dips in the FRFs are difficult to determine theantiresonances can be identified by a curve-fitting techniquebased on the rational fraction polynomial representation ofthe FRFs [38]

32 Numerical Antiresonances For a lightly damped struc-ture the anti-resonant frequencies are nearly unaffected bydamping and therefore they can be obtained from theundamped system using only the stiffness andmass matrices

By definition the FRFmatrix is the inverse of the dynamicstiffness matrix

H (120596) = (K minus 1205962M)minus1

=adj (K minus 120596

2M)

det (K minus 1205962M) (14)

The operators adj(sdot) and det(sdot) indicate the adjointand determinant respectively The anti-resonant frequenciescorrespond to the zeros of the FRFs The zeros of the 119894119896th FRF are the values of 120596 for which the numerator of119867119894119896

(120596)vanishes The numerator of 119867119894119896

(120596) is the 119894 119896th termof adj(Kminus 120596

2M) which is given by (minus1)119894+119896 det(K

119894119896minus 1205962M119894119896

)The subscripts 119894 and 119896 denote that the 119894th row and 119896th columnhave been deleted As a consequence the antiresonances ofthe 119894 119896th FRF are the frequency values that satisfy

det (K119894119896

minus 1205962M119894119896

) = 0 (15)

which is equivalent to solving the eigenvalue problem

(K119894119896

minus 1205962M119894119896

) u = 0 (16)

If 119894 = 119896 (16) represents a physical system obtained bygrounding the 119894th degree of freedom Therefore the anti-resonant frequencies obtained from point FRFs (119894 = 119896) areequivalent to the resonant frequencies of the structure withthe 119894th degree of freedom grounded

Because we are working with point FRFs matching theexperimental and numerical anti-resonant frequencies is aneasy task because each anti-resonant frequency lies betweentwo resonant frequencies

4 Eight-DOF Spring-Mass System

The structure shown in Figure 4 consists of an 8-DOF spring-mass system The Los Alamos National Laboratory (LANL)designed and constructed this system to study the effec-tiveness of various vibration-based damage identificationtechniques [39]

The system consists of eight translatingmasses connectedby springs and each mass consists of a disc of aluminumwith a diameter of 762mm and a thickness of 254mm Themasses slide on a highly polished steel rod and are fastenedtogether with coil springs The spring and mass locationsare designated sequentially with the first located the closestto the shaker attachment In the undamaged configurationall springs are identical and have linear stiffness coefficientsDamage is simulated by replacing the fifth spring with adifferent spring with a lower stiffness (55 stiffness reduc-tion)The acceleration is measured horizontally at eachmassgiving a total of eightmeasuredDOFsThe structure is excitedrandomly by an electrodynamic shaker The responses aremeasured in the eight DOFs in the undamaged and damagedcases

Seven anti-resonant frequencies are identified from theFRF corresponding to the first mass which corresponds to

8 Shock and Vibration

Mag

nitu

de (a

bs)

102

100

104

10minus2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

2855988

10261549

8125

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

Phas

e (de

g)

50

0

minus50

minus100

minus150

minus200

(b)

Figure 3 Identification of experimental anti-resonant frequencies

m8 m7 m6 m5 m4 m3 m2 m1 f(t)

k7 k6 k5 k4 k3 k2 k1x8 x7 x6 x5 x4 x3 x2 x1

Figure 4 Experimental system with eight degrees of freedom

a point FRF Given that the anti-resonant frequencies areindependent seven antiresonances should be sufficient toassess the stiffness of each of the seven springs

Table 1 lists the anti-resonant frequencies identified inthe undamaged and damaged cases The second and fourthantiresonances provide the larger variation due to the exper-imental damage

The numerical model is built in Matlab with springs andconcentrated masses and the initial parameters are set asfollows

(i) mass 1 5593 g (this mass is greater than the othersbecause of the hardware required to attach the shaker)

(ii) masses 2 to 8 4194 g(iii) spring constants 567 kNmminus1

Table 2 shows the numerical and experimental anti-resonant frequencies The maximum difference between theexperimental and numerical antiresonances is 07 As aconsequence the numerical model provides a faithful repre-sentation of the experimental structure and can be used totrain the neural network

41 Construction of the Neural Network The network con-tains seven inputs and seven outputsThe best parameters forthe neural network are defined after performing a sensitivity

Table 1 Experimental antiresonances

Antiresonant frequencies (Hz) Variation ()Undamaged Damaged1231 1166 minus5283694 3244 minus12185961 598 0327996 7174 minus10289612 9302 minus32310944 10686 minus23611572 11267 minus264

Table 2 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical123 123 043369 367 054596 592 070800 795 055961 956 0521094 1090 0401157 1155 019

analysis The network is trained under different configura-tions and the selected configuration is the one that providesthe lowest validation error

First the effects of five different transfer functions usedfor the output layer are studied and these transfer functionsare formulated as follows (a) hyperbolic tangent sigmoid(b) logarithm sigmoid (c) linear (d) saturating linear and(e) symmetric saturating linear Figure 2 illustrates thesefunctions During the experimental runs the remainingparameters of the network are characterized as follows thehidden transfer function is a logarithm sigmoid the numberof hidden neurons is 30 the training data are polluted with2 random noise and the Levenberg-Marquardt algorithmis used to train the network The network is trained 10times with each of the output transfer functions Table 3summarizes the average validation performance obtained

Shock and Vibration 9

Table 3 Network validation performance with different transfer functions

Output transfer function 119864119904

Hidden transfer function 119864119904

Hyperbolic tangent sigmoid 00067 Hyperbolic tangent sigmoid 00045Logarithm sigmoid 00134 Logarithm sigmoid 00044Linear 00060 Linear 00130Saturating linear 00127 Saturating linear 00050Symmetric saturating linear 00046 Symmetric saturating linear 00047

Table 4 Network validation performance with different trainingalgorithms

Training algorithms Time (s) 119864119904

Gradient descent 237 00467Gradient descent with adaptive learning 9 00289Scaled conjugate gradient 12 00158Levenberg-Marquardt 15 00046

for each function The symmetric saturating linear functionprovides the best performance

Second the performance of each of these five transferfunctions in the hidden layer is investigated The remainingnetwork parameters are the symmetric saturating linearoutput transfer function 30 hidden neurons 2 noise andthe Levenberg-Marquardt training algorithmThe network istrained 10 times with each of these output transfer functionsTable 3 summarizes the average validation performance Thelogarithm sigmoid function provides the best performance

Next the performance of four training algorithms isstudied Table 4 summarizes the average results for 10 runsThe gradient descent algorithm provides the worst perfor-mance in terms of training time and validation mean squareerror An adaptive learning rate substantially improves theresults the training time is 26 times lower and the meanerror is reduced by a factor of 16 The scale conjugate-gradient reaches a lower mean square error with a slightlyhigher training time Nevertheless the Levenberg-Marquardtalgorithm provides the lowest mean square validation errorby far and its training time is nearly the same as that with thescale conjugate gradient algorithm

Figure 5 shows the average validation error for differentnumbers of hidden nodes Seventy nodes in the hidden layergive the lower validation error 119864

119904 Less than 70 neurons are

not sufficient to fit the data whereas greater than 70 neuronsoverfit the neural network

The last parameter corresponds to the level of noise inthe training data The selection of this parameter is relatedto the level of experimental noise Hence this parameterwas defined by studying the performance of the trainednetwork with the experimental data and different levels ofperturbation noise Figure 6 shows the mean sizing error(MSE) obtained when the network is tested with experimen-tal dataThis case corresponds to a 55 stiffness reduction inelement 5 The MSE is the difference between the predicteddamage level on each element and the actual damage level asdefined in (9) A data perturbation noise of 15 provides theminimum error in the predicted damage

20 40 60 80 100 120 1403638

442444648

5

Number of hidden nodes

Es

times10minus3

Figure 5 Network validation performance as a function of thenumber of hidden nodes

0 05 1 15 2 25 3 35 40

00501

01502

02503

035

Noise level ()

MSE

Figure 6 Experimental mean sizing error for different levels ofinjected noise

42 Network Validation The selected network parametersare

(i) network three layers of multilayer perceptron(ii) number of input nodes 7(iii) number of hidden nodes 70(iv) number of output nodes 7(v) transfer function in the hidden layer Logarithm

sigmoid(vi) transfer function in the output layer Symmetric

saturating linear(vii) training method Levenberg-Marquardt

The validation patterns verify the performance of the net-work and these patterns are polluted with 15 noise Theresulting mean sizing error is MSE = 337 and thus thenetwork quantification accuracy is 97 Figure 7 shows thedamage missing error (DME) and false alarm error (FAE)divided by the damage levelsThe results of the DME indicate

10 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

Damage level ()

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(b)

Figure 7 Damage missing error (DME) and false alarm error (FAE) for different damage levels

1 2 3 4 5 6 70

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Figure 8 Assessment of experimental damage (55 stiffness reduc-tion in element 5)

that the detection of damage with low severities is poor Infact 80 of the damage with severities lower than 10 and40of the damagewith severities of 10ndash20 are not detectedThe algorithm can detect damage with severities larger than30with confidence it correctly detects 996 of the damagewith severities over 30The false alarm results indicate thatmost of the damage detected with levels lower than 10are false damage (near 75) The amount of false damagedetection is drastically reduced with an increment in the levelof damage In fact 9924 of the damage detected with levelsover 30 is true damage

Figure 8 shows the results of the damage detected usingthe experimental case The damaged element is correctlyassessed the location is correct and the quantification erroris 5

5 Experimental Beam

The structure consists of a steel beam with a rectangularcross-section The dimensions of the beam are length = 1mand section = 25 times 10mm2 As shown in Figure 9 soft springssuspend the structure to simulate a ldquofree-freerdquo boundarycondition

A hammer excites the beam at four points distributedalong the beam and an accelerometer measures the responseat each excitation location Both the excitation force and themeasured responses lie in the horizontal direction In thisdirection the anti-resonant frequencies are more sensitive to

Table 5 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical

FRF1

903 89 1102934 289 1356063 604 04010328 1033 00015528 1576 151

FRF2

1288 128 0692609 258 1005831 586 05611128 1127 12616219 1650 170

FRF3

1288 128 069260 258 0665844 586 03411138 1127 11716175 1650 198

FRF4

903 89 1102929 289 1356061 604 04010328 1033 00015534 1576 151

Figure 9 Experimental beam

the experimental damage Five anti-resonant frequencies areidentified at each of the four experimental FRFs

The numerical model is built in Matlab with 2D beamelements The model contains 20 beam elements and 40degrees of freedom as shown in Figure 10 The shadowedelements indicate possible locations of damage Table 5 showsthe experimental and numerical frequencies and their differ-ences and the maximum difference is 198

The structure is subjected to four different damagescenarios containing single and double cracks Cracks are

Shock and Vibration 11

Table 6 Damage cases introduced to the beam

Case Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

1 315 7 5 mdash mdash mdash2 635 13 10 mdash mdash mdash3 303 7 5 640 13 104 360 8 5 810 17 15

1 2 3 4 5 6 7 8 9 10 1211 13 14 15 16 17 18 19 20

Figure 10 Numerical model and element numbering

C

Figure 11 Saw cut introduced to the beam

introduced to the structure by saw cuts of length 119897119888 as

illustrated in Figure 11 Table 6 summarizes the differentdamage scenarios and indicates the distance from the left endto the cut the corresponding element in the numericalmodeland the cut length

51 Network Validation The network uses the same param-eters selected for the first structure except for the number ofinput hidden and output nodes The number of input nodescorresponds to the number of anti-resonant frequencieswhich is 20 Elements 2 to 19 were defined as possiblelocations of damage thus giving 18 output nodes (damageindices) The number of hidden nodes was defined afterthe sensitivity analysis shown in Figure 12 The minimumvalidation error is obtained with 80 hidden nodes

Hence the network parameters are

(i) network three layers of multilayer perceptron(ii) number of input nodes 20(iii) number of hidden nodes 80(iv) number of output nodes 18(v) transfer function in the hidden layer logarithm sig-

moid(vi) transfer function in the output layer symmetric

saturating linear(vii) training method Levenberg-Marquardt

The training and validation patterns were polluted with15 random noise The mean sizing error obtained for thevalidation patterns is MSE = 153 and thus the networkquantification accuracy is 98 Figure 13 shows the damagemissing error (DME) and false alarm error (FAE) separatedby damage levels The results of the DME indicate that thedetection of damage with low severities is poor In fact

20 40 60 80 100 120 140 160 180 200

28

3

32

34

36

Number of hidden nodes

times10minus3

Es

Figure 12 Validation performance as a function of the number ofhidden nodes

near 90 of the damage with severities lower than 10 and70 of the damage with severities between 10 and 20 isnot detected The algorithm is able to detect damage withseverities larger than 40with confidence it correctly detects995 of the damage with severities over 40The false alarmresults indicate that most of the damage detected with levelslower than 10 is false damage (near 80) The amount offalse damage detection decreases with an increment of thelevel of damage 99 of the damage detected with levels over40 is true damage

Note that these results are slightly worse than the resultsobtained from the 8-DOF system This result is explained bythe fact that the anti-resonant frequencies are less sensitiveto damage in the free beam than the 8-DOF structure thusmaking it more difficult to detect small damage

Figure 14 shows the results of the damage detected in thefour experimental scenarios An arrow indicates the actualdamage location In the first three cases the damage iscorrectly located although false damage appears next to theactual locations Nevertheless the assumption that a crackonly affects the stiffness of the corresponding element mightnot always be true For instance in the fourth case the effectof the larger cut is represented more accurately by a stiffnessreduction of two elements rather than one In the last case thenetwork does not detect the small crack in element 8 becausethe effect of the larger cut hides the effect of the smaller cut

6 Conclusions

This paper presents a new methodology designed to assessexperimental damage using neural networks and anti-resonant frequencies A multilayer perceptron network wastrained with data obtained from a numerical model and

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 3

method could be used to roughly locate the crack andsubsequently other methods can be applied to determinethe crack characteristics more precisely The changes inthe resonant and anti-resonant frequencies were used byInada et al [26] to locate and quantify the delaminationof a composite beam This group implemented a two-stepprocedure first the delamination domain is identified fromthe anti-resonant changes and next the location and sizeare defined using natural frequency changes This methodwas effective in identifying the delamination locations andsizes Wang and Zhu [27] suggested a method for identifyingcracks in beams which makes use of the natural frequenciesand antiresonances from the point FRFsThemethodology issimilar to that proposed by Bamnios et al [25] in that the shiftin the first anti-resonant frequency versus the driving pointlocation is used to locate damage They validated the methodwith a numerical example of a simply supported beam withthree cracks Their method was able to accurately predict thelocations and qualitatively estimate the crack size Meruaneand Heylen [28] showed that antiresonances are a goodalternative to mode shapes in damage assessment Howeverthey stated that further research is required for identificationof the experimental antiresonances and for the matching ofexperimental and numerical antiresonances Thus antires-onances present an attractive alternative to mode shapes asthe input values of neural networks for the following reasonsthey require a lower number of inputs are less contaminatedby noise and provide the same information Neverthelessmethods that use antiresonances are still under developmentand the application of antiresonances to structural damagedetection has not been fully investigated primarily becausethe inverse optimization problem used with antiresonances isparticularly challenging and robust optimization algorithmsare needed However these limitations should not present aproblem for methods based on neural networks

11 Artificial Neural Networks An ANN is a data processingalgorithm that attempts to emulate the processing scheme ofthe human brain [29] An ANN is formed by ldquoneuronsrdquo thatare interconnected to build a complex network Knowledge isacquired by a learning process and stored in the interneuronconnections known as the ldquosynaptic weightsrdquo

Different types of network architectures exist and amongthem themultilayer perceptron (MLP) is themost frequentlyused An MLP network consists of an array of input neuronsknown as the input layer an array of output neurons known asthe output layer and a number of hidden layers Each neuronreceives a weighted sum from the neurons in the precedinglayer and provides an input to every neuron of the next layerThe activation of each neuron is governed by a functionknown as the transfer function Typical selections for thetransfer function are linear log-sigmoid and hyperbolictangent-sigmoid as shown in Figure 1

Figure 2 illustrates the principle of a damage assessmentalgorithm using ANN The vibration characteristics of thestructure act as the inputs to the neural network and theoutputs are the damage indices of each element in thestructure

The outputs of a three-layer MLP are given by

119910119894= 119891 (

NHsum

119895=1

(119908119894119895119891 (

NIsum

119896=1

V119895119896

119909119896

+ 119887V119896) + 119887119908119894

)) (1)

where 119909 = 1199091 1199092 119909

119899 and 119910 = 119910

1 1199102 119910

119899 are the

input and output vectors 119908119894119895and V119895119896are the interconnection

weights 119887 represents the bias (or threshold) terms 119891(sdot) is thetransfer function and NI and NH are the number of inputand hidden nodes respectively

The values of the weights are updated by training theneural network with data from the structure The trainingproblem consists of finding theweights that willminimize themean square error 119864

119904

119864119904

=1

NO

NOsum

119896=1

(119910119896

minus 119900119896)2 (2)

where 119900119896is the desired output at the 119896th output node and

NO is the number of output nodes Many optimizationtechniques are able to address the network-training problemThe simplest method is gradient descent also known as thesteepest descent This algorithm begins with an initial weightvector and iteratively updates it by moving in the direction ofthe greatest error decrease The gradient descent rule can bewritten as

Δ119908120591

119894= minus120578

120597119864119904

120597119908120591

119894

(3)

where Δ119908120591

119894is the change for the 119894th weight at step 120591 The

backpropagation algorithm calculates the gradient at eachstep The convergence of this algorithm strongly dependson the learning rate 120578 If 120578 is too large the algorithm mayovershoot leading to divergent oscillations However if 120578 istoo small the search might proceed quite slowly Fang etal [15] showed that an adaptive learning rate significantlyimproves the training performance Another improvementto the gradient descent algorithm involves ensuring thateach search direction is conjugate to all previous directionsthus avoiding unnecessary loops in the search process Thescale-conjugated gradient algorithm proposed byMoslashller [30]combines this concept and variable steps Many additionaloptimization algorithms have been proposed to train neuralnetworks but the Levenberg-Marquant algorithm has beenthe most efficient [31] This method is a gradient-basedalgorithm specifically designed to minimize the sum-of-squares error [32]

A disadvantage of ANN is the need for large trainingsets It is highly difficult and time-consuming to producesufficiently large training data sets from experiments Analternative to generating training samples is to use a numer-ical model of the structure Castellini and Revel [13] showedthat it is possible to produce correct damage predictions inan experimental structure using a neural network that wastrained with samples generated by a finite element modelNevertheless this approach is highly dependent on theaccuracy of the numerical model There are two approaches

4 Shock and Vibration

f(n)

+1

minus1

n

(a)

f(n)

+1

minus1

n

(b)

f(n)

+1

minus1

n

(c)

f(n)

+1

minus1

n

(d)

f(n)

+1

minus1

n

(e)

Figure 1 Transfer functions (a) hyperbolic tangent sigmoid (b) logarithm sigmoid (c) linear (d) saturating linear and (e) symmetricsaturating linear

Shock and Vibration 5

to overcome this problem The first is to update the numer-ical model using experimental data from the undamagedstructure However even after updating differences will stillremain between the numerical and experimental modelsThe second alternative is to define an input parameter thatconsiders the initial errors in the numerical model thusavoiding the need for an accurate numerical modelThis goalis achieved using the changes in the data instead of theirabsolute values The main assumption is that any changein the structural properties is caused by damage Thus anyerror in the undamaged model of the structure that is alsopresent in the damaged model will be removed [33] Leeet al [34] showed that natural frequency changes becausestructural damage in a system without modeling errors isapproximated the same as those in a system with modelingerror Hence changes in the natural frequencies are lesssensitive to modeling errors than the natural frequenciesthemselves This group demonstrated the applicability ofa neural network trained with mode shape changes Thedamage locations were estimated with reasonable accuracyalthough false alarms were detected at several locations

Simulated data derived from a numerical model arenoise-free whereas actual measurements are never free fromexperimental noise Noise in themeasurements will cause thenetwork to estimate parameters that are different from theactual properties of the structure A solution is to introduceartificial noise into the numerical data used to train thenetwork This process is known as data perturbation scheme[35] Yun et al [19] used a noise-injection learning algorithmand a data perturbation schemeThey implemented a neural-network-based damage assessment algorithm to detect thedamage in structural joints The joints are modeled assemirigid connections using rotational springs and accurateresults are obtained in the cases of moderated noise Sahinand Shenoi [21] trained a neural network with artificiallyadded noise to detect single damage in beam-like compositelaminates The network contains two outputs the locationand the amount of damage They studied different combina-tions of features extracted from the resonant frequencies andmode shape curvatures as inputs to the neural network andtheir results show that feature selection plays a crucial rolein the predictions accuracy Zang and Imregun [14] proposeda different approach to address experimental noise and usedprincipal component projection of the FRF data as an inputto a neural network with two outputs of healthy or damagedThe authors stated that reconstructing the response using thehighest principal components should not only achieve datacompression but also remove a proportion of the noise ThePCA compression acts as a noise filter which can be useful inmodal analysis as well

The successful application of a neural network dependson the representation and the learning algorithms Never-theless their selection is problem-dependent and is usuallydetermined by trial and error [36] Sahoo and Maity [7]used a Genetic Algorithm (GA) to automate the trial-and-error process The network parameters (number of neuronslearning rate etc) are set as variables in an optimizationproblem handled by a GA They used an MLP network withtwo hidden layers trained by a backpropagation algorithm

Structure

Vibration characteristics

Artificial neural networks

Input layer

Hidden layers

Output layer

Stiffness reduction factors

Figure 2 Damage assessment with artificial neural networks

The inputs are the natural frequencies and strains in certainlocations of the structure Fang et al [15] explored the use ofa tunable steepest descent (TSD) algorithm that dynamicallyadjusts the learning speed during the training process Theyshowed that this methodology significantly increases thetraining speed while maintaining the learning stabilityTheseauthors used FRF data measured at key spectral pointsaround the resonant frequencies

The primary contribution of this research is the devel-opment of a real time damage assessment algorithm usingANN and anti-resonant frequencies In literature similarlyto mode shapes anti-resonant frequencies are used as com-plementary information to natural frequencies NeverthelessMotthershead [37] demonstrated that the sensitivity of anti-resonant frequencies can be expressed as a combination of thesensitivities of mode shapes and natural frequencies Henceanti-resonant frequencies contain the same information asnatural frequencies and mode shapes together which is thehypothesis of the present studyThis study is restricted to anti-resonant frequencies obtained from point FRFs Althoughmeasurement of the point FRFs is time-consuming becausethe excitation point is moved together with the response

6 Shock and Vibration

location this method offers several advantages for damageassessment as follows

(i) All antiresonances contain independent informationbecause they correspond to the resonant frequenciesof the system grounded at different degrees of free-dom

(ii) For a given FRF the number of anti-resonant frequen-cies does not change from one damage scenario tothe next this is a valuable property if we require thesefrequencies to act as inputs to a neural network

(iii) An anti-resonant frequency always lies betweentwo resonant frequencies Hence there is no doubtwhether a dip in an FRF is an anti-resonant frequencyor a minimum

This study addresses the setup of the neural networkparameters and provides guidelines for their selection insimilar damage detection problems The proposed method-ology is evaluated with two experimental structures an 8-DOF mass-spring system and a beam with multiple damagescenarios

2 Neural Network for Damage Assessment

This study intends to train a neural network using anti-resonant frequencies and to determine its feasibility in assess-ing experimental damage Hence this study works with thesimplest neural network that has been able to detect locateand quantify structural damage the multilayer perceptron(MLP) with three layers (input hidden and output) aspresented by Fang et al [15] The outputs of the networkcorrespond to the damage indices of each element whereasthe inputs are the changes in anti-resonant frequencies Thenumber of hidden nodes is defined after a sensitivity analysisfor each application case

21 Inputs As proposed by Lee et al [34] the inputs tothe neural network are defined as the changes in the modalparameters rather than their absolute values With thisapproach the network is less sensitive to errors in the baselineFE model

Therefore the inputs correspond to the experimentalchange in the anti-resonant frequencies with respect to theintact case

119909119894times119899

=120596119863

119903119894119899minus 120596119880

119903119894119899

120596119880

119903119894119899

(4)

The superscripts 119863 and 119880 denote undamaged and damagedrespectively and120596

119903119894119899is the 119894th anti-resonant frequency of the

119899th FRFTo reduce the effects of experimental noise the simulated

data are polluted with random noise As proposed by Hjelm-stad and Shin [35] each set of perturbed data is created byadding uniformly distributed random noise to the numericaldata

120596119903119894119899

= 120596119903119894119899 (1 + 120585) (5)

where 120585 is a uniform random number with a specifiedamplitudeThe variance of the perturbing noise should be thesame as the variance of the measurement noise

22 Outputs The damage indices are represented by theelemental stiffness reduction factors 120573

119894defined as the ratios

between the stiffness reduction and the initial stiffness

119910119894= 120573119894 (6)

The stiffnessmatrix of the damaged structureK119889 is expressed

as a sum of element matrices multiplied by reduction factors

K119889

= sum

119894

(1 minus 120573119894)K119894 (7)

The value 120573119894= 0 indicates that the element is undamaged

whereas 0 lt 120573119894le 1 implies partial or complete damage

23 Training and Validation Patterns The distribution ofthe training patterns plays a crucial role in the success of aneural network The relationship between the anti-resonantfrequencies and the different damage levels is not linearand as a consequence the network might be unable tointerpolate Fang et al [15] recommend the use of trainingpatterns with evenly distributed damaged levels Combiningsimultaneous damages into the training increases the numberof training patterns and a large number of training patternscould overwhelm the training procedure In this study thetraining patterns were generated by considering up to twosimultaneous damage incidences with nine damage levelsevenly distributed between 0 and 80 Hence the totalnumber of training patterns depends on the number ofelements 119873 as

number of training patterns = 92

times 119862 (119873 2) (8)

where 119862(119873 2) is the number of combinations of 119873 elementstaken two at a time

The algorithm trains the network using the early stoppingtechnique [37] In this technique two sets of data are usedduring training a training set and a validation set Thetraining set is used for updating the network weights andbiases As defined in (2) the mean square error evaluated onthe validation set ismonitored during trainingWhen the val-idation error increases for six iterations the algorithm stopsthe training The weights of the network at the minimumvalidation error become the final weights

The validation set is a group of patterns that must bedifferent from the training patterns To ensure this conditionthe validation patterns were created with nine damage levelsevenly distributed between 5 and 85 and considered up totwo simultaneous damage incidences Note that the numberof training patterns is equal to the number of validationpatterns This set of validation patterns is also used to checkthe network performance after training

24 Measures of Network Performance The performance ofthe network is measured by three indicators the mean sizing

Shock and Vibration 7

error the damage missing error and the false alarm error asdefined by Yun et al [19] The mean sizing error (MSE) is theaverage quantification error

MSE =1

NOsum

119894

1003816100381610038161003816119910119894 minus 119900119894

1003816100381610038161003816 (9)

where119910119894and 119900119894are the estimated and desired outputs for node

119894 respectively and NO is the number of output nodesThe damage missing error (DME) is given by

DME =1

NTsum

119894

120576119868

119894 0 le DME le 1 (10)

where 120576119868

119894= 0 if the 119894th damaged element is correctly detected

and 120576119868

119894= 1 otherwise The NT corresponds to the number

of true damage locations If DME = 0 all damage locationsare correctly detected The value of 120576

119868

119894is calculated using the

following equation

120576119868

119894=

1 if 119900119894gt 0 119910

119894le 120572119888

0 otherwise(11)

It is assumed that an element is detected as damaged if theestimated damage 119910

119894is greater than a prescribed critical value

120572119888 The critical damage level 120572

119888is defined as being equal to

the average MSEThis value is the minimum damage that thenetwork can reasonably assess

The false alarm error (FAE) is defined as

FAE =1

NFsum

119894

120576119868119868

119894 0 le FAE le 1 (12)

where 120576119868119868

119894= 0 if the 119894th detected damage is truly damaged

and 120576119868119868

119894= 1 otherwise The NF is the number of predicted

damage locations If FAE = 0 all of the detected locations areactual damage locations The value of 120576

119868119868

119894is calculated using

the following equation

120576119868119868

119894=

1 if119910119894ge 120572119888 119900119894= 0

0 otherwise(13)

3 Identification and Matching ofAntiresonant Frequencies

31 Experimental Antiresonances The experimental antires-onances are identified from experimental point FRFs byldquodip-pickingrdquo [22] In this technique the antiresonances areselected by picking the dips from the magnitude plot ofa given FRF that have an associated change of +180∘ inthe phase plot This method is similar to the ldquopeak pick-ingrdquo technique for resonant frequencies Figure 3 shows anexample of the anti-resonant picking procedure When usingthis technique the errors in the estimated antiresonancesoriginate primarily from the frequency resolution of theFRFs If the dips in the FRFs are difficult to determine theantiresonances can be identified by a curve-fitting techniquebased on the rational fraction polynomial representation ofthe FRFs [38]

32 Numerical Antiresonances For a lightly damped struc-ture the anti-resonant frequencies are nearly unaffected bydamping and therefore they can be obtained from theundamped system using only the stiffness andmass matrices

By definition the FRFmatrix is the inverse of the dynamicstiffness matrix

H (120596) = (K minus 1205962M)minus1

=adj (K minus 120596

2M)

det (K minus 1205962M) (14)

The operators adj(sdot) and det(sdot) indicate the adjointand determinant respectively The anti-resonant frequenciescorrespond to the zeros of the FRFs The zeros of the 119894119896th FRF are the values of 120596 for which the numerator of119867119894119896

(120596)vanishes The numerator of 119867119894119896

(120596) is the 119894 119896th termof adj(Kminus 120596

2M) which is given by (minus1)119894+119896 det(K

119894119896minus 1205962M119894119896

)The subscripts 119894 and 119896 denote that the 119894th row and 119896th columnhave been deleted As a consequence the antiresonances ofthe 119894 119896th FRF are the frequency values that satisfy

det (K119894119896

minus 1205962M119894119896

) = 0 (15)

which is equivalent to solving the eigenvalue problem

(K119894119896

minus 1205962M119894119896

) u = 0 (16)

If 119894 = 119896 (16) represents a physical system obtained bygrounding the 119894th degree of freedom Therefore the anti-resonant frequencies obtained from point FRFs (119894 = 119896) areequivalent to the resonant frequencies of the structure withthe 119894th degree of freedom grounded

Because we are working with point FRFs matching theexperimental and numerical anti-resonant frequencies is aneasy task because each anti-resonant frequency lies betweentwo resonant frequencies

4 Eight-DOF Spring-Mass System

The structure shown in Figure 4 consists of an 8-DOF spring-mass system The Los Alamos National Laboratory (LANL)designed and constructed this system to study the effec-tiveness of various vibration-based damage identificationtechniques [39]

The system consists of eight translatingmasses connectedby springs and each mass consists of a disc of aluminumwith a diameter of 762mm and a thickness of 254mm Themasses slide on a highly polished steel rod and are fastenedtogether with coil springs The spring and mass locationsare designated sequentially with the first located the closestto the shaker attachment In the undamaged configurationall springs are identical and have linear stiffness coefficientsDamage is simulated by replacing the fifth spring with adifferent spring with a lower stiffness (55 stiffness reduc-tion)The acceleration is measured horizontally at eachmassgiving a total of eightmeasuredDOFsThe structure is excitedrandomly by an electrodynamic shaker The responses aremeasured in the eight DOFs in the undamaged and damagedcases

Seven anti-resonant frequencies are identified from theFRF corresponding to the first mass which corresponds to

8 Shock and Vibration

Mag

nitu

de (a

bs)

102

100

104

10minus2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

2855988

10261549

8125

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

Phas

e (de

g)

50

0

minus50

minus100

minus150

minus200

(b)

Figure 3 Identification of experimental anti-resonant frequencies

m8 m7 m6 m5 m4 m3 m2 m1 f(t)

k7 k6 k5 k4 k3 k2 k1x8 x7 x6 x5 x4 x3 x2 x1

Figure 4 Experimental system with eight degrees of freedom

a point FRF Given that the anti-resonant frequencies areindependent seven antiresonances should be sufficient toassess the stiffness of each of the seven springs

Table 1 lists the anti-resonant frequencies identified inthe undamaged and damaged cases The second and fourthantiresonances provide the larger variation due to the exper-imental damage

The numerical model is built in Matlab with springs andconcentrated masses and the initial parameters are set asfollows

(i) mass 1 5593 g (this mass is greater than the othersbecause of the hardware required to attach the shaker)

(ii) masses 2 to 8 4194 g(iii) spring constants 567 kNmminus1

Table 2 shows the numerical and experimental anti-resonant frequencies The maximum difference between theexperimental and numerical antiresonances is 07 As aconsequence the numerical model provides a faithful repre-sentation of the experimental structure and can be used totrain the neural network

41 Construction of the Neural Network The network con-tains seven inputs and seven outputsThe best parameters forthe neural network are defined after performing a sensitivity

Table 1 Experimental antiresonances

Antiresonant frequencies (Hz) Variation ()Undamaged Damaged1231 1166 minus5283694 3244 minus12185961 598 0327996 7174 minus10289612 9302 minus32310944 10686 minus23611572 11267 minus264

Table 2 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical123 123 043369 367 054596 592 070800 795 055961 956 0521094 1090 0401157 1155 019

analysis The network is trained under different configura-tions and the selected configuration is the one that providesthe lowest validation error

First the effects of five different transfer functions usedfor the output layer are studied and these transfer functionsare formulated as follows (a) hyperbolic tangent sigmoid(b) logarithm sigmoid (c) linear (d) saturating linear and(e) symmetric saturating linear Figure 2 illustrates thesefunctions During the experimental runs the remainingparameters of the network are characterized as follows thehidden transfer function is a logarithm sigmoid the numberof hidden neurons is 30 the training data are polluted with2 random noise and the Levenberg-Marquardt algorithmis used to train the network The network is trained 10times with each of the output transfer functions Table 3summarizes the average validation performance obtained

Shock and Vibration 9

Table 3 Network validation performance with different transfer functions

Output transfer function 119864119904

Hidden transfer function 119864119904

Hyperbolic tangent sigmoid 00067 Hyperbolic tangent sigmoid 00045Logarithm sigmoid 00134 Logarithm sigmoid 00044Linear 00060 Linear 00130Saturating linear 00127 Saturating linear 00050Symmetric saturating linear 00046 Symmetric saturating linear 00047

Table 4 Network validation performance with different trainingalgorithms

Training algorithms Time (s) 119864119904

Gradient descent 237 00467Gradient descent with adaptive learning 9 00289Scaled conjugate gradient 12 00158Levenberg-Marquardt 15 00046

for each function The symmetric saturating linear functionprovides the best performance

Second the performance of each of these five transferfunctions in the hidden layer is investigated The remainingnetwork parameters are the symmetric saturating linearoutput transfer function 30 hidden neurons 2 noise andthe Levenberg-Marquardt training algorithmThe network istrained 10 times with each of these output transfer functionsTable 3 summarizes the average validation performance Thelogarithm sigmoid function provides the best performance

Next the performance of four training algorithms isstudied Table 4 summarizes the average results for 10 runsThe gradient descent algorithm provides the worst perfor-mance in terms of training time and validation mean squareerror An adaptive learning rate substantially improves theresults the training time is 26 times lower and the meanerror is reduced by a factor of 16 The scale conjugate-gradient reaches a lower mean square error with a slightlyhigher training time Nevertheless the Levenberg-Marquardtalgorithm provides the lowest mean square validation errorby far and its training time is nearly the same as that with thescale conjugate gradient algorithm

Figure 5 shows the average validation error for differentnumbers of hidden nodes Seventy nodes in the hidden layergive the lower validation error 119864

119904 Less than 70 neurons are

not sufficient to fit the data whereas greater than 70 neuronsoverfit the neural network

The last parameter corresponds to the level of noise inthe training data The selection of this parameter is relatedto the level of experimental noise Hence this parameterwas defined by studying the performance of the trainednetwork with the experimental data and different levels ofperturbation noise Figure 6 shows the mean sizing error(MSE) obtained when the network is tested with experimen-tal dataThis case corresponds to a 55 stiffness reduction inelement 5 The MSE is the difference between the predicteddamage level on each element and the actual damage level asdefined in (9) A data perturbation noise of 15 provides theminimum error in the predicted damage

20 40 60 80 100 120 1403638

442444648

5

Number of hidden nodes

Es

times10minus3

Figure 5 Network validation performance as a function of thenumber of hidden nodes

0 05 1 15 2 25 3 35 40

00501

01502

02503

035

Noise level ()

MSE

Figure 6 Experimental mean sizing error for different levels ofinjected noise

42 Network Validation The selected network parametersare

(i) network three layers of multilayer perceptron(ii) number of input nodes 7(iii) number of hidden nodes 70(iv) number of output nodes 7(v) transfer function in the hidden layer Logarithm

sigmoid(vi) transfer function in the output layer Symmetric

saturating linear(vii) training method Levenberg-Marquardt

The validation patterns verify the performance of the net-work and these patterns are polluted with 15 noise Theresulting mean sizing error is MSE = 337 and thus thenetwork quantification accuracy is 97 Figure 7 shows thedamage missing error (DME) and false alarm error (FAE)divided by the damage levelsThe results of the DME indicate

10 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

Damage level ()

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(b)

Figure 7 Damage missing error (DME) and false alarm error (FAE) for different damage levels

1 2 3 4 5 6 70

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Figure 8 Assessment of experimental damage (55 stiffness reduc-tion in element 5)

that the detection of damage with low severities is poor Infact 80 of the damage with severities lower than 10 and40of the damagewith severities of 10ndash20 are not detectedThe algorithm can detect damage with severities larger than30with confidence it correctly detects 996 of the damagewith severities over 30The false alarm results indicate thatmost of the damage detected with levels lower than 10are false damage (near 75) The amount of false damagedetection is drastically reduced with an increment in the levelof damage In fact 9924 of the damage detected with levelsover 30 is true damage

Figure 8 shows the results of the damage detected usingthe experimental case The damaged element is correctlyassessed the location is correct and the quantification erroris 5

5 Experimental Beam

The structure consists of a steel beam with a rectangularcross-section The dimensions of the beam are length = 1mand section = 25 times 10mm2 As shown in Figure 9 soft springssuspend the structure to simulate a ldquofree-freerdquo boundarycondition

A hammer excites the beam at four points distributedalong the beam and an accelerometer measures the responseat each excitation location Both the excitation force and themeasured responses lie in the horizontal direction In thisdirection the anti-resonant frequencies are more sensitive to

Table 5 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical

FRF1

903 89 1102934 289 1356063 604 04010328 1033 00015528 1576 151

FRF2

1288 128 0692609 258 1005831 586 05611128 1127 12616219 1650 170

FRF3

1288 128 069260 258 0665844 586 03411138 1127 11716175 1650 198

FRF4

903 89 1102929 289 1356061 604 04010328 1033 00015534 1576 151

Figure 9 Experimental beam

the experimental damage Five anti-resonant frequencies areidentified at each of the four experimental FRFs

The numerical model is built in Matlab with 2D beamelements The model contains 20 beam elements and 40degrees of freedom as shown in Figure 10 The shadowedelements indicate possible locations of damage Table 5 showsthe experimental and numerical frequencies and their differ-ences and the maximum difference is 198

The structure is subjected to four different damagescenarios containing single and double cracks Cracks are

Shock and Vibration 11

Table 6 Damage cases introduced to the beam

Case Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

1 315 7 5 mdash mdash mdash2 635 13 10 mdash mdash mdash3 303 7 5 640 13 104 360 8 5 810 17 15

1 2 3 4 5 6 7 8 9 10 1211 13 14 15 16 17 18 19 20

Figure 10 Numerical model and element numbering

C

Figure 11 Saw cut introduced to the beam

introduced to the structure by saw cuts of length 119897119888 as

illustrated in Figure 11 Table 6 summarizes the differentdamage scenarios and indicates the distance from the left endto the cut the corresponding element in the numericalmodeland the cut length

51 Network Validation The network uses the same param-eters selected for the first structure except for the number ofinput hidden and output nodes The number of input nodescorresponds to the number of anti-resonant frequencieswhich is 20 Elements 2 to 19 were defined as possiblelocations of damage thus giving 18 output nodes (damageindices) The number of hidden nodes was defined afterthe sensitivity analysis shown in Figure 12 The minimumvalidation error is obtained with 80 hidden nodes

Hence the network parameters are

(i) network three layers of multilayer perceptron(ii) number of input nodes 20(iii) number of hidden nodes 80(iv) number of output nodes 18(v) transfer function in the hidden layer logarithm sig-

moid(vi) transfer function in the output layer symmetric

saturating linear(vii) training method Levenberg-Marquardt

The training and validation patterns were polluted with15 random noise The mean sizing error obtained for thevalidation patterns is MSE = 153 and thus the networkquantification accuracy is 98 Figure 13 shows the damagemissing error (DME) and false alarm error (FAE) separatedby damage levels The results of the DME indicate that thedetection of damage with low severities is poor In fact

20 40 60 80 100 120 140 160 180 200

28

3

32

34

36

Number of hidden nodes

times10minus3

Es

Figure 12 Validation performance as a function of the number ofhidden nodes

near 90 of the damage with severities lower than 10 and70 of the damage with severities between 10 and 20 isnot detected The algorithm is able to detect damage withseverities larger than 40with confidence it correctly detects995 of the damage with severities over 40The false alarmresults indicate that most of the damage detected with levelslower than 10 is false damage (near 80) The amount offalse damage detection decreases with an increment of thelevel of damage 99 of the damage detected with levels over40 is true damage

Note that these results are slightly worse than the resultsobtained from the 8-DOF system This result is explained bythe fact that the anti-resonant frequencies are less sensitiveto damage in the free beam than the 8-DOF structure thusmaking it more difficult to detect small damage

Figure 14 shows the results of the damage detected in thefour experimental scenarios An arrow indicates the actualdamage location In the first three cases the damage iscorrectly located although false damage appears next to theactual locations Nevertheless the assumption that a crackonly affects the stiffness of the corresponding element mightnot always be true For instance in the fourth case the effectof the larger cut is represented more accurately by a stiffnessreduction of two elements rather than one In the last case thenetwork does not detect the small crack in element 8 becausethe effect of the larger cut hides the effect of the smaller cut

6 Conclusions

This paper presents a new methodology designed to assessexperimental damage using neural networks and anti-resonant frequencies A multilayer perceptron network wastrained with data obtained from a numerical model and

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

f(n)

+1

minus1

n

(a)

f(n)

+1

minus1

n

(b)

f(n)

+1

minus1

n

(c)

f(n)

+1

minus1

n

(d)

f(n)

+1

minus1

n

(e)

Figure 1 Transfer functions (a) hyperbolic tangent sigmoid (b) logarithm sigmoid (c) linear (d) saturating linear and (e) symmetricsaturating linear

Shock and Vibration 5

to overcome this problem The first is to update the numer-ical model using experimental data from the undamagedstructure However even after updating differences will stillremain between the numerical and experimental modelsThe second alternative is to define an input parameter thatconsiders the initial errors in the numerical model thusavoiding the need for an accurate numerical modelThis goalis achieved using the changes in the data instead of theirabsolute values The main assumption is that any changein the structural properties is caused by damage Thus anyerror in the undamaged model of the structure that is alsopresent in the damaged model will be removed [33] Leeet al [34] showed that natural frequency changes becausestructural damage in a system without modeling errors isapproximated the same as those in a system with modelingerror Hence changes in the natural frequencies are lesssensitive to modeling errors than the natural frequenciesthemselves This group demonstrated the applicability ofa neural network trained with mode shape changes Thedamage locations were estimated with reasonable accuracyalthough false alarms were detected at several locations

Simulated data derived from a numerical model arenoise-free whereas actual measurements are never free fromexperimental noise Noise in themeasurements will cause thenetwork to estimate parameters that are different from theactual properties of the structure A solution is to introduceartificial noise into the numerical data used to train thenetwork This process is known as data perturbation scheme[35] Yun et al [19] used a noise-injection learning algorithmand a data perturbation schemeThey implemented a neural-network-based damage assessment algorithm to detect thedamage in structural joints The joints are modeled assemirigid connections using rotational springs and accurateresults are obtained in the cases of moderated noise Sahinand Shenoi [21] trained a neural network with artificiallyadded noise to detect single damage in beam-like compositelaminates The network contains two outputs the locationand the amount of damage They studied different combina-tions of features extracted from the resonant frequencies andmode shape curvatures as inputs to the neural network andtheir results show that feature selection plays a crucial rolein the predictions accuracy Zang and Imregun [14] proposeda different approach to address experimental noise and usedprincipal component projection of the FRF data as an inputto a neural network with two outputs of healthy or damagedThe authors stated that reconstructing the response using thehighest principal components should not only achieve datacompression but also remove a proportion of the noise ThePCA compression acts as a noise filter which can be useful inmodal analysis as well

The successful application of a neural network dependson the representation and the learning algorithms Never-theless their selection is problem-dependent and is usuallydetermined by trial and error [36] Sahoo and Maity [7]used a Genetic Algorithm (GA) to automate the trial-and-error process The network parameters (number of neuronslearning rate etc) are set as variables in an optimizationproblem handled by a GA They used an MLP network withtwo hidden layers trained by a backpropagation algorithm

Structure

Vibration characteristics

Artificial neural networks

Input layer

Hidden layers

Output layer

Stiffness reduction factors

Figure 2 Damage assessment with artificial neural networks

The inputs are the natural frequencies and strains in certainlocations of the structure Fang et al [15] explored the use ofa tunable steepest descent (TSD) algorithm that dynamicallyadjusts the learning speed during the training process Theyshowed that this methodology significantly increases thetraining speed while maintaining the learning stabilityTheseauthors used FRF data measured at key spectral pointsaround the resonant frequencies

The primary contribution of this research is the devel-opment of a real time damage assessment algorithm usingANN and anti-resonant frequencies In literature similarlyto mode shapes anti-resonant frequencies are used as com-plementary information to natural frequencies NeverthelessMotthershead [37] demonstrated that the sensitivity of anti-resonant frequencies can be expressed as a combination of thesensitivities of mode shapes and natural frequencies Henceanti-resonant frequencies contain the same information asnatural frequencies and mode shapes together which is thehypothesis of the present studyThis study is restricted to anti-resonant frequencies obtained from point FRFs Althoughmeasurement of the point FRFs is time-consuming becausethe excitation point is moved together with the response

6 Shock and Vibration

location this method offers several advantages for damageassessment as follows

(i) All antiresonances contain independent informationbecause they correspond to the resonant frequenciesof the system grounded at different degrees of free-dom

(ii) For a given FRF the number of anti-resonant frequen-cies does not change from one damage scenario tothe next this is a valuable property if we require thesefrequencies to act as inputs to a neural network

(iii) An anti-resonant frequency always lies betweentwo resonant frequencies Hence there is no doubtwhether a dip in an FRF is an anti-resonant frequencyor a minimum

This study addresses the setup of the neural networkparameters and provides guidelines for their selection insimilar damage detection problems The proposed method-ology is evaluated with two experimental structures an 8-DOF mass-spring system and a beam with multiple damagescenarios

2 Neural Network for Damage Assessment

This study intends to train a neural network using anti-resonant frequencies and to determine its feasibility in assess-ing experimental damage Hence this study works with thesimplest neural network that has been able to detect locateand quantify structural damage the multilayer perceptron(MLP) with three layers (input hidden and output) aspresented by Fang et al [15] The outputs of the networkcorrespond to the damage indices of each element whereasthe inputs are the changes in anti-resonant frequencies Thenumber of hidden nodes is defined after a sensitivity analysisfor each application case

21 Inputs As proposed by Lee et al [34] the inputs tothe neural network are defined as the changes in the modalparameters rather than their absolute values With thisapproach the network is less sensitive to errors in the baselineFE model

Therefore the inputs correspond to the experimentalchange in the anti-resonant frequencies with respect to theintact case

119909119894times119899

=120596119863

119903119894119899minus 120596119880

119903119894119899

120596119880

119903119894119899

(4)

The superscripts 119863 and 119880 denote undamaged and damagedrespectively and120596

119903119894119899is the 119894th anti-resonant frequency of the

119899th FRFTo reduce the effects of experimental noise the simulated

data are polluted with random noise As proposed by Hjelm-stad and Shin [35] each set of perturbed data is created byadding uniformly distributed random noise to the numericaldata

120596119903119894119899

= 120596119903119894119899 (1 + 120585) (5)

where 120585 is a uniform random number with a specifiedamplitudeThe variance of the perturbing noise should be thesame as the variance of the measurement noise

22 Outputs The damage indices are represented by theelemental stiffness reduction factors 120573

119894defined as the ratios

between the stiffness reduction and the initial stiffness

119910119894= 120573119894 (6)

The stiffnessmatrix of the damaged structureK119889 is expressed

as a sum of element matrices multiplied by reduction factors

K119889

= sum

119894

(1 minus 120573119894)K119894 (7)

The value 120573119894= 0 indicates that the element is undamaged

whereas 0 lt 120573119894le 1 implies partial or complete damage

23 Training and Validation Patterns The distribution ofthe training patterns plays a crucial role in the success of aneural network The relationship between the anti-resonantfrequencies and the different damage levels is not linearand as a consequence the network might be unable tointerpolate Fang et al [15] recommend the use of trainingpatterns with evenly distributed damaged levels Combiningsimultaneous damages into the training increases the numberof training patterns and a large number of training patternscould overwhelm the training procedure In this study thetraining patterns were generated by considering up to twosimultaneous damage incidences with nine damage levelsevenly distributed between 0 and 80 Hence the totalnumber of training patterns depends on the number ofelements 119873 as

number of training patterns = 92

times 119862 (119873 2) (8)

where 119862(119873 2) is the number of combinations of 119873 elementstaken two at a time

The algorithm trains the network using the early stoppingtechnique [37] In this technique two sets of data are usedduring training a training set and a validation set Thetraining set is used for updating the network weights andbiases As defined in (2) the mean square error evaluated onthe validation set ismonitored during trainingWhen the val-idation error increases for six iterations the algorithm stopsthe training The weights of the network at the minimumvalidation error become the final weights

The validation set is a group of patterns that must bedifferent from the training patterns To ensure this conditionthe validation patterns were created with nine damage levelsevenly distributed between 5 and 85 and considered up totwo simultaneous damage incidences Note that the numberof training patterns is equal to the number of validationpatterns This set of validation patterns is also used to checkthe network performance after training

24 Measures of Network Performance The performance ofthe network is measured by three indicators the mean sizing

Shock and Vibration 7

error the damage missing error and the false alarm error asdefined by Yun et al [19] The mean sizing error (MSE) is theaverage quantification error

MSE =1

NOsum

119894

1003816100381610038161003816119910119894 minus 119900119894

1003816100381610038161003816 (9)

where119910119894and 119900119894are the estimated and desired outputs for node

119894 respectively and NO is the number of output nodesThe damage missing error (DME) is given by

DME =1

NTsum

119894

120576119868

119894 0 le DME le 1 (10)

where 120576119868

119894= 0 if the 119894th damaged element is correctly detected

and 120576119868

119894= 1 otherwise The NT corresponds to the number

of true damage locations If DME = 0 all damage locationsare correctly detected The value of 120576

119868

119894is calculated using the

following equation

120576119868

119894=

1 if 119900119894gt 0 119910

119894le 120572119888

0 otherwise(11)

It is assumed that an element is detected as damaged if theestimated damage 119910

119894is greater than a prescribed critical value

120572119888 The critical damage level 120572

119888is defined as being equal to

the average MSEThis value is the minimum damage that thenetwork can reasonably assess

The false alarm error (FAE) is defined as

FAE =1

NFsum

119894

120576119868119868

119894 0 le FAE le 1 (12)

where 120576119868119868

119894= 0 if the 119894th detected damage is truly damaged

and 120576119868119868

119894= 1 otherwise The NF is the number of predicted

damage locations If FAE = 0 all of the detected locations areactual damage locations The value of 120576

119868119868

119894is calculated using

the following equation

120576119868119868

119894=

1 if119910119894ge 120572119888 119900119894= 0

0 otherwise(13)

3 Identification and Matching ofAntiresonant Frequencies

31 Experimental Antiresonances The experimental antires-onances are identified from experimental point FRFs byldquodip-pickingrdquo [22] In this technique the antiresonances areselected by picking the dips from the magnitude plot ofa given FRF that have an associated change of +180∘ inthe phase plot This method is similar to the ldquopeak pick-ingrdquo technique for resonant frequencies Figure 3 shows anexample of the anti-resonant picking procedure When usingthis technique the errors in the estimated antiresonancesoriginate primarily from the frequency resolution of theFRFs If the dips in the FRFs are difficult to determine theantiresonances can be identified by a curve-fitting techniquebased on the rational fraction polynomial representation ofthe FRFs [38]

32 Numerical Antiresonances For a lightly damped struc-ture the anti-resonant frequencies are nearly unaffected bydamping and therefore they can be obtained from theundamped system using only the stiffness andmass matrices

By definition the FRFmatrix is the inverse of the dynamicstiffness matrix

H (120596) = (K minus 1205962M)minus1

=adj (K minus 120596

2M)

det (K minus 1205962M) (14)

The operators adj(sdot) and det(sdot) indicate the adjointand determinant respectively The anti-resonant frequenciescorrespond to the zeros of the FRFs The zeros of the 119894119896th FRF are the values of 120596 for which the numerator of119867119894119896

(120596)vanishes The numerator of 119867119894119896

(120596) is the 119894 119896th termof adj(Kminus 120596

2M) which is given by (minus1)119894+119896 det(K

119894119896minus 1205962M119894119896

)The subscripts 119894 and 119896 denote that the 119894th row and 119896th columnhave been deleted As a consequence the antiresonances ofthe 119894 119896th FRF are the frequency values that satisfy

det (K119894119896

minus 1205962M119894119896

) = 0 (15)

which is equivalent to solving the eigenvalue problem

(K119894119896

minus 1205962M119894119896

) u = 0 (16)

If 119894 = 119896 (16) represents a physical system obtained bygrounding the 119894th degree of freedom Therefore the anti-resonant frequencies obtained from point FRFs (119894 = 119896) areequivalent to the resonant frequencies of the structure withthe 119894th degree of freedom grounded

Because we are working with point FRFs matching theexperimental and numerical anti-resonant frequencies is aneasy task because each anti-resonant frequency lies betweentwo resonant frequencies

4 Eight-DOF Spring-Mass System

The structure shown in Figure 4 consists of an 8-DOF spring-mass system The Los Alamos National Laboratory (LANL)designed and constructed this system to study the effec-tiveness of various vibration-based damage identificationtechniques [39]

The system consists of eight translatingmasses connectedby springs and each mass consists of a disc of aluminumwith a diameter of 762mm and a thickness of 254mm Themasses slide on a highly polished steel rod and are fastenedtogether with coil springs The spring and mass locationsare designated sequentially with the first located the closestto the shaker attachment In the undamaged configurationall springs are identical and have linear stiffness coefficientsDamage is simulated by replacing the fifth spring with adifferent spring with a lower stiffness (55 stiffness reduc-tion)The acceleration is measured horizontally at eachmassgiving a total of eightmeasuredDOFsThe structure is excitedrandomly by an electrodynamic shaker The responses aremeasured in the eight DOFs in the undamaged and damagedcases

Seven anti-resonant frequencies are identified from theFRF corresponding to the first mass which corresponds to

8 Shock and Vibration

Mag

nitu

de (a

bs)

102

100

104

10minus2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

2855988

10261549

8125

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

Phas

e (de

g)

50

0

minus50

minus100

minus150

minus200

(b)

Figure 3 Identification of experimental anti-resonant frequencies

m8 m7 m6 m5 m4 m3 m2 m1 f(t)

k7 k6 k5 k4 k3 k2 k1x8 x7 x6 x5 x4 x3 x2 x1

Figure 4 Experimental system with eight degrees of freedom

a point FRF Given that the anti-resonant frequencies areindependent seven antiresonances should be sufficient toassess the stiffness of each of the seven springs

Table 1 lists the anti-resonant frequencies identified inthe undamaged and damaged cases The second and fourthantiresonances provide the larger variation due to the exper-imental damage

The numerical model is built in Matlab with springs andconcentrated masses and the initial parameters are set asfollows

(i) mass 1 5593 g (this mass is greater than the othersbecause of the hardware required to attach the shaker)

(ii) masses 2 to 8 4194 g(iii) spring constants 567 kNmminus1

Table 2 shows the numerical and experimental anti-resonant frequencies The maximum difference between theexperimental and numerical antiresonances is 07 As aconsequence the numerical model provides a faithful repre-sentation of the experimental structure and can be used totrain the neural network

41 Construction of the Neural Network The network con-tains seven inputs and seven outputsThe best parameters forthe neural network are defined after performing a sensitivity

Table 1 Experimental antiresonances

Antiresonant frequencies (Hz) Variation ()Undamaged Damaged1231 1166 minus5283694 3244 minus12185961 598 0327996 7174 minus10289612 9302 minus32310944 10686 minus23611572 11267 minus264

Table 2 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical123 123 043369 367 054596 592 070800 795 055961 956 0521094 1090 0401157 1155 019

analysis The network is trained under different configura-tions and the selected configuration is the one that providesthe lowest validation error

First the effects of five different transfer functions usedfor the output layer are studied and these transfer functionsare formulated as follows (a) hyperbolic tangent sigmoid(b) logarithm sigmoid (c) linear (d) saturating linear and(e) symmetric saturating linear Figure 2 illustrates thesefunctions During the experimental runs the remainingparameters of the network are characterized as follows thehidden transfer function is a logarithm sigmoid the numberof hidden neurons is 30 the training data are polluted with2 random noise and the Levenberg-Marquardt algorithmis used to train the network The network is trained 10times with each of the output transfer functions Table 3summarizes the average validation performance obtained

Shock and Vibration 9

Table 3 Network validation performance with different transfer functions

Output transfer function 119864119904

Hidden transfer function 119864119904

Hyperbolic tangent sigmoid 00067 Hyperbolic tangent sigmoid 00045Logarithm sigmoid 00134 Logarithm sigmoid 00044Linear 00060 Linear 00130Saturating linear 00127 Saturating linear 00050Symmetric saturating linear 00046 Symmetric saturating linear 00047

Table 4 Network validation performance with different trainingalgorithms

Training algorithms Time (s) 119864119904

Gradient descent 237 00467Gradient descent with adaptive learning 9 00289Scaled conjugate gradient 12 00158Levenberg-Marquardt 15 00046

for each function The symmetric saturating linear functionprovides the best performance

Second the performance of each of these five transferfunctions in the hidden layer is investigated The remainingnetwork parameters are the symmetric saturating linearoutput transfer function 30 hidden neurons 2 noise andthe Levenberg-Marquardt training algorithmThe network istrained 10 times with each of these output transfer functionsTable 3 summarizes the average validation performance Thelogarithm sigmoid function provides the best performance

Next the performance of four training algorithms isstudied Table 4 summarizes the average results for 10 runsThe gradient descent algorithm provides the worst perfor-mance in terms of training time and validation mean squareerror An adaptive learning rate substantially improves theresults the training time is 26 times lower and the meanerror is reduced by a factor of 16 The scale conjugate-gradient reaches a lower mean square error with a slightlyhigher training time Nevertheless the Levenberg-Marquardtalgorithm provides the lowest mean square validation errorby far and its training time is nearly the same as that with thescale conjugate gradient algorithm

Figure 5 shows the average validation error for differentnumbers of hidden nodes Seventy nodes in the hidden layergive the lower validation error 119864

119904 Less than 70 neurons are

not sufficient to fit the data whereas greater than 70 neuronsoverfit the neural network

The last parameter corresponds to the level of noise inthe training data The selection of this parameter is relatedto the level of experimental noise Hence this parameterwas defined by studying the performance of the trainednetwork with the experimental data and different levels ofperturbation noise Figure 6 shows the mean sizing error(MSE) obtained when the network is tested with experimen-tal dataThis case corresponds to a 55 stiffness reduction inelement 5 The MSE is the difference between the predicteddamage level on each element and the actual damage level asdefined in (9) A data perturbation noise of 15 provides theminimum error in the predicted damage

20 40 60 80 100 120 1403638

442444648

5

Number of hidden nodes

Es

times10minus3

Figure 5 Network validation performance as a function of thenumber of hidden nodes

0 05 1 15 2 25 3 35 40

00501

01502

02503

035

Noise level ()

MSE

Figure 6 Experimental mean sizing error for different levels ofinjected noise

42 Network Validation The selected network parametersare

(i) network three layers of multilayer perceptron(ii) number of input nodes 7(iii) number of hidden nodes 70(iv) number of output nodes 7(v) transfer function in the hidden layer Logarithm

sigmoid(vi) transfer function in the output layer Symmetric

saturating linear(vii) training method Levenberg-Marquardt

The validation patterns verify the performance of the net-work and these patterns are polluted with 15 noise Theresulting mean sizing error is MSE = 337 and thus thenetwork quantification accuracy is 97 Figure 7 shows thedamage missing error (DME) and false alarm error (FAE)divided by the damage levelsThe results of the DME indicate

10 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

Damage level ()

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(b)

Figure 7 Damage missing error (DME) and false alarm error (FAE) for different damage levels

1 2 3 4 5 6 70

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Figure 8 Assessment of experimental damage (55 stiffness reduc-tion in element 5)

that the detection of damage with low severities is poor Infact 80 of the damage with severities lower than 10 and40of the damagewith severities of 10ndash20 are not detectedThe algorithm can detect damage with severities larger than30with confidence it correctly detects 996 of the damagewith severities over 30The false alarm results indicate thatmost of the damage detected with levels lower than 10are false damage (near 75) The amount of false damagedetection is drastically reduced with an increment in the levelof damage In fact 9924 of the damage detected with levelsover 30 is true damage

Figure 8 shows the results of the damage detected usingthe experimental case The damaged element is correctlyassessed the location is correct and the quantification erroris 5

5 Experimental Beam

The structure consists of a steel beam with a rectangularcross-section The dimensions of the beam are length = 1mand section = 25 times 10mm2 As shown in Figure 9 soft springssuspend the structure to simulate a ldquofree-freerdquo boundarycondition

A hammer excites the beam at four points distributedalong the beam and an accelerometer measures the responseat each excitation location Both the excitation force and themeasured responses lie in the horizontal direction In thisdirection the anti-resonant frequencies are more sensitive to

Table 5 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical

FRF1

903 89 1102934 289 1356063 604 04010328 1033 00015528 1576 151

FRF2

1288 128 0692609 258 1005831 586 05611128 1127 12616219 1650 170

FRF3

1288 128 069260 258 0665844 586 03411138 1127 11716175 1650 198

FRF4

903 89 1102929 289 1356061 604 04010328 1033 00015534 1576 151

Figure 9 Experimental beam

the experimental damage Five anti-resonant frequencies areidentified at each of the four experimental FRFs

The numerical model is built in Matlab with 2D beamelements The model contains 20 beam elements and 40degrees of freedom as shown in Figure 10 The shadowedelements indicate possible locations of damage Table 5 showsthe experimental and numerical frequencies and their differ-ences and the maximum difference is 198

The structure is subjected to four different damagescenarios containing single and double cracks Cracks are

Shock and Vibration 11

Table 6 Damage cases introduced to the beam

Case Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

1 315 7 5 mdash mdash mdash2 635 13 10 mdash mdash mdash3 303 7 5 640 13 104 360 8 5 810 17 15

1 2 3 4 5 6 7 8 9 10 1211 13 14 15 16 17 18 19 20

Figure 10 Numerical model and element numbering

C

Figure 11 Saw cut introduced to the beam

introduced to the structure by saw cuts of length 119897119888 as

illustrated in Figure 11 Table 6 summarizes the differentdamage scenarios and indicates the distance from the left endto the cut the corresponding element in the numericalmodeland the cut length

51 Network Validation The network uses the same param-eters selected for the first structure except for the number ofinput hidden and output nodes The number of input nodescorresponds to the number of anti-resonant frequencieswhich is 20 Elements 2 to 19 were defined as possiblelocations of damage thus giving 18 output nodes (damageindices) The number of hidden nodes was defined afterthe sensitivity analysis shown in Figure 12 The minimumvalidation error is obtained with 80 hidden nodes

Hence the network parameters are

(i) network three layers of multilayer perceptron(ii) number of input nodes 20(iii) number of hidden nodes 80(iv) number of output nodes 18(v) transfer function in the hidden layer logarithm sig-

moid(vi) transfer function in the output layer symmetric

saturating linear(vii) training method Levenberg-Marquardt

The training and validation patterns were polluted with15 random noise The mean sizing error obtained for thevalidation patterns is MSE = 153 and thus the networkquantification accuracy is 98 Figure 13 shows the damagemissing error (DME) and false alarm error (FAE) separatedby damage levels The results of the DME indicate that thedetection of damage with low severities is poor In fact

20 40 60 80 100 120 140 160 180 200

28

3

32

34

36

Number of hidden nodes

times10minus3

Es

Figure 12 Validation performance as a function of the number ofhidden nodes

near 90 of the damage with severities lower than 10 and70 of the damage with severities between 10 and 20 isnot detected The algorithm is able to detect damage withseverities larger than 40with confidence it correctly detects995 of the damage with severities over 40The false alarmresults indicate that most of the damage detected with levelslower than 10 is false damage (near 80) The amount offalse damage detection decreases with an increment of thelevel of damage 99 of the damage detected with levels over40 is true damage

Note that these results are slightly worse than the resultsobtained from the 8-DOF system This result is explained bythe fact that the anti-resonant frequencies are less sensitiveto damage in the free beam than the 8-DOF structure thusmaking it more difficult to detect small damage

Figure 14 shows the results of the damage detected in thefour experimental scenarios An arrow indicates the actualdamage location In the first three cases the damage iscorrectly located although false damage appears next to theactual locations Nevertheless the assumption that a crackonly affects the stiffness of the corresponding element mightnot always be true For instance in the fourth case the effectof the larger cut is represented more accurately by a stiffnessreduction of two elements rather than one In the last case thenetwork does not detect the small crack in element 8 becausethe effect of the larger cut hides the effect of the smaller cut

6 Conclusions

This paper presents a new methodology designed to assessexperimental damage using neural networks and anti-resonant frequencies A multilayer perceptron network wastrained with data obtained from a numerical model and

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

to overcome this problem The first is to update the numer-ical model using experimental data from the undamagedstructure However even after updating differences will stillremain between the numerical and experimental modelsThe second alternative is to define an input parameter thatconsiders the initial errors in the numerical model thusavoiding the need for an accurate numerical modelThis goalis achieved using the changes in the data instead of theirabsolute values The main assumption is that any changein the structural properties is caused by damage Thus anyerror in the undamaged model of the structure that is alsopresent in the damaged model will be removed [33] Leeet al [34] showed that natural frequency changes becausestructural damage in a system without modeling errors isapproximated the same as those in a system with modelingerror Hence changes in the natural frequencies are lesssensitive to modeling errors than the natural frequenciesthemselves This group demonstrated the applicability ofa neural network trained with mode shape changes Thedamage locations were estimated with reasonable accuracyalthough false alarms were detected at several locations

Simulated data derived from a numerical model arenoise-free whereas actual measurements are never free fromexperimental noise Noise in themeasurements will cause thenetwork to estimate parameters that are different from theactual properties of the structure A solution is to introduceartificial noise into the numerical data used to train thenetwork This process is known as data perturbation scheme[35] Yun et al [19] used a noise-injection learning algorithmand a data perturbation schemeThey implemented a neural-network-based damage assessment algorithm to detect thedamage in structural joints The joints are modeled assemirigid connections using rotational springs and accurateresults are obtained in the cases of moderated noise Sahinand Shenoi [21] trained a neural network with artificiallyadded noise to detect single damage in beam-like compositelaminates The network contains two outputs the locationand the amount of damage They studied different combina-tions of features extracted from the resonant frequencies andmode shape curvatures as inputs to the neural network andtheir results show that feature selection plays a crucial rolein the predictions accuracy Zang and Imregun [14] proposeda different approach to address experimental noise and usedprincipal component projection of the FRF data as an inputto a neural network with two outputs of healthy or damagedThe authors stated that reconstructing the response using thehighest principal components should not only achieve datacompression but also remove a proportion of the noise ThePCA compression acts as a noise filter which can be useful inmodal analysis as well

The successful application of a neural network dependson the representation and the learning algorithms Never-theless their selection is problem-dependent and is usuallydetermined by trial and error [36] Sahoo and Maity [7]used a Genetic Algorithm (GA) to automate the trial-and-error process The network parameters (number of neuronslearning rate etc) are set as variables in an optimizationproblem handled by a GA They used an MLP network withtwo hidden layers trained by a backpropagation algorithm

Structure

Vibration characteristics

Artificial neural networks

Input layer

Hidden layers

Output layer

Stiffness reduction factors

Figure 2 Damage assessment with artificial neural networks

The inputs are the natural frequencies and strains in certainlocations of the structure Fang et al [15] explored the use ofa tunable steepest descent (TSD) algorithm that dynamicallyadjusts the learning speed during the training process Theyshowed that this methodology significantly increases thetraining speed while maintaining the learning stabilityTheseauthors used FRF data measured at key spectral pointsaround the resonant frequencies

The primary contribution of this research is the devel-opment of a real time damage assessment algorithm usingANN and anti-resonant frequencies In literature similarlyto mode shapes anti-resonant frequencies are used as com-plementary information to natural frequencies NeverthelessMotthershead [37] demonstrated that the sensitivity of anti-resonant frequencies can be expressed as a combination of thesensitivities of mode shapes and natural frequencies Henceanti-resonant frequencies contain the same information asnatural frequencies and mode shapes together which is thehypothesis of the present studyThis study is restricted to anti-resonant frequencies obtained from point FRFs Althoughmeasurement of the point FRFs is time-consuming becausethe excitation point is moved together with the response

6 Shock and Vibration

location this method offers several advantages for damageassessment as follows

(i) All antiresonances contain independent informationbecause they correspond to the resonant frequenciesof the system grounded at different degrees of free-dom

(ii) For a given FRF the number of anti-resonant frequen-cies does not change from one damage scenario tothe next this is a valuable property if we require thesefrequencies to act as inputs to a neural network

(iii) An anti-resonant frequency always lies betweentwo resonant frequencies Hence there is no doubtwhether a dip in an FRF is an anti-resonant frequencyor a minimum

This study addresses the setup of the neural networkparameters and provides guidelines for their selection insimilar damage detection problems The proposed method-ology is evaluated with two experimental structures an 8-DOF mass-spring system and a beam with multiple damagescenarios

2 Neural Network for Damage Assessment

This study intends to train a neural network using anti-resonant frequencies and to determine its feasibility in assess-ing experimental damage Hence this study works with thesimplest neural network that has been able to detect locateand quantify structural damage the multilayer perceptron(MLP) with three layers (input hidden and output) aspresented by Fang et al [15] The outputs of the networkcorrespond to the damage indices of each element whereasthe inputs are the changes in anti-resonant frequencies Thenumber of hidden nodes is defined after a sensitivity analysisfor each application case

21 Inputs As proposed by Lee et al [34] the inputs tothe neural network are defined as the changes in the modalparameters rather than their absolute values With thisapproach the network is less sensitive to errors in the baselineFE model

Therefore the inputs correspond to the experimentalchange in the anti-resonant frequencies with respect to theintact case

119909119894times119899

=120596119863

119903119894119899minus 120596119880

119903119894119899

120596119880

119903119894119899

(4)

The superscripts 119863 and 119880 denote undamaged and damagedrespectively and120596

119903119894119899is the 119894th anti-resonant frequency of the

119899th FRFTo reduce the effects of experimental noise the simulated

data are polluted with random noise As proposed by Hjelm-stad and Shin [35] each set of perturbed data is created byadding uniformly distributed random noise to the numericaldata

120596119903119894119899

= 120596119903119894119899 (1 + 120585) (5)

where 120585 is a uniform random number with a specifiedamplitudeThe variance of the perturbing noise should be thesame as the variance of the measurement noise

22 Outputs The damage indices are represented by theelemental stiffness reduction factors 120573

119894defined as the ratios

between the stiffness reduction and the initial stiffness

119910119894= 120573119894 (6)

The stiffnessmatrix of the damaged structureK119889 is expressed

as a sum of element matrices multiplied by reduction factors

K119889

= sum

119894

(1 minus 120573119894)K119894 (7)

The value 120573119894= 0 indicates that the element is undamaged

whereas 0 lt 120573119894le 1 implies partial or complete damage

23 Training and Validation Patterns The distribution ofthe training patterns plays a crucial role in the success of aneural network The relationship between the anti-resonantfrequencies and the different damage levels is not linearand as a consequence the network might be unable tointerpolate Fang et al [15] recommend the use of trainingpatterns with evenly distributed damaged levels Combiningsimultaneous damages into the training increases the numberof training patterns and a large number of training patternscould overwhelm the training procedure In this study thetraining patterns were generated by considering up to twosimultaneous damage incidences with nine damage levelsevenly distributed between 0 and 80 Hence the totalnumber of training patterns depends on the number ofelements 119873 as

number of training patterns = 92

times 119862 (119873 2) (8)

where 119862(119873 2) is the number of combinations of 119873 elementstaken two at a time

The algorithm trains the network using the early stoppingtechnique [37] In this technique two sets of data are usedduring training a training set and a validation set Thetraining set is used for updating the network weights andbiases As defined in (2) the mean square error evaluated onthe validation set ismonitored during trainingWhen the val-idation error increases for six iterations the algorithm stopsthe training The weights of the network at the minimumvalidation error become the final weights

The validation set is a group of patterns that must bedifferent from the training patterns To ensure this conditionthe validation patterns were created with nine damage levelsevenly distributed between 5 and 85 and considered up totwo simultaneous damage incidences Note that the numberof training patterns is equal to the number of validationpatterns This set of validation patterns is also used to checkthe network performance after training

24 Measures of Network Performance The performance ofthe network is measured by three indicators the mean sizing

Shock and Vibration 7

error the damage missing error and the false alarm error asdefined by Yun et al [19] The mean sizing error (MSE) is theaverage quantification error

MSE =1

NOsum

119894

1003816100381610038161003816119910119894 minus 119900119894

1003816100381610038161003816 (9)

where119910119894and 119900119894are the estimated and desired outputs for node

119894 respectively and NO is the number of output nodesThe damage missing error (DME) is given by

DME =1

NTsum

119894

120576119868

119894 0 le DME le 1 (10)

where 120576119868

119894= 0 if the 119894th damaged element is correctly detected

and 120576119868

119894= 1 otherwise The NT corresponds to the number

of true damage locations If DME = 0 all damage locationsare correctly detected The value of 120576

119868

119894is calculated using the

following equation

120576119868

119894=

1 if 119900119894gt 0 119910

119894le 120572119888

0 otherwise(11)

It is assumed that an element is detected as damaged if theestimated damage 119910

119894is greater than a prescribed critical value

120572119888 The critical damage level 120572

119888is defined as being equal to

the average MSEThis value is the minimum damage that thenetwork can reasonably assess

The false alarm error (FAE) is defined as

FAE =1

NFsum

119894

120576119868119868

119894 0 le FAE le 1 (12)

where 120576119868119868

119894= 0 if the 119894th detected damage is truly damaged

and 120576119868119868

119894= 1 otherwise The NF is the number of predicted

damage locations If FAE = 0 all of the detected locations areactual damage locations The value of 120576

119868119868

119894is calculated using

the following equation

120576119868119868

119894=

1 if119910119894ge 120572119888 119900119894= 0

0 otherwise(13)

3 Identification and Matching ofAntiresonant Frequencies

31 Experimental Antiresonances The experimental antires-onances are identified from experimental point FRFs byldquodip-pickingrdquo [22] In this technique the antiresonances areselected by picking the dips from the magnitude plot ofa given FRF that have an associated change of +180∘ inthe phase plot This method is similar to the ldquopeak pick-ingrdquo technique for resonant frequencies Figure 3 shows anexample of the anti-resonant picking procedure When usingthis technique the errors in the estimated antiresonancesoriginate primarily from the frequency resolution of theFRFs If the dips in the FRFs are difficult to determine theantiresonances can be identified by a curve-fitting techniquebased on the rational fraction polynomial representation ofthe FRFs [38]

32 Numerical Antiresonances For a lightly damped struc-ture the anti-resonant frequencies are nearly unaffected bydamping and therefore they can be obtained from theundamped system using only the stiffness andmass matrices

By definition the FRFmatrix is the inverse of the dynamicstiffness matrix

H (120596) = (K minus 1205962M)minus1

=adj (K minus 120596

2M)

det (K minus 1205962M) (14)

The operators adj(sdot) and det(sdot) indicate the adjointand determinant respectively The anti-resonant frequenciescorrespond to the zeros of the FRFs The zeros of the 119894119896th FRF are the values of 120596 for which the numerator of119867119894119896

(120596)vanishes The numerator of 119867119894119896

(120596) is the 119894 119896th termof adj(Kminus 120596

2M) which is given by (minus1)119894+119896 det(K

119894119896minus 1205962M119894119896

)The subscripts 119894 and 119896 denote that the 119894th row and 119896th columnhave been deleted As a consequence the antiresonances ofthe 119894 119896th FRF are the frequency values that satisfy

det (K119894119896

minus 1205962M119894119896

) = 0 (15)

which is equivalent to solving the eigenvalue problem

(K119894119896

minus 1205962M119894119896

) u = 0 (16)

If 119894 = 119896 (16) represents a physical system obtained bygrounding the 119894th degree of freedom Therefore the anti-resonant frequencies obtained from point FRFs (119894 = 119896) areequivalent to the resonant frequencies of the structure withthe 119894th degree of freedom grounded

Because we are working with point FRFs matching theexperimental and numerical anti-resonant frequencies is aneasy task because each anti-resonant frequency lies betweentwo resonant frequencies

4 Eight-DOF Spring-Mass System

The structure shown in Figure 4 consists of an 8-DOF spring-mass system The Los Alamos National Laboratory (LANL)designed and constructed this system to study the effec-tiveness of various vibration-based damage identificationtechniques [39]

The system consists of eight translatingmasses connectedby springs and each mass consists of a disc of aluminumwith a diameter of 762mm and a thickness of 254mm Themasses slide on a highly polished steel rod and are fastenedtogether with coil springs The spring and mass locationsare designated sequentially with the first located the closestto the shaker attachment In the undamaged configurationall springs are identical and have linear stiffness coefficientsDamage is simulated by replacing the fifth spring with adifferent spring with a lower stiffness (55 stiffness reduc-tion)The acceleration is measured horizontally at eachmassgiving a total of eightmeasuredDOFsThe structure is excitedrandomly by an electrodynamic shaker The responses aremeasured in the eight DOFs in the undamaged and damagedcases

Seven anti-resonant frequencies are identified from theFRF corresponding to the first mass which corresponds to

8 Shock and Vibration

Mag

nitu

de (a

bs)

102

100

104

10minus2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

2855988

10261549

8125

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

Phas

e (de

g)

50

0

minus50

minus100

minus150

minus200

(b)

Figure 3 Identification of experimental anti-resonant frequencies

m8 m7 m6 m5 m4 m3 m2 m1 f(t)

k7 k6 k5 k4 k3 k2 k1x8 x7 x6 x5 x4 x3 x2 x1

Figure 4 Experimental system with eight degrees of freedom

a point FRF Given that the anti-resonant frequencies areindependent seven antiresonances should be sufficient toassess the stiffness of each of the seven springs

Table 1 lists the anti-resonant frequencies identified inthe undamaged and damaged cases The second and fourthantiresonances provide the larger variation due to the exper-imental damage

The numerical model is built in Matlab with springs andconcentrated masses and the initial parameters are set asfollows

(i) mass 1 5593 g (this mass is greater than the othersbecause of the hardware required to attach the shaker)

(ii) masses 2 to 8 4194 g(iii) spring constants 567 kNmminus1

Table 2 shows the numerical and experimental anti-resonant frequencies The maximum difference between theexperimental and numerical antiresonances is 07 As aconsequence the numerical model provides a faithful repre-sentation of the experimental structure and can be used totrain the neural network

41 Construction of the Neural Network The network con-tains seven inputs and seven outputsThe best parameters forthe neural network are defined after performing a sensitivity

Table 1 Experimental antiresonances

Antiresonant frequencies (Hz) Variation ()Undamaged Damaged1231 1166 minus5283694 3244 minus12185961 598 0327996 7174 minus10289612 9302 minus32310944 10686 minus23611572 11267 minus264

Table 2 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical123 123 043369 367 054596 592 070800 795 055961 956 0521094 1090 0401157 1155 019

analysis The network is trained under different configura-tions and the selected configuration is the one that providesthe lowest validation error

First the effects of five different transfer functions usedfor the output layer are studied and these transfer functionsare formulated as follows (a) hyperbolic tangent sigmoid(b) logarithm sigmoid (c) linear (d) saturating linear and(e) symmetric saturating linear Figure 2 illustrates thesefunctions During the experimental runs the remainingparameters of the network are characterized as follows thehidden transfer function is a logarithm sigmoid the numberof hidden neurons is 30 the training data are polluted with2 random noise and the Levenberg-Marquardt algorithmis used to train the network The network is trained 10times with each of the output transfer functions Table 3summarizes the average validation performance obtained

Shock and Vibration 9

Table 3 Network validation performance with different transfer functions

Output transfer function 119864119904

Hidden transfer function 119864119904

Hyperbolic tangent sigmoid 00067 Hyperbolic tangent sigmoid 00045Logarithm sigmoid 00134 Logarithm sigmoid 00044Linear 00060 Linear 00130Saturating linear 00127 Saturating linear 00050Symmetric saturating linear 00046 Symmetric saturating linear 00047

Table 4 Network validation performance with different trainingalgorithms

Training algorithms Time (s) 119864119904

Gradient descent 237 00467Gradient descent with adaptive learning 9 00289Scaled conjugate gradient 12 00158Levenberg-Marquardt 15 00046

for each function The symmetric saturating linear functionprovides the best performance

Second the performance of each of these five transferfunctions in the hidden layer is investigated The remainingnetwork parameters are the symmetric saturating linearoutput transfer function 30 hidden neurons 2 noise andthe Levenberg-Marquardt training algorithmThe network istrained 10 times with each of these output transfer functionsTable 3 summarizes the average validation performance Thelogarithm sigmoid function provides the best performance

Next the performance of four training algorithms isstudied Table 4 summarizes the average results for 10 runsThe gradient descent algorithm provides the worst perfor-mance in terms of training time and validation mean squareerror An adaptive learning rate substantially improves theresults the training time is 26 times lower and the meanerror is reduced by a factor of 16 The scale conjugate-gradient reaches a lower mean square error with a slightlyhigher training time Nevertheless the Levenberg-Marquardtalgorithm provides the lowest mean square validation errorby far and its training time is nearly the same as that with thescale conjugate gradient algorithm

Figure 5 shows the average validation error for differentnumbers of hidden nodes Seventy nodes in the hidden layergive the lower validation error 119864

119904 Less than 70 neurons are

not sufficient to fit the data whereas greater than 70 neuronsoverfit the neural network

The last parameter corresponds to the level of noise inthe training data The selection of this parameter is relatedto the level of experimental noise Hence this parameterwas defined by studying the performance of the trainednetwork with the experimental data and different levels ofperturbation noise Figure 6 shows the mean sizing error(MSE) obtained when the network is tested with experimen-tal dataThis case corresponds to a 55 stiffness reduction inelement 5 The MSE is the difference between the predicteddamage level on each element and the actual damage level asdefined in (9) A data perturbation noise of 15 provides theminimum error in the predicted damage

20 40 60 80 100 120 1403638

442444648

5

Number of hidden nodes

Es

times10minus3

Figure 5 Network validation performance as a function of thenumber of hidden nodes

0 05 1 15 2 25 3 35 40

00501

01502

02503

035

Noise level ()

MSE

Figure 6 Experimental mean sizing error for different levels ofinjected noise

42 Network Validation The selected network parametersare

(i) network three layers of multilayer perceptron(ii) number of input nodes 7(iii) number of hidden nodes 70(iv) number of output nodes 7(v) transfer function in the hidden layer Logarithm

sigmoid(vi) transfer function in the output layer Symmetric

saturating linear(vii) training method Levenberg-Marquardt

The validation patterns verify the performance of the net-work and these patterns are polluted with 15 noise Theresulting mean sizing error is MSE = 337 and thus thenetwork quantification accuracy is 97 Figure 7 shows thedamage missing error (DME) and false alarm error (FAE)divided by the damage levelsThe results of the DME indicate

10 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

Damage level ()

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(b)

Figure 7 Damage missing error (DME) and false alarm error (FAE) for different damage levels

1 2 3 4 5 6 70

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Figure 8 Assessment of experimental damage (55 stiffness reduc-tion in element 5)

that the detection of damage with low severities is poor Infact 80 of the damage with severities lower than 10 and40of the damagewith severities of 10ndash20 are not detectedThe algorithm can detect damage with severities larger than30with confidence it correctly detects 996 of the damagewith severities over 30The false alarm results indicate thatmost of the damage detected with levels lower than 10are false damage (near 75) The amount of false damagedetection is drastically reduced with an increment in the levelof damage In fact 9924 of the damage detected with levelsover 30 is true damage

Figure 8 shows the results of the damage detected usingthe experimental case The damaged element is correctlyassessed the location is correct and the quantification erroris 5

5 Experimental Beam

The structure consists of a steel beam with a rectangularcross-section The dimensions of the beam are length = 1mand section = 25 times 10mm2 As shown in Figure 9 soft springssuspend the structure to simulate a ldquofree-freerdquo boundarycondition

A hammer excites the beam at four points distributedalong the beam and an accelerometer measures the responseat each excitation location Both the excitation force and themeasured responses lie in the horizontal direction In thisdirection the anti-resonant frequencies are more sensitive to

Table 5 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical

FRF1

903 89 1102934 289 1356063 604 04010328 1033 00015528 1576 151

FRF2

1288 128 0692609 258 1005831 586 05611128 1127 12616219 1650 170

FRF3

1288 128 069260 258 0665844 586 03411138 1127 11716175 1650 198

FRF4

903 89 1102929 289 1356061 604 04010328 1033 00015534 1576 151

Figure 9 Experimental beam

the experimental damage Five anti-resonant frequencies areidentified at each of the four experimental FRFs

The numerical model is built in Matlab with 2D beamelements The model contains 20 beam elements and 40degrees of freedom as shown in Figure 10 The shadowedelements indicate possible locations of damage Table 5 showsthe experimental and numerical frequencies and their differ-ences and the maximum difference is 198

The structure is subjected to four different damagescenarios containing single and double cracks Cracks are

Shock and Vibration 11

Table 6 Damage cases introduced to the beam

Case Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

1 315 7 5 mdash mdash mdash2 635 13 10 mdash mdash mdash3 303 7 5 640 13 104 360 8 5 810 17 15

1 2 3 4 5 6 7 8 9 10 1211 13 14 15 16 17 18 19 20

Figure 10 Numerical model and element numbering

C

Figure 11 Saw cut introduced to the beam

introduced to the structure by saw cuts of length 119897119888 as

illustrated in Figure 11 Table 6 summarizes the differentdamage scenarios and indicates the distance from the left endto the cut the corresponding element in the numericalmodeland the cut length

51 Network Validation The network uses the same param-eters selected for the first structure except for the number ofinput hidden and output nodes The number of input nodescorresponds to the number of anti-resonant frequencieswhich is 20 Elements 2 to 19 were defined as possiblelocations of damage thus giving 18 output nodes (damageindices) The number of hidden nodes was defined afterthe sensitivity analysis shown in Figure 12 The minimumvalidation error is obtained with 80 hidden nodes

Hence the network parameters are

(i) network three layers of multilayer perceptron(ii) number of input nodes 20(iii) number of hidden nodes 80(iv) number of output nodes 18(v) transfer function in the hidden layer logarithm sig-

moid(vi) transfer function in the output layer symmetric

saturating linear(vii) training method Levenberg-Marquardt

The training and validation patterns were polluted with15 random noise The mean sizing error obtained for thevalidation patterns is MSE = 153 and thus the networkquantification accuracy is 98 Figure 13 shows the damagemissing error (DME) and false alarm error (FAE) separatedby damage levels The results of the DME indicate that thedetection of damage with low severities is poor In fact

20 40 60 80 100 120 140 160 180 200

28

3

32

34

36

Number of hidden nodes

times10minus3

Es

Figure 12 Validation performance as a function of the number ofhidden nodes

near 90 of the damage with severities lower than 10 and70 of the damage with severities between 10 and 20 isnot detected The algorithm is able to detect damage withseverities larger than 40with confidence it correctly detects995 of the damage with severities over 40The false alarmresults indicate that most of the damage detected with levelslower than 10 is false damage (near 80) The amount offalse damage detection decreases with an increment of thelevel of damage 99 of the damage detected with levels over40 is true damage

Note that these results are slightly worse than the resultsobtained from the 8-DOF system This result is explained bythe fact that the anti-resonant frequencies are less sensitiveto damage in the free beam than the 8-DOF structure thusmaking it more difficult to detect small damage

Figure 14 shows the results of the damage detected in thefour experimental scenarios An arrow indicates the actualdamage location In the first three cases the damage iscorrectly located although false damage appears next to theactual locations Nevertheless the assumption that a crackonly affects the stiffness of the corresponding element mightnot always be true For instance in the fourth case the effectof the larger cut is represented more accurately by a stiffnessreduction of two elements rather than one In the last case thenetwork does not detect the small crack in element 8 becausethe effect of the larger cut hides the effect of the smaller cut

6 Conclusions

This paper presents a new methodology designed to assessexperimental damage using neural networks and anti-resonant frequencies A multilayer perceptron network wastrained with data obtained from a numerical model and

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

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DistributedSensor Networks

International Journal of

6 Shock and Vibration

location this method offers several advantages for damageassessment as follows

(i) All antiresonances contain independent informationbecause they correspond to the resonant frequenciesof the system grounded at different degrees of free-dom

(ii) For a given FRF the number of anti-resonant frequen-cies does not change from one damage scenario tothe next this is a valuable property if we require thesefrequencies to act as inputs to a neural network

(iii) An anti-resonant frequency always lies betweentwo resonant frequencies Hence there is no doubtwhether a dip in an FRF is an anti-resonant frequencyor a minimum

This study addresses the setup of the neural networkparameters and provides guidelines for their selection insimilar damage detection problems The proposed method-ology is evaluated with two experimental structures an 8-DOF mass-spring system and a beam with multiple damagescenarios

2 Neural Network for Damage Assessment

This study intends to train a neural network using anti-resonant frequencies and to determine its feasibility in assess-ing experimental damage Hence this study works with thesimplest neural network that has been able to detect locateand quantify structural damage the multilayer perceptron(MLP) with three layers (input hidden and output) aspresented by Fang et al [15] The outputs of the networkcorrespond to the damage indices of each element whereasthe inputs are the changes in anti-resonant frequencies Thenumber of hidden nodes is defined after a sensitivity analysisfor each application case

21 Inputs As proposed by Lee et al [34] the inputs tothe neural network are defined as the changes in the modalparameters rather than their absolute values With thisapproach the network is less sensitive to errors in the baselineFE model

Therefore the inputs correspond to the experimentalchange in the anti-resonant frequencies with respect to theintact case

119909119894times119899

=120596119863

119903119894119899minus 120596119880

119903119894119899

120596119880

119903119894119899

(4)

The superscripts 119863 and 119880 denote undamaged and damagedrespectively and120596

119903119894119899is the 119894th anti-resonant frequency of the

119899th FRFTo reduce the effects of experimental noise the simulated

data are polluted with random noise As proposed by Hjelm-stad and Shin [35] each set of perturbed data is created byadding uniformly distributed random noise to the numericaldata

120596119903119894119899

= 120596119903119894119899 (1 + 120585) (5)

where 120585 is a uniform random number with a specifiedamplitudeThe variance of the perturbing noise should be thesame as the variance of the measurement noise

22 Outputs The damage indices are represented by theelemental stiffness reduction factors 120573

119894defined as the ratios

between the stiffness reduction and the initial stiffness

119910119894= 120573119894 (6)

The stiffnessmatrix of the damaged structureK119889 is expressed

as a sum of element matrices multiplied by reduction factors

K119889

= sum

119894

(1 minus 120573119894)K119894 (7)

The value 120573119894= 0 indicates that the element is undamaged

whereas 0 lt 120573119894le 1 implies partial or complete damage

23 Training and Validation Patterns The distribution ofthe training patterns plays a crucial role in the success of aneural network The relationship between the anti-resonantfrequencies and the different damage levels is not linearand as a consequence the network might be unable tointerpolate Fang et al [15] recommend the use of trainingpatterns with evenly distributed damaged levels Combiningsimultaneous damages into the training increases the numberof training patterns and a large number of training patternscould overwhelm the training procedure In this study thetraining patterns were generated by considering up to twosimultaneous damage incidences with nine damage levelsevenly distributed between 0 and 80 Hence the totalnumber of training patterns depends on the number ofelements 119873 as

number of training patterns = 92

times 119862 (119873 2) (8)

where 119862(119873 2) is the number of combinations of 119873 elementstaken two at a time

The algorithm trains the network using the early stoppingtechnique [37] In this technique two sets of data are usedduring training a training set and a validation set Thetraining set is used for updating the network weights andbiases As defined in (2) the mean square error evaluated onthe validation set ismonitored during trainingWhen the val-idation error increases for six iterations the algorithm stopsthe training The weights of the network at the minimumvalidation error become the final weights

The validation set is a group of patterns that must bedifferent from the training patterns To ensure this conditionthe validation patterns were created with nine damage levelsevenly distributed between 5 and 85 and considered up totwo simultaneous damage incidences Note that the numberof training patterns is equal to the number of validationpatterns This set of validation patterns is also used to checkthe network performance after training

24 Measures of Network Performance The performance ofthe network is measured by three indicators the mean sizing

Shock and Vibration 7

error the damage missing error and the false alarm error asdefined by Yun et al [19] The mean sizing error (MSE) is theaverage quantification error

MSE =1

NOsum

119894

1003816100381610038161003816119910119894 minus 119900119894

1003816100381610038161003816 (9)

where119910119894and 119900119894are the estimated and desired outputs for node

119894 respectively and NO is the number of output nodesThe damage missing error (DME) is given by

DME =1

NTsum

119894

120576119868

119894 0 le DME le 1 (10)

where 120576119868

119894= 0 if the 119894th damaged element is correctly detected

and 120576119868

119894= 1 otherwise The NT corresponds to the number

of true damage locations If DME = 0 all damage locationsare correctly detected The value of 120576

119868

119894is calculated using the

following equation

120576119868

119894=

1 if 119900119894gt 0 119910

119894le 120572119888

0 otherwise(11)

It is assumed that an element is detected as damaged if theestimated damage 119910

119894is greater than a prescribed critical value

120572119888 The critical damage level 120572

119888is defined as being equal to

the average MSEThis value is the minimum damage that thenetwork can reasonably assess

The false alarm error (FAE) is defined as

FAE =1

NFsum

119894

120576119868119868

119894 0 le FAE le 1 (12)

where 120576119868119868

119894= 0 if the 119894th detected damage is truly damaged

and 120576119868119868

119894= 1 otherwise The NF is the number of predicted

damage locations If FAE = 0 all of the detected locations areactual damage locations The value of 120576

119868119868

119894is calculated using

the following equation

120576119868119868

119894=

1 if119910119894ge 120572119888 119900119894= 0

0 otherwise(13)

3 Identification and Matching ofAntiresonant Frequencies

31 Experimental Antiresonances The experimental antires-onances are identified from experimental point FRFs byldquodip-pickingrdquo [22] In this technique the antiresonances areselected by picking the dips from the magnitude plot ofa given FRF that have an associated change of +180∘ inthe phase plot This method is similar to the ldquopeak pick-ingrdquo technique for resonant frequencies Figure 3 shows anexample of the anti-resonant picking procedure When usingthis technique the errors in the estimated antiresonancesoriginate primarily from the frequency resolution of theFRFs If the dips in the FRFs are difficult to determine theantiresonances can be identified by a curve-fitting techniquebased on the rational fraction polynomial representation ofthe FRFs [38]

32 Numerical Antiresonances For a lightly damped struc-ture the anti-resonant frequencies are nearly unaffected bydamping and therefore they can be obtained from theundamped system using only the stiffness andmass matrices

By definition the FRFmatrix is the inverse of the dynamicstiffness matrix

H (120596) = (K minus 1205962M)minus1

=adj (K minus 120596

2M)

det (K minus 1205962M) (14)

The operators adj(sdot) and det(sdot) indicate the adjointand determinant respectively The anti-resonant frequenciescorrespond to the zeros of the FRFs The zeros of the 119894119896th FRF are the values of 120596 for which the numerator of119867119894119896

(120596)vanishes The numerator of 119867119894119896

(120596) is the 119894 119896th termof adj(Kminus 120596

2M) which is given by (minus1)119894+119896 det(K

119894119896minus 1205962M119894119896

)The subscripts 119894 and 119896 denote that the 119894th row and 119896th columnhave been deleted As a consequence the antiresonances ofthe 119894 119896th FRF are the frequency values that satisfy

det (K119894119896

minus 1205962M119894119896

) = 0 (15)

which is equivalent to solving the eigenvalue problem

(K119894119896

minus 1205962M119894119896

) u = 0 (16)

If 119894 = 119896 (16) represents a physical system obtained bygrounding the 119894th degree of freedom Therefore the anti-resonant frequencies obtained from point FRFs (119894 = 119896) areequivalent to the resonant frequencies of the structure withthe 119894th degree of freedom grounded

Because we are working with point FRFs matching theexperimental and numerical anti-resonant frequencies is aneasy task because each anti-resonant frequency lies betweentwo resonant frequencies

4 Eight-DOF Spring-Mass System

The structure shown in Figure 4 consists of an 8-DOF spring-mass system The Los Alamos National Laboratory (LANL)designed and constructed this system to study the effec-tiveness of various vibration-based damage identificationtechniques [39]

The system consists of eight translatingmasses connectedby springs and each mass consists of a disc of aluminumwith a diameter of 762mm and a thickness of 254mm Themasses slide on a highly polished steel rod and are fastenedtogether with coil springs The spring and mass locationsare designated sequentially with the first located the closestto the shaker attachment In the undamaged configurationall springs are identical and have linear stiffness coefficientsDamage is simulated by replacing the fifth spring with adifferent spring with a lower stiffness (55 stiffness reduc-tion)The acceleration is measured horizontally at eachmassgiving a total of eightmeasuredDOFsThe structure is excitedrandomly by an electrodynamic shaker The responses aremeasured in the eight DOFs in the undamaged and damagedcases

Seven anti-resonant frequencies are identified from theFRF corresponding to the first mass which corresponds to

8 Shock and Vibration

Mag

nitu

de (a

bs)

102

100

104

10minus2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

2855988

10261549

8125

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

Phas

e (de

g)

50

0

minus50

minus100

minus150

minus200

(b)

Figure 3 Identification of experimental anti-resonant frequencies

m8 m7 m6 m5 m4 m3 m2 m1 f(t)

k7 k6 k5 k4 k3 k2 k1x8 x7 x6 x5 x4 x3 x2 x1

Figure 4 Experimental system with eight degrees of freedom

a point FRF Given that the anti-resonant frequencies areindependent seven antiresonances should be sufficient toassess the stiffness of each of the seven springs

Table 1 lists the anti-resonant frequencies identified inthe undamaged and damaged cases The second and fourthantiresonances provide the larger variation due to the exper-imental damage

The numerical model is built in Matlab with springs andconcentrated masses and the initial parameters are set asfollows

(i) mass 1 5593 g (this mass is greater than the othersbecause of the hardware required to attach the shaker)

(ii) masses 2 to 8 4194 g(iii) spring constants 567 kNmminus1

Table 2 shows the numerical and experimental anti-resonant frequencies The maximum difference between theexperimental and numerical antiresonances is 07 As aconsequence the numerical model provides a faithful repre-sentation of the experimental structure and can be used totrain the neural network

41 Construction of the Neural Network The network con-tains seven inputs and seven outputsThe best parameters forthe neural network are defined after performing a sensitivity

Table 1 Experimental antiresonances

Antiresonant frequencies (Hz) Variation ()Undamaged Damaged1231 1166 minus5283694 3244 minus12185961 598 0327996 7174 minus10289612 9302 minus32310944 10686 minus23611572 11267 minus264

Table 2 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical123 123 043369 367 054596 592 070800 795 055961 956 0521094 1090 0401157 1155 019

analysis The network is trained under different configura-tions and the selected configuration is the one that providesthe lowest validation error

First the effects of five different transfer functions usedfor the output layer are studied and these transfer functionsare formulated as follows (a) hyperbolic tangent sigmoid(b) logarithm sigmoid (c) linear (d) saturating linear and(e) symmetric saturating linear Figure 2 illustrates thesefunctions During the experimental runs the remainingparameters of the network are characterized as follows thehidden transfer function is a logarithm sigmoid the numberof hidden neurons is 30 the training data are polluted with2 random noise and the Levenberg-Marquardt algorithmis used to train the network The network is trained 10times with each of the output transfer functions Table 3summarizes the average validation performance obtained

Shock and Vibration 9

Table 3 Network validation performance with different transfer functions

Output transfer function 119864119904

Hidden transfer function 119864119904

Hyperbolic tangent sigmoid 00067 Hyperbolic tangent sigmoid 00045Logarithm sigmoid 00134 Logarithm sigmoid 00044Linear 00060 Linear 00130Saturating linear 00127 Saturating linear 00050Symmetric saturating linear 00046 Symmetric saturating linear 00047

Table 4 Network validation performance with different trainingalgorithms

Training algorithms Time (s) 119864119904

Gradient descent 237 00467Gradient descent with adaptive learning 9 00289Scaled conjugate gradient 12 00158Levenberg-Marquardt 15 00046

for each function The symmetric saturating linear functionprovides the best performance

Second the performance of each of these five transferfunctions in the hidden layer is investigated The remainingnetwork parameters are the symmetric saturating linearoutput transfer function 30 hidden neurons 2 noise andthe Levenberg-Marquardt training algorithmThe network istrained 10 times with each of these output transfer functionsTable 3 summarizes the average validation performance Thelogarithm sigmoid function provides the best performance

Next the performance of four training algorithms isstudied Table 4 summarizes the average results for 10 runsThe gradient descent algorithm provides the worst perfor-mance in terms of training time and validation mean squareerror An adaptive learning rate substantially improves theresults the training time is 26 times lower and the meanerror is reduced by a factor of 16 The scale conjugate-gradient reaches a lower mean square error with a slightlyhigher training time Nevertheless the Levenberg-Marquardtalgorithm provides the lowest mean square validation errorby far and its training time is nearly the same as that with thescale conjugate gradient algorithm

Figure 5 shows the average validation error for differentnumbers of hidden nodes Seventy nodes in the hidden layergive the lower validation error 119864

119904 Less than 70 neurons are

not sufficient to fit the data whereas greater than 70 neuronsoverfit the neural network

The last parameter corresponds to the level of noise inthe training data The selection of this parameter is relatedto the level of experimental noise Hence this parameterwas defined by studying the performance of the trainednetwork with the experimental data and different levels ofperturbation noise Figure 6 shows the mean sizing error(MSE) obtained when the network is tested with experimen-tal dataThis case corresponds to a 55 stiffness reduction inelement 5 The MSE is the difference between the predicteddamage level on each element and the actual damage level asdefined in (9) A data perturbation noise of 15 provides theminimum error in the predicted damage

20 40 60 80 100 120 1403638

442444648

5

Number of hidden nodes

Es

times10minus3

Figure 5 Network validation performance as a function of thenumber of hidden nodes

0 05 1 15 2 25 3 35 40

00501

01502

02503

035

Noise level ()

MSE

Figure 6 Experimental mean sizing error for different levels ofinjected noise

42 Network Validation The selected network parametersare

(i) network three layers of multilayer perceptron(ii) number of input nodes 7(iii) number of hidden nodes 70(iv) number of output nodes 7(v) transfer function in the hidden layer Logarithm

sigmoid(vi) transfer function in the output layer Symmetric

saturating linear(vii) training method Levenberg-Marquardt

The validation patterns verify the performance of the net-work and these patterns are polluted with 15 noise Theresulting mean sizing error is MSE = 337 and thus thenetwork quantification accuracy is 97 Figure 7 shows thedamage missing error (DME) and false alarm error (FAE)divided by the damage levelsThe results of the DME indicate

10 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

Damage level ()

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(b)

Figure 7 Damage missing error (DME) and false alarm error (FAE) for different damage levels

1 2 3 4 5 6 70

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Figure 8 Assessment of experimental damage (55 stiffness reduc-tion in element 5)

that the detection of damage with low severities is poor Infact 80 of the damage with severities lower than 10 and40of the damagewith severities of 10ndash20 are not detectedThe algorithm can detect damage with severities larger than30with confidence it correctly detects 996 of the damagewith severities over 30The false alarm results indicate thatmost of the damage detected with levels lower than 10are false damage (near 75) The amount of false damagedetection is drastically reduced with an increment in the levelof damage In fact 9924 of the damage detected with levelsover 30 is true damage

Figure 8 shows the results of the damage detected usingthe experimental case The damaged element is correctlyassessed the location is correct and the quantification erroris 5

5 Experimental Beam

The structure consists of a steel beam with a rectangularcross-section The dimensions of the beam are length = 1mand section = 25 times 10mm2 As shown in Figure 9 soft springssuspend the structure to simulate a ldquofree-freerdquo boundarycondition

A hammer excites the beam at four points distributedalong the beam and an accelerometer measures the responseat each excitation location Both the excitation force and themeasured responses lie in the horizontal direction In thisdirection the anti-resonant frequencies are more sensitive to

Table 5 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical

FRF1

903 89 1102934 289 1356063 604 04010328 1033 00015528 1576 151

FRF2

1288 128 0692609 258 1005831 586 05611128 1127 12616219 1650 170

FRF3

1288 128 069260 258 0665844 586 03411138 1127 11716175 1650 198

FRF4

903 89 1102929 289 1356061 604 04010328 1033 00015534 1576 151

Figure 9 Experimental beam

the experimental damage Five anti-resonant frequencies areidentified at each of the four experimental FRFs

The numerical model is built in Matlab with 2D beamelements The model contains 20 beam elements and 40degrees of freedom as shown in Figure 10 The shadowedelements indicate possible locations of damage Table 5 showsthe experimental and numerical frequencies and their differ-ences and the maximum difference is 198

The structure is subjected to four different damagescenarios containing single and double cracks Cracks are

Shock and Vibration 11

Table 6 Damage cases introduced to the beam

Case Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

1 315 7 5 mdash mdash mdash2 635 13 10 mdash mdash mdash3 303 7 5 640 13 104 360 8 5 810 17 15

1 2 3 4 5 6 7 8 9 10 1211 13 14 15 16 17 18 19 20

Figure 10 Numerical model and element numbering

C

Figure 11 Saw cut introduced to the beam

introduced to the structure by saw cuts of length 119897119888 as

illustrated in Figure 11 Table 6 summarizes the differentdamage scenarios and indicates the distance from the left endto the cut the corresponding element in the numericalmodeland the cut length

51 Network Validation The network uses the same param-eters selected for the first structure except for the number ofinput hidden and output nodes The number of input nodescorresponds to the number of anti-resonant frequencieswhich is 20 Elements 2 to 19 were defined as possiblelocations of damage thus giving 18 output nodes (damageindices) The number of hidden nodes was defined afterthe sensitivity analysis shown in Figure 12 The minimumvalidation error is obtained with 80 hidden nodes

Hence the network parameters are

(i) network three layers of multilayer perceptron(ii) number of input nodes 20(iii) number of hidden nodes 80(iv) number of output nodes 18(v) transfer function in the hidden layer logarithm sig-

moid(vi) transfer function in the output layer symmetric

saturating linear(vii) training method Levenberg-Marquardt

The training and validation patterns were polluted with15 random noise The mean sizing error obtained for thevalidation patterns is MSE = 153 and thus the networkquantification accuracy is 98 Figure 13 shows the damagemissing error (DME) and false alarm error (FAE) separatedby damage levels The results of the DME indicate that thedetection of damage with low severities is poor In fact

20 40 60 80 100 120 140 160 180 200

28

3

32

34

36

Number of hidden nodes

times10minus3

Es

Figure 12 Validation performance as a function of the number ofhidden nodes

near 90 of the damage with severities lower than 10 and70 of the damage with severities between 10 and 20 isnot detected The algorithm is able to detect damage withseverities larger than 40with confidence it correctly detects995 of the damage with severities over 40The false alarmresults indicate that most of the damage detected with levelslower than 10 is false damage (near 80) The amount offalse damage detection decreases with an increment of thelevel of damage 99 of the damage detected with levels over40 is true damage

Note that these results are slightly worse than the resultsobtained from the 8-DOF system This result is explained bythe fact that the anti-resonant frequencies are less sensitiveto damage in the free beam than the 8-DOF structure thusmaking it more difficult to detect small damage

Figure 14 shows the results of the damage detected in thefour experimental scenarios An arrow indicates the actualdamage location In the first three cases the damage iscorrectly located although false damage appears next to theactual locations Nevertheless the assumption that a crackonly affects the stiffness of the corresponding element mightnot always be true For instance in the fourth case the effectof the larger cut is represented more accurately by a stiffnessreduction of two elements rather than one In the last case thenetwork does not detect the small crack in element 8 becausethe effect of the larger cut hides the effect of the smaller cut

6 Conclusions

This paper presents a new methodology designed to assessexperimental damage using neural networks and anti-resonant frequencies A multilayer perceptron network wastrained with data obtained from a numerical model and

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

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SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

error the damage missing error and the false alarm error asdefined by Yun et al [19] The mean sizing error (MSE) is theaverage quantification error

MSE =1

NOsum

119894

1003816100381610038161003816119910119894 minus 119900119894

1003816100381610038161003816 (9)

where119910119894and 119900119894are the estimated and desired outputs for node

119894 respectively and NO is the number of output nodesThe damage missing error (DME) is given by

DME =1

NTsum

119894

120576119868

119894 0 le DME le 1 (10)

where 120576119868

119894= 0 if the 119894th damaged element is correctly detected

and 120576119868

119894= 1 otherwise The NT corresponds to the number

of true damage locations If DME = 0 all damage locationsare correctly detected The value of 120576

119868

119894is calculated using the

following equation

120576119868

119894=

1 if 119900119894gt 0 119910

119894le 120572119888

0 otherwise(11)

It is assumed that an element is detected as damaged if theestimated damage 119910

119894is greater than a prescribed critical value

120572119888 The critical damage level 120572

119888is defined as being equal to

the average MSEThis value is the minimum damage that thenetwork can reasonably assess

The false alarm error (FAE) is defined as

FAE =1

NFsum

119894

120576119868119868

119894 0 le FAE le 1 (12)

where 120576119868119868

119894= 0 if the 119894th detected damage is truly damaged

and 120576119868119868

119894= 1 otherwise The NF is the number of predicted

damage locations If FAE = 0 all of the detected locations areactual damage locations The value of 120576

119868119868

119894is calculated using

the following equation

120576119868119868

119894=

1 if119910119894ge 120572119888 119900119894= 0

0 otherwise(13)

3 Identification and Matching ofAntiresonant Frequencies

31 Experimental Antiresonances The experimental antires-onances are identified from experimental point FRFs byldquodip-pickingrdquo [22] In this technique the antiresonances areselected by picking the dips from the magnitude plot ofa given FRF that have an associated change of +180∘ inthe phase plot This method is similar to the ldquopeak pick-ingrdquo technique for resonant frequencies Figure 3 shows anexample of the anti-resonant picking procedure When usingthis technique the errors in the estimated antiresonancesoriginate primarily from the frequency resolution of theFRFs If the dips in the FRFs are difficult to determine theantiresonances can be identified by a curve-fitting techniquebased on the rational fraction polynomial representation ofthe FRFs [38]

32 Numerical Antiresonances For a lightly damped struc-ture the anti-resonant frequencies are nearly unaffected bydamping and therefore they can be obtained from theundamped system using only the stiffness andmass matrices

By definition the FRFmatrix is the inverse of the dynamicstiffness matrix

H (120596) = (K minus 1205962M)minus1

=adj (K minus 120596

2M)

det (K minus 1205962M) (14)

The operators adj(sdot) and det(sdot) indicate the adjointand determinant respectively The anti-resonant frequenciescorrespond to the zeros of the FRFs The zeros of the 119894119896th FRF are the values of 120596 for which the numerator of119867119894119896

(120596)vanishes The numerator of 119867119894119896

(120596) is the 119894 119896th termof adj(Kminus 120596

2M) which is given by (minus1)119894+119896 det(K

119894119896minus 1205962M119894119896

)The subscripts 119894 and 119896 denote that the 119894th row and 119896th columnhave been deleted As a consequence the antiresonances ofthe 119894 119896th FRF are the frequency values that satisfy

det (K119894119896

minus 1205962M119894119896

) = 0 (15)

which is equivalent to solving the eigenvalue problem

(K119894119896

minus 1205962M119894119896

) u = 0 (16)

If 119894 = 119896 (16) represents a physical system obtained bygrounding the 119894th degree of freedom Therefore the anti-resonant frequencies obtained from point FRFs (119894 = 119896) areequivalent to the resonant frequencies of the structure withthe 119894th degree of freedom grounded

Because we are working with point FRFs matching theexperimental and numerical anti-resonant frequencies is aneasy task because each anti-resonant frequency lies betweentwo resonant frequencies

4 Eight-DOF Spring-Mass System

The structure shown in Figure 4 consists of an 8-DOF spring-mass system The Los Alamos National Laboratory (LANL)designed and constructed this system to study the effec-tiveness of various vibration-based damage identificationtechniques [39]

The system consists of eight translatingmasses connectedby springs and each mass consists of a disc of aluminumwith a diameter of 762mm and a thickness of 254mm Themasses slide on a highly polished steel rod and are fastenedtogether with coil springs The spring and mass locationsare designated sequentially with the first located the closestto the shaker attachment In the undamaged configurationall springs are identical and have linear stiffness coefficientsDamage is simulated by replacing the fifth spring with adifferent spring with a lower stiffness (55 stiffness reduc-tion)The acceleration is measured horizontally at eachmassgiving a total of eightmeasuredDOFsThe structure is excitedrandomly by an electrodynamic shaker The responses aremeasured in the eight DOFs in the undamaged and damagedcases

Seven anti-resonant frequencies are identified from theFRF corresponding to the first mass which corresponds to

8 Shock and Vibration

Mag

nitu

de (a

bs)

102

100

104

10minus2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

2855988

10261549

8125

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

Phas

e (de

g)

50

0

minus50

minus100

minus150

minus200

(b)

Figure 3 Identification of experimental anti-resonant frequencies

m8 m7 m6 m5 m4 m3 m2 m1 f(t)

k7 k6 k5 k4 k3 k2 k1x8 x7 x6 x5 x4 x3 x2 x1

Figure 4 Experimental system with eight degrees of freedom

a point FRF Given that the anti-resonant frequencies areindependent seven antiresonances should be sufficient toassess the stiffness of each of the seven springs

Table 1 lists the anti-resonant frequencies identified inthe undamaged and damaged cases The second and fourthantiresonances provide the larger variation due to the exper-imental damage

The numerical model is built in Matlab with springs andconcentrated masses and the initial parameters are set asfollows

(i) mass 1 5593 g (this mass is greater than the othersbecause of the hardware required to attach the shaker)

(ii) masses 2 to 8 4194 g(iii) spring constants 567 kNmminus1

Table 2 shows the numerical and experimental anti-resonant frequencies The maximum difference between theexperimental and numerical antiresonances is 07 As aconsequence the numerical model provides a faithful repre-sentation of the experimental structure and can be used totrain the neural network

41 Construction of the Neural Network The network con-tains seven inputs and seven outputsThe best parameters forthe neural network are defined after performing a sensitivity

Table 1 Experimental antiresonances

Antiresonant frequencies (Hz) Variation ()Undamaged Damaged1231 1166 minus5283694 3244 minus12185961 598 0327996 7174 minus10289612 9302 minus32310944 10686 minus23611572 11267 minus264

Table 2 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical123 123 043369 367 054596 592 070800 795 055961 956 0521094 1090 0401157 1155 019

analysis The network is trained under different configura-tions and the selected configuration is the one that providesthe lowest validation error

First the effects of five different transfer functions usedfor the output layer are studied and these transfer functionsare formulated as follows (a) hyperbolic tangent sigmoid(b) logarithm sigmoid (c) linear (d) saturating linear and(e) symmetric saturating linear Figure 2 illustrates thesefunctions During the experimental runs the remainingparameters of the network are characterized as follows thehidden transfer function is a logarithm sigmoid the numberof hidden neurons is 30 the training data are polluted with2 random noise and the Levenberg-Marquardt algorithmis used to train the network The network is trained 10times with each of the output transfer functions Table 3summarizes the average validation performance obtained

Shock and Vibration 9

Table 3 Network validation performance with different transfer functions

Output transfer function 119864119904

Hidden transfer function 119864119904

Hyperbolic tangent sigmoid 00067 Hyperbolic tangent sigmoid 00045Logarithm sigmoid 00134 Logarithm sigmoid 00044Linear 00060 Linear 00130Saturating linear 00127 Saturating linear 00050Symmetric saturating linear 00046 Symmetric saturating linear 00047

Table 4 Network validation performance with different trainingalgorithms

Training algorithms Time (s) 119864119904

Gradient descent 237 00467Gradient descent with adaptive learning 9 00289Scaled conjugate gradient 12 00158Levenberg-Marquardt 15 00046

for each function The symmetric saturating linear functionprovides the best performance

Second the performance of each of these five transferfunctions in the hidden layer is investigated The remainingnetwork parameters are the symmetric saturating linearoutput transfer function 30 hidden neurons 2 noise andthe Levenberg-Marquardt training algorithmThe network istrained 10 times with each of these output transfer functionsTable 3 summarizes the average validation performance Thelogarithm sigmoid function provides the best performance

Next the performance of four training algorithms isstudied Table 4 summarizes the average results for 10 runsThe gradient descent algorithm provides the worst perfor-mance in terms of training time and validation mean squareerror An adaptive learning rate substantially improves theresults the training time is 26 times lower and the meanerror is reduced by a factor of 16 The scale conjugate-gradient reaches a lower mean square error with a slightlyhigher training time Nevertheless the Levenberg-Marquardtalgorithm provides the lowest mean square validation errorby far and its training time is nearly the same as that with thescale conjugate gradient algorithm

Figure 5 shows the average validation error for differentnumbers of hidden nodes Seventy nodes in the hidden layergive the lower validation error 119864

119904 Less than 70 neurons are

not sufficient to fit the data whereas greater than 70 neuronsoverfit the neural network

The last parameter corresponds to the level of noise inthe training data The selection of this parameter is relatedto the level of experimental noise Hence this parameterwas defined by studying the performance of the trainednetwork with the experimental data and different levels ofperturbation noise Figure 6 shows the mean sizing error(MSE) obtained when the network is tested with experimen-tal dataThis case corresponds to a 55 stiffness reduction inelement 5 The MSE is the difference between the predicteddamage level on each element and the actual damage level asdefined in (9) A data perturbation noise of 15 provides theminimum error in the predicted damage

20 40 60 80 100 120 1403638

442444648

5

Number of hidden nodes

Es

times10minus3

Figure 5 Network validation performance as a function of thenumber of hidden nodes

0 05 1 15 2 25 3 35 40

00501

01502

02503

035

Noise level ()

MSE

Figure 6 Experimental mean sizing error for different levels ofinjected noise

42 Network Validation The selected network parametersare

(i) network three layers of multilayer perceptron(ii) number of input nodes 7(iii) number of hidden nodes 70(iv) number of output nodes 7(v) transfer function in the hidden layer Logarithm

sigmoid(vi) transfer function in the output layer Symmetric

saturating linear(vii) training method Levenberg-Marquardt

The validation patterns verify the performance of the net-work and these patterns are polluted with 15 noise Theresulting mean sizing error is MSE = 337 and thus thenetwork quantification accuracy is 97 Figure 7 shows thedamage missing error (DME) and false alarm error (FAE)divided by the damage levelsThe results of the DME indicate

10 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

Damage level ()

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(b)

Figure 7 Damage missing error (DME) and false alarm error (FAE) for different damage levels

1 2 3 4 5 6 70

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Figure 8 Assessment of experimental damage (55 stiffness reduc-tion in element 5)

that the detection of damage with low severities is poor Infact 80 of the damage with severities lower than 10 and40of the damagewith severities of 10ndash20 are not detectedThe algorithm can detect damage with severities larger than30with confidence it correctly detects 996 of the damagewith severities over 30The false alarm results indicate thatmost of the damage detected with levels lower than 10are false damage (near 75) The amount of false damagedetection is drastically reduced with an increment in the levelof damage In fact 9924 of the damage detected with levelsover 30 is true damage

Figure 8 shows the results of the damage detected usingthe experimental case The damaged element is correctlyassessed the location is correct and the quantification erroris 5

5 Experimental Beam

The structure consists of a steel beam with a rectangularcross-section The dimensions of the beam are length = 1mand section = 25 times 10mm2 As shown in Figure 9 soft springssuspend the structure to simulate a ldquofree-freerdquo boundarycondition

A hammer excites the beam at four points distributedalong the beam and an accelerometer measures the responseat each excitation location Both the excitation force and themeasured responses lie in the horizontal direction In thisdirection the anti-resonant frequencies are more sensitive to

Table 5 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical

FRF1

903 89 1102934 289 1356063 604 04010328 1033 00015528 1576 151

FRF2

1288 128 0692609 258 1005831 586 05611128 1127 12616219 1650 170

FRF3

1288 128 069260 258 0665844 586 03411138 1127 11716175 1650 198

FRF4

903 89 1102929 289 1356061 604 04010328 1033 00015534 1576 151

Figure 9 Experimental beam

the experimental damage Five anti-resonant frequencies areidentified at each of the four experimental FRFs

The numerical model is built in Matlab with 2D beamelements The model contains 20 beam elements and 40degrees of freedom as shown in Figure 10 The shadowedelements indicate possible locations of damage Table 5 showsthe experimental and numerical frequencies and their differ-ences and the maximum difference is 198

The structure is subjected to four different damagescenarios containing single and double cracks Cracks are

Shock and Vibration 11

Table 6 Damage cases introduced to the beam

Case Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

1 315 7 5 mdash mdash mdash2 635 13 10 mdash mdash mdash3 303 7 5 640 13 104 360 8 5 810 17 15

1 2 3 4 5 6 7 8 9 10 1211 13 14 15 16 17 18 19 20

Figure 10 Numerical model and element numbering

C

Figure 11 Saw cut introduced to the beam

introduced to the structure by saw cuts of length 119897119888 as

illustrated in Figure 11 Table 6 summarizes the differentdamage scenarios and indicates the distance from the left endto the cut the corresponding element in the numericalmodeland the cut length

51 Network Validation The network uses the same param-eters selected for the first structure except for the number ofinput hidden and output nodes The number of input nodescorresponds to the number of anti-resonant frequencieswhich is 20 Elements 2 to 19 were defined as possiblelocations of damage thus giving 18 output nodes (damageindices) The number of hidden nodes was defined afterthe sensitivity analysis shown in Figure 12 The minimumvalidation error is obtained with 80 hidden nodes

Hence the network parameters are

(i) network three layers of multilayer perceptron(ii) number of input nodes 20(iii) number of hidden nodes 80(iv) number of output nodes 18(v) transfer function in the hidden layer logarithm sig-

moid(vi) transfer function in the output layer symmetric

saturating linear(vii) training method Levenberg-Marquardt

The training and validation patterns were polluted with15 random noise The mean sizing error obtained for thevalidation patterns is MSE = 153 and thus the networkquantification accuracy is 98 Figure 13 shows the damagemissing error (DME) and false alarm error (FAE) separatedby damage levels The results of the DME indicate that thedetection of damage with low severities is poor In fact

20 40 60 80 100 120 140 160 180 200

28

3

32

34

36

Number of hidden nodes

times10minus3

Es

Figure 12 Validation performance as a function of the number ofhidden nodes

near 90 of the damage with severities lower than 10 and70 of the damage with severities between 10 and 20 isnot detected The algorithm is able to detect damage withseverities larger than 40with confidence it correctly detects995 of the damage with severities over 40The false alarmresults indicate that most of the damage detected with levelslower than 10 is false damage (near 80) The amount offalse damage detection decreases with an increment of thelevel of damage 99 of the damage detected with levels over40 is true damage

Note that these results are slightly worse than the resultsobtained from the 8-DOF system This result is explained bythe fact that the anti-resonant frequencies are less sensitiveto damage in the free beam than the 8-DOF structure thusmaking it more difficult to detect small damage

Figure 14 shows the results of the damage detected in thefour experimental scenarios An arrow indicates the actualdamage location In the first three cases the damage iscorrectly located although false damage appears next to theactual locations Nevertheless the assumption that a crackonly affects the stiffness of the corresponding element mightnot always be true For instance in the fourth case the effectof the larger cut is represented more accurately by a stiffnessreduction of two elements rather than one In the last case thenetwork does not detect the small crack in element 8 becausethe effect of the larger cut hides the effect of the smaller cut

6 Conclusions

This paper presents a new methodology designed to assessexperimental damage using neural networks and anti-resonant frequencies A multilayer perceptron network wastrained with data obtained from a numerical model and

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

Mag

nitu

de (a

bs)

102

100

104

10minus2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

2855988

10261549

8125

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

Phas

e (de

g)

50

0

minus50

minus100

minus150

minus200

(b)

Figure 3 Identification of experimental anti-resonant frequencies

m8 m7 m6 m5 m4 m3 m2 m1 f(t)

k7 k6 k5 k4 k3 k2 k1x8 x7 x6 x5 x4 x3 x2 x1

Figure 4 Experimental system with eight degrees of freedom

a point FRF Given that the anti-resonant frequencies areindependent seven antiresonances should be sufficient toassess the stiffness of each of the seven springs

Table 1 lists the anti-resonant frequencies identified inthe undamaged and damaged cases The second and fourthantiresonances provide the larger variation due to the exper-imental damage

The numerical model is built in Matlab with springs andconcentrated masses and the initial parameters are set asfollows

(i) mass 1 5593 g (this mass is greater than the othersbecause of the hardware required to attach the shaker)

(ii) masses 2 to 8 4194 g(iii) spring constants 567 kNmminus1

Table 2 shows the numerical and experimental anti-resonant frequencies The maximum difference between theexperimental and numerical antiresonances is 07 As aconsequence the numerical model provides a faithful repre-sentation of the experimental structure and can be used totrain the neural network

41 Construction of the Neural Network The network con-tains seven inputs and seven outputsThe best parameters forthe neural network are defined after performing a sensitivity

Table 1 Experimental antiresonances

Antiresonant frequencies (Hz) Variation ()Undamaged Damaged1231 1166 minus5283694 3244 minus12185961 598 0327996 7174 minus10289612 9302 minus32310944 10686 minus23611572 11267 minus264

Table 2 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical123 123 043369 367 054596 592 070800 795 055961 956 0521094 1090 0401157 1155 019

analysis The network is trained under different configura-tions and the selected configuration is the one that providesthe lowest validation error

First the effects of five different transfer functions usedfor the output layer are studied and these transfer functionsare formulated as follows (a) hyperbolic tangent sigmoid(b) logarithm sigmoid (c) linear (d) saturating linear and(e) symmetric saturating linear Figure 2 illustrates thesefunctions During the experimental runs the remainingparameters of the network are characterized as follows thehidden transfer function is a logarithm sigmoid the numberof hidden neurons is 30 the training data are polluted with2 random noise and the Levenberg-Marquardt algorithmis used to train the network The network is trained 10times with each of the output transfer functions Table 3summarizes the average validation performance obtained

Shock and Vibration 9

Table 3 Network validation performance with different transfer functions

Output transfer function 119864119904

Hidden transfer function 119864119904

Hyperbolic tangent sigmoid 00067 Hyperbolic tangent sigmoid 00045Logarithm sigmoid 00134 Logarithm sigmoid 00044Linear 00060 Linear 00130Saturating linear 00127 Saturating linear 00050Symmetric saturating linear 00046 Symmetric saturating linear 00047

Table 4 Network validation performance with different trainingalgorithms

Training algorithms Time (s) 119864119904

Gradient descent 237 00467Gradient descent with adaptive learning 9 00289Scaled conjugate gradient 12 00158Levenberg-Marquardt 15 00046

for each function The symmetric saturating linear functionprovides the best performance

Second the performance of each of these five transferfunctions in the hidden layer is investigated The remainingnetwork parameters are the symmetric saturating linearoutput transfer function 30 hidden neurons 2 noise andthe Levenberg-Marquardt training algorithmThe network istrained 10 times with each of these output transfer functionsTable 3 summarizes the average validation performance Thelogarithm sigmoid function provides the best performance

Next the performance of four training algorithms isstudied Table 4 summarizes the average results for 10 runsThe gradient descent algorithm provides the worst perfor-mance in terms of training time and validation mean squareerror An adaptive learning rate substantially improves theresults the training time is 26 times lower and the meanerror is reduced by a factor of 16 The scale conjugate-gradient reaches a lower mean square error with a slightlyhigher training time Nevertheless the Levenberg-Marquardtalgorithm provides the lowest mean square validation errorby far and its training time is nearly the same as that with thescale conjugate gradient algorithm

Figure 5 shows the average validation error for differentnumbers of hidden nodes Seventy nodes in the hidden layergive the lower validation error 119864

119904 Less than 70 neurons are

not sufficient to fit the data whereas greater than 70 neuronsoverfit the neural network

The last parameter corresponds to the level of noise inthe training data The selection of this parameter is relatedto the level of experimental noise Hence this parameterwas defined by studying the performance of the trainednetwork with the experimental data and different levels ofperturbation noise Figure 6 shows the mean sizing error(MSE) obtained when the network is tested with experimen-tal dataThis case corresponds to a 55 stiffness reduction inelement 5 The MSE is the difference between the predicteddamage level on each element and the actual damage level asdefined in (9) A data perturbation noise of 15 provides theminimum error in the predicted damage

20 40 60 80 100 120 1403638

442444648

5

Number of hidden nodes

Es

times10minus3

Figure 5 Network validation performance as a function of thenumber of hidden nodes

0 05 1 15 2 25 3 35 40

00501

01502

02503

035

Noise level ()

MSE

Figure 6 Experimental mean sizing error for different levels ofinjected noise

42 Network Validation The selected network parametersare

(i) network three layers of multilayer perceptron(ii) number of input nodes 7(iii) number of hidden nodes 70(iv) number of output nodes 7(v) transfer function in the hidden layer Logarithm

sigmoid(vi) transfer function in the output layer Symmetric

saturating linear(vii) training method Levenberg-Marquardt

The validation patterns verify the performance of the net-work and these patterns are polluted with 15 noise Theresulting mean sizing error is MSE = 337 and thus thenetwork quantification accuracy is 97 Figure 7 shows thedamage missing error (DME) and false alarm error (FAE)divided by the damage levelsThe results of the DME indicate

10 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

Damage level ()

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(b)

Figure 7 Damage missing error (DME) and false alarm error (FAE) for different damage levels

1 2 3 4 5 6 70

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Figure 8 Assessment of experimental damage (55 stiffness reduc-tion in element 5)

that the detection of damage with low severities is poor Infact 80 of the damage with severities lower than 10 and40of the damagewith severities of 10ndash20 are not detectedThe algorithm can detect damage with severities larger than30with confidence it correctly detects 996 of the damagewith severities over 30The false alarm results indicate thatmost of the damage detected with levels lower than 10are false damage (near 75) The amount of false damagedetection is drastically reduced with an increment in the levelof damage In fact 9924 of the damage detected with levelsover 30 is true damage

Figure 8 shows the results of the damage detected usingthe experimental case The damaged element is correctlyassessed the location is correct and the quantification erroris 5

5 Experimental Beam

The structure consists of a steel beam with a rectangularcross-section The dimensions of the beam are length = 1mand section = 25 times 10mm2 As shown in Figure 9 soft springssuspend the structure to simulate a ldquofree-freerdquo boundarycondition

A hammer excites the beam at four points distributedalong the beam and an accelerometer measures the responseat each excitation location Both the excitation force and themeasured responses lie in the horizontal direction In thisdirection the anti-resonant frequencies are more sensitive to

Table 5 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical

FRF1

903 89 1102934 289 1356063 604 04010328 1033 00015528 1576 151

FRF2

1288 128 0692609 258 1005831 586 05611128 1127 12616219 1650 170

FRF3

1288 128 069260 258 0665844 586 03411138 1127 11716175 1650 198

FRF4

903 89 1102929 289 1356061 604 04010328 1033 00015534 1576 151

Figure 9 Experimental beam

the experimental damage Five anti-resonant frequencies areidentified at each of the four experimental FRFs

The numerical model is built in Matlab with 2D beamelements The model contains 20 beam elements and 40degrees of freedom as shown in Figure 10 The shadowedelements indicate possible locations of damage Table 5 showsthe experimental and numerical frequencies and their differ-ences and the maximum difference is 198

The structure is subjected to four different damagescenarios containing single and double cracks Cracks are

Shock and Vibration 11

Table 6 Damage cases introduced to the beam

Case Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

1 315 7 5 mdash mdash mdash2 635 13 10 mdash mdash mdash3 303 7 5 640 13 104 360 8 5 810 17 15

1 2 3 4 5 6 7 8 9 10 1211 13 14 15 16 17 18 19 20

Figure 10 Numerical model and element numbering

C

Figure 11 Saw cut introduced to the beam

introduced to the structure by saw cuts of length 119897119888 as

illustrated in Figure 11 Table 6 summarizes the differentdamage scenarios and indicates the distance from the left endto the cut the corresponding element in the numericalmodeland the cut length

51 Network Validation The network uses the same param-eters selected for the first structure except for the number ofinput hidden and output nodes The number of input nodescorresponds to the number of anti-resonant frequencieswhich is 20 Elements 2 to 19 were defined as possiblelocations of damage thus giving 18 output nodes (damageindices) The number of hidden nodes was defined afterthe sensitivity analysis shown in Figure 12 The minimumvalidation error is obtained with 80 hidden nodes

Hence the network parameters are

(i) network three layers of multilayer perceptron(ii) number of input nodes 20(iii) number of hidden nodes 80(iv) number of output nodes 18(v) transfer function in the hidden layer logarithm sig-

moid(vi) transfer function in the output layer symmetric

saturating linear(vii) training method Levenberg-Marquardt

The training and validation patterns were polluted with15 random noise The mean sizing error obtained for thevalidation patterns is MSE = 153 and thus the networkquantification accuracy is 98 Figure 13 shows the damagemissing error (DME) and false alarm error (FAE) separatedby damage levels The results of the DME indicate that thedetection of damage with low severities is poor In fact

20 40 60 80 100 120 140 160 180 200

28

3

32

34

36

Number of hidden nodes

times10minus3

Es

Figure 12 Validation performance as a function of the number ofhidden nodes

near 90 of the damage with severities lower than 10 and70 of the damage with severities between 10 and 20 isnot detected The algorithm is able to detect damage withseverities larger than 40with confidence it correctly detects995 of the damage with severities over 40The false alarmresults indicate that most of the damage detected with levelslower than 10 is false damage (near 80) The amount offalse damage detection decreases with an increment of thelevel of damage 99 of the damage detected with levels over40 is true damage

Note that these results are slightly worse than the resultsobtained from the 8-DOF system This result is explained bythe fact that the anti-resonant frequencies are less sensitiveto damage in the free beam than the 8-DOF structure thusmaking it more difficult to detect small damage

Figure 14 shows the results of the damage detected in thefour experimental scenarios An arrow indicates the actualdamage location In the first three cases the damage iscorrectly located although false damage appears next to theactual locations Nevertheless the assumption that a crackonly affects the stiffness of the corresponding element mightnot always be true For instance in the fourth case the effectof the larger cut is represented more accurately by a stiffnessreduction of two elements rather than one In the last case thenetwork does not detect the small crack in element 8 becausethe effect of the larger cut hides the effect of the smaller cut

6 Conclusions

This paper presents a new methodology designed to assessexperimental damage using neural networks and anti-resonant frequencies A multilayer perceptron network wastrained with data obtained from a numerical model and

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

Table 3 Network validation performance with different transfer functions

Output transfer function 119864119904

Hidden transfer function 119864119904

Hyperbolic tangent sigmoid 00067 Hyperbolic tangent sigmoid 00045Logarithm sigmoid 00134 Logarithm sigmoid 00044Linear 00060 Linear 00130Saturating linear 00127 Saturating linear 00050Symmetric saturating linear 00046 Symmetric saturating linear 00047

Table 4 Network validation performance with different trainingalgorithms

Training algorithms Time (s) 119864119904

Gradient descent 237 00467Gradient descent with adaptive learning 9 00289Scaled conjugate gradient 12 00158Levenberg-Marquardt 15 00046

for each function The symmetric saturating linear functionprovides the best performance

Second the performance of each of these five transferfunctions in the hidden layer is investigated The remainingnetwork parameters are the symmetric saturating linearoutput transfer function 30 hidden neurons 2 noise andthe Levenberg-Marquardt training algorithmThe network istrained 10 times with each of these output transfer functionsTable 3 summarizes the average validation performance Thelogarithm sigmoid function provides the best performance

Next the performance of four training algorithms isstudied Table 4 summarizes the average results for 10 runsThe gradient descent algorithm provides the worst perfor-mance in terms of training time and validation mean squareerror An adaptive learning rate substantially improves theresults the training time is 26 times lower and the meanerror is reduced by a factor of 16 The scale conjugate-gradient reaches a lower mean square error with a slightlyhigher training time Nevertheless the Levenberg-Marquardtalgorithm provides the lowest mean square validation errorby far and its training time is nearly the same as that with thescale conjugate gradient algorithm

Figure 5 shows the average validation error for differentnumbers of hidden nodes Seventy nodes in the hidden layergive the lower validation error 119864

119904 Less than 70 neurons are

not sufficient to fit the data whereas greater than 70 neuronsoverfit the neural network

The last parameter corresponds to the level of noise inthe training data The selection of this parameter is relatedto the level of experimental noise Hence this parameterwas defined by studying the performance of the trainednetwork with the experimental data and different levels ofperturbation noise Figure 6 shows the mean sizing error(MSE) obtained when the network is tested with experimen-tal dataThis case corresponds to a 55 stiffness reduction inelement 5 The MSE is the difference between the predicteddamage level on each element and the actual damage level asdefined in (9) A data perturbation noise of 15 provides theminimum error in the predicted damage

20 40 60 80 100 120 1403638

442444648

5

Number of hidden nodes

Es

times10minus3

Figure 5 Network validation performance as a function of thenumber of hidden nodes

0 05 1 15 2 25 3 35 40

00501

01502

02503

035

Noise level ()

MSE

Figure 6 Experimental mean sizing error for different levels ofinjected noise

42 Network Validation The selected network parametersare

(i) network three layers of multilayer perceptron(ii) number of input nodes 7(iii) number of hidden nodes 70(iv) number of output nodes 7(v) transfer function in the hidden layer Logarithm

sigmoid(vi) transfer function in the output layer Symmetric

saturating linear(vii) training method Levenberg-Marquardt

The validation patterns verify the performance of the net-work and these patterns are polluted with 15 noise Theresulting mean sizing error is MSE = 337 and thus thenetwork quantification accuracy is 97 Figure 7 shows thedamage missing error (DME) and false alarm error (FAE)divided by the damage levelsThe results of the DME indicate

10 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

Damage level ()

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(b)

Figure 7 Damage missing error (DME) and false alarm error (FAE) for different damage levels

1 2 3 4 5 6 70

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Figure 8 Assessment of experimental damage (55 stiffness reduc-tion in element 5)

that the detection of damage with low severities is poor Infact 80 of the damage with severities lower than 10 and40of the damagewith severities of 10ndash20 are not detectedThe algorithm can detect damage with severities larger than30with confidence it correctly detects 996 of the damagewith severities over 30The false alarm results indicate thatmost of the damage detected with levels lower than 10are false damage (near 75) The amount of false damagedetection is drastically reduced with an increment in the levelof damage In fact 9924 of the damage detected with levelsover 30 is true damage

Figure 8 shows the results of the damage detected usingthe experimental case The damaged element is correctlyassessed the location is correct and the quantification erroris 5

5 Experimental Beam

The structure consists of a steel beam with a rectangularcross-section The dimensions of the beam are length = 1mand section = 25 times 10mm2 As shown in Figure 9 soft springssuspend the structure to simulate a ldquofree-freerdquo boundarycondition

A hammer excites the beam at four points distributedalong the beam and an accelerometer measures the responseat each excitation location Both the excitation force and themeasured responses lie in the horizontal direction In thisdirection the anti-resonant frequencies are more sensitive to

Table 5 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical

FRF1

903 89 1102934 289 1356063 604 04010328 1033 00015528 1576 151

FRF2

1288 128 0692609 258 1005831 586 05611128 1127 12616219 1650 170

FRF3

1288 128 069260 258 0665844 586 03411138 1127 11716175 1650 198

FRF4

903 89 1102929 289 1356061 604 04010328 1033 00015534 1576 151

Figure 9 Experimental beam

the experimental damage Five anti-resonant frequencies areidentified at each of the four experimental FRFs

The numerical model is built in Matlab with 2D beamelements The model contains 20 beam elements and 40degrees of freedom as shown in Figure 10 The shadowedelements indicate possible locations of damage Table 5 showsthe experimental and numerical frequencies and their differ-ences and the maximum difference is 198

The structure is subjected to four different damagescenarios containing single and double cracks Cracks are

Shock and Vibration 11

Table 6 Damage cases introduced to the beam

Case Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

1 315 7 5 mdash mdash mdash2 635 13 10 mdash mdash mdash3 303 7 5 640 13 104 360 8 5 810 17 15

1 2 3 4 5 6 7 8 9 10 1211 13 14 15 16 17 18 19 20

Figure 10 Numerical model and element numbering

C

Figure 11 Saw cut introduced to the beam

introduced to the structure by saw cuts of length 119897119888 as

illustrated in Figure 11 Table 6 summarizes the differentdamage scenarios and indicates the distance from the left endto the cut the corresponding element in the numericalmodeland the cut length

51 Network Validation The network uses the same param-eters selected for the first structure except for the number ofinput hidden and output nodes The number of input nodescorresponds to the number of anti-resonant frequencieswhich is 20 Elements 2 to 19 were defined as possiblelocations of damage thus giving 18 output nodes (damageindices) The number of hidden nodes was defined afterthe sensitivity analysis shown in Figure 12 The minimumvalidation error is obtained with 80 hidden nodes

Hence the network parameters are

(i) network three layers of multilayer perceptron(ii) number of input nodes 20(iii) number of hidden nodes 80(iv) number of output nodes 18(v) transfer function in the hidden layer logarithm sig-

moid(vi) transfer function in the output layer symmetric

saturating linear(vii) training method Levenberg-Marquardt

The training and validation patterns were polluted with15 random noise The mean sizing error obtained for thevalidation patterns is MSE = 153 and thus the networkquantification accuracy is 98 Figure 13 shows the damagemissing error (DME) and false alarm error (FAE) separatedby damage levels The results of the DME indicate that thedetection of damage with low severities is poor In fact

20 40 60 80 100 120 140 160 180 200

28

3

32

34

36

Number of hidden nodes

times10minus3

Es

Figure 12 Validation performance as a function of the number ofhidden nodes

near 90 of the damage with severities lower than 10 and70 of the damage with severities between 10 and 20 isnot detected The algorithm is able to detect damage withseverities larger than 40with confidence it correctly detects995 of the damage with severities over 40The false alarmresults indicate that most of the damage detected with levelslower than 10 is false damage (near 80) The amount offalse damage detection decreases with an increment of thelevel of damage 99 of the damage detected with levels over40 is true damage

Note that these results are slightly worse than the resultsobtained from the 8-DOF system This result is explained bythe fact that the anti-resonant frequencies are less sensitiveto damage in the free beam than the 8-DOF structure thusmaking it more difficult to detect small damage

Figure 14 shows the results of the damage detected in thefour experimental scenarios An arrow indicates the actualdamage location In the first three cases the damage iscorrectly located although false damage appears next to theactual locations Nevertheless the assumption that a crackonly affects the stiffness of the corresponding element mightnot always be true For instance in the fourth case the effectof the larger cut is represented more accurately by a stiffnessreduction of two elements rather than one In the last case thenetwork does not detect the small crack in element 8 becausethe effect of the larger cut hides the effect of the smaller cut

6 Conclusions

This paper presents a new methodology designed to assessexperimental damage using neural networks and anti-resonant frequencies A multilayer perceptron network wastrained with data obtained from a numerical model and

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

Damage level ()

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(b)

Figure 7 Damage missing error (DME) and false alarm error (FAE) for different damage levels

1 2 3 4 5 6 70

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Figure 8 Assessment of experimental damage (55 stiffness reduc-tion in element 5)

that the detection of damage with low severities is poor Infact 80 of the damage with severities lower than 10 and40of the damagewith severities of 10ndash20 are not detectedThe algorithm can detect damage with severities larger than30with confidence it correctly detects 996 of the damagewith severities over 30The false alarm results indicate thatmost of the damage detected with levels lower than 10are false damage (near 75) The amount of false damagedetection is drastically reduced with an increment in the levelof damage In fact 9924 of the damage detected with levelsover 30 is true damage

Figure 8 shows the results of the damage detected usingthe experimental case The damaged element is correctlyassessed the location is correct and the quantification erroris 5

5 Experimental Beam

The structure consists of a steel beam with a rectangularcross-section The dimensions of the beam are length = 1mand section = 25 times 10mm2 As shown in Figure 9 soft springssuspend the structure to simulate a ldquofree-freerdquo boundarycondition

A hammer excites the beam at four points distributedalong the beam and an accelerometer measures the responseat each excitation location Both the excitation force and themeasured responses lie in the horizontal direction In thisdirection the anti-resonant frequencies are more sensitive to

Table 5 Experimental and numerical antiresonances

Antiresonant frequencies (Hz) Difference ()Experimental Numerical

FRF1

903 89 1102934 289 1356063 604 04010328 1033 00015528 1576 151

FRF2

1288 128 0692609 258 1005831 586 05611128 1127 12616219 1650 170

FRF3

1288 128 069260 258 0665844 586 03411138 1127 11716175 1650 198

FRF4

903 89 1102929 289 1356061 604 04010328 1033 00015534 1576 151

Figure 9 Experimental beam

the experimental damage Five anti-resonant frequencies areidentified at each of the four experimental FRFs

The numerical model is built in Matlab with 2D beamelements The model contains 20 beam elements and 40degrees of freedom as shown in Figure 10 The shadowedelements indicate possible locations of damage Table 5 showsthe experimental and numerical frequencies and their differ-ences and the maximum difference is 198

The structure is subjected to four different damagescenarios containing single and double cracks Cracks are

Shock and Vibration 11

Table 6 Damage cases introduced to the beam

Case Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

1 315 7 5 mdash mdash mdash2 635 13 10 mdash mdash mdash3 303 7 5 640 13 104 360 8 5 810 17 15

1 2 3 4 5 6 7 8 9 10 1211 13 14 15 16 17 18 19 20

Figure 10 Numerical model and element numbering

C

Figure 11 Saw cut introduced to the beam

introduced to the structure by saw cuts of length 119897119888 as

illustrated in Figure 11 Table 6 summarizes the differentdamage scenarios and indicates the distance from the left endto the cut the corresponding element in the numericalmodeland the cut length

51 Network Validation The network uses the same param-eters selected for the first structure except for the number ofinput hidden and output nodes The number of input nodescorresponds to the number of anti-resonant frequencieswhich is 20 Elements 2 to 19 were defined as possiblelocations of damage thus giving 18 output nodes (damageindices) The number of hidden nodes was defined afterthe sensitivity analysis shown in Figure 12 The minimumvalidation error is obtained with 80 hidden nodes

Hence the network parameters are

(i) network three layers of multilayer perceptron(ii) number of input nodes 20(iii) number of hidden nodes 80(iv) number of output nodes 18(v) transfer function in the hidden layer logarithm sig-

moid(vi) transfer function in the output layer symmetric

saturating linear(vii) training method Levenberg-Marquardt

The training and validation patterns were polluted with15 random noise The mean sizing error obtained for thevalidation patterns is MSE = 153 and thus the networkquantification accuracy is 98 Figure 13 shows the damagemissing error (DME) and false alarm error (FAE) separatedby damage levels The results of the DME indicate that thedetection of damage with low severities is poor In fact

20 40 60 80 100 120 140 160 180 200

28

3

32

34

36

Number of hidden nodes

times10minus3

Es

Figure 12 Validation performance as a function of the number ofhidden nodes

near 90 of the damage with severities lower than 10 and70 of the damage with severities between 10 and 20 isnot detected The algorithm is able to detect damage withseverities larger than 40with confidence it correctly detects995 of the damage with severities over 40The false alarmresults indicate that most of the damage detected with levelslower than 10 is false damage (near 80) The amount offalse damage detection decreases with an increment of thelevel of damage 99 of the damage detected with levels over40 is true damage

Note that these results are slightly worse than the resultsobtained from the 8-DOF system This result is explained bythe fact that the anti-resonant frequencies are less sensitiveto damage in the free beam than the 8-DOF structure thusmaking it more difficult to detect small damage

Figure 14 shows the results of the damage detected in thefour experimental scenarios An arrow indicates the actualdamage location In the first three cases the damage iscorrectly located although false damage appears next to theactual locations Nevertheless the assumption that a crackonly affects the stiffness of the corresponding element mightnot always be true For instance in the fourth case the effectof the larger cut is represented more accurately by a stiffnessreduction of two elements rather than one In the last case thenetwork does not detect the small crack in element 8 becausethe effect of the larger cut hides the effect of the smaller cut

6 Conclusions

This paper presents a new methodology designed to assessexperimental damage using neural networks and anti-resonant frequencies A multilayer perceptron network wastrained with data obtained from a numerical model and

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

Table 6 Damage cases introduced to the beam

Case Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

Distance from theleft end (mm)

Elementnumber

Saw cutlength (mm)

1 315 7 5 mdash mdash mdash2 635 13 10 mdash mdash mdash3 303 7 5 640 13 104 360 8 5 810 17 15

1 2 3 4 5 6 7 8 9 10 1211 13 14 15 16 17 18 19 20

Figure 10 Numerical model and element numbering

C

Figure 11 Saw cut introduced to the beam

introduced to the structure by saw cuts of length 119897119888 as

illustrated in Figure 11 Table 6 summarizes the differentdamage scenarios and indicates the distance from the left endto the cut the corresponding element in the numericalmodeland the cut length

51 Network Validation The network uses the same param-eters selected for the first structure except for the number ofinput hidden and output nodes The number of input nodescorresponds to the number of anti-resonant frequencieswhich is 20 Elements 2 to 19 were defined as possiblelocations of damage thus giving 18 output nodes (damageindices) The number of hidden nodes was defined afterthe sensitivity analysis shown in Figure 12 The minimumvalidation error is obtained with 80 hidden nodes

Hence the network parameters are

(i) network three layers of multilayer perceptron(ii) number of input nodes 20(iii) number of hidden nodes 80(iv) number of output nodes 18(v) transfer function in the hidden layer logarithm sig-

moid(vi) transfer function in the output layer symmetric

saturating linear(vii) training method Levenberg-Marquardt

The training and validation patterns were polluted with15 random noise The mean sizing error obtained for thevalidation patterns is MSE = 153 and thus the networkquantification accuracy is 98 Figure 13 shows the damagemissing error (DME) and false alarm error (FAE) separatedby damage levels The results of the DME indicate that thedetection of damage with low severities is poor In fact

20 40 60 80 100 120 140 160 180 200

28

3

32

34

36

Number of hidden nodes

times10minus3

Es

Figure 12 Validation performance as a function of the number ofhidden nodes

near 90 of the damage with severities lower than 10 and70 of the damage with severities between 10 and 20 isnot detected The algorithm is able to detect damage withseverities larger than 40with confidence it correctly detects995 of the damage with severities over 40The false alarmresults indicate that most of the damage detected with levelslower than 10 is false damage (near 80) The amount offalse damage detection decreases with an increment of thelevel of damage 99 of the damage detected with levels over40 is true damage

Note that these results are slightly worse than the resultsobtained from the 8-DOF system This result is explained bythe fact that the anti-resonant frequencies are less sensitiveto damage in the free beam than the 8-DOF structure thusmaking it more difficult to detect small damage

Figure 14 shows the results of the damage detected in thefour experimental scenarios An arrow indicates the actualdamage location In the first three cases the damage iscorrectly located although false damage appears next to theactual locations Nevertheless the assumption that a crackonly affects the stiffness of the corresponding element mightnot always be true For instance in the fourth case the effectof the larger cut is represented more accurately by a stiffnessreduction of two elements rather than one In the last case thenetwork does not detect the small crack in element 8 becausethe effect of the larger cut hides the effect of the smaller cut

6 Conclusions

This paper presents a new methodology designed to assessexperimental damage using neural networks and anti-resonant frequencies A multilayer perceptron network wastrained with data obtained from a numerical model and

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

80

0

20

40

60

100

Damage level ()

Dam

age m

issin

g er

ror

lt10

gt80

10-20

20-30

30

-40

40

-50

50

-60

60

-70

70

-80

(a)

80

0

20

40

60

100

False

alar

m er

ror

10ndash20

20ndash3

0

30

ndash40

40

ndash50

50

ndash60

60

ndash70

Damage level ()

70

ndash80

lt10

gt80

(b)

Figure 13 Damage missing error (DME) and false alarm error (FAE) for different damage levels

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 1

(a)

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 2

(b)

0 2 4 6 8 10 12 14 16 18 20Element number

0

20

40

60

80

100

Stifn

ess r

educ

tion

()

Case 3

(c)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Element number

Stifn

ess r

educ

tion

()

Case 4

(d)

Figure 14 Experimental reduction of stiffness detected at each case

tested with experimental data The study addresses the setupof the neural network parameters and provides guidelines fortheir selection in similar damage assessment problems Twoexperimental cases verify the algorithm an 8-DOF mass-spring system and a beam with multiple damage scenarios

In both structures the algorithm is successful in assessingthe experimental damage The damage detected correspondsclosely with the experimental damage in all cases Theseresults show that it is possible to locate and quantify stru-ctural damage using only the anti-resonant informationobtained from point frequency response functions Henceanti-resonant frequencies are an attractive feature for use indamage assessment

This study shows that it is possible to assess experimentaldamage in real time in two different structures and hence

this work demonstrates the possibility of continuous moni-toring of the structural condition Nevertheless according tothe validation results the algorithmdisplays aminimum levelof damage that can be assessed with confidence which is 30in the case of an 8-DOF system and 40 for the beamTheseresults can be improved using a larger training database butthese changes imply larger training times

It is important to note that in this study anti-resonantfrequencies were identified with the same precision as reso-nant frequencies which is true for lightly damped structureswith low levels of experimental noise If this is not thecase anti-resonant frequencies might be difficult to identifyand the damage identification algorithm will be affected Anarea of further research is the identification of anti-resonantfrequencies in structures with high levels of damping or high

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

levels of experimental noise Another area of further researchis the use of anti-resonant frequencies obtained from thetransfer frequency response functions which are more sen-sitive to structural changes but require additional attentionespecially when pairing experimental and numerical anti-resonant frequencies

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been partially funded by Program U-INICIA VID 2011 Grant U-INICIA 1101 University ofChile and by the Fondo Nacional de Desarrollo Cientıficoy Tecnologico (FONDECYT) of the Chilean GovernmentProject 11110046

References

[1] V Meruane and W Heylen ldquoDamage detection with parallelgenetic algorithms and operational modesrdquo Structural HealthMonitoring vol 9 no 6 pp 481ndash496 2010

[2] VMeruane andWHeylen ldquoAn hybrid real genetic algorithm todetect structural damage using modal propertiesrdquo MechanicalSystems and Signal Processing vol 25 no 5 pp 1559ndash1573 2011

[3] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[4] B Kouchmeshky W Aquino J C Bongard and H LipsonldquoCo-evolutionary algorithm for structural damage identifica-tion using minimal physical testingrdquo International Journal forNumerical Methods in Engineering vol 69 no 5 pp 1085ndash11072007

[5] A Teughels G de Roeck and J A K Suykens ldquoGlobaloptimization by coupled local minimizers and its applicationto FE model updatingrdquo Computers and Structures vol 81 no24-25 pp 2337ndash2351 2003

[6] C Valdes-Gonzalez and J Gonzalez-Perez ldquoIdentification ofstructural damage in a vehicular bridge using artificial neuralnetworksrdquo Structural Health Monitoring vol 10 no 1 pp 33ndash48 2011

[7] B Sahoo and D Maity ldquoDamage assessment of structuresusing hybrid neuro-genetic algorithmrdquo Applied Soft ComputingJournal vol 7 no 1 pp 89ndash104 2007

[8] S Arangio and J L Beck ldquoBayesian neural networks for bridgeintegrity assessmentrdquo Structural Control andHealthMonitoringvol 19 no 1 pp 3ndash21 2012

[9] AMessina A Jones and E JWilliams ldquoDamage detection andlocalisation using natural frequency changesrdquo in Proceedings ofthe 1st Conference on Identification in Engineering Systems pp67ndash76 Swansea UK 1996

[10] R M Lim and D J Ewins ldquoModel updating using FRF datardquo inProceedings of the 15th International Seminar onModal Analysispp 141ndash163 Leuven Belgium 1990

[11] N Stubbs J T Kim and K Topole ldquoAn efficient and robustalgorithm for damage localization in offshore platformsrdquo in

Proceedings of the ASCE 10th Structures Congress pp 543ndash546San Antonio Tex USA 1992

[12] S F Masri A W Smyth A G Chassiakos T K Caughey andN F Hunter ldquoApplication of neural networks for detection ofchanges in nonlinear systemsrdquo Journal of Engineering Mechan-ics vol 126 no 7 pp 666ndash676 2000

[13] P Castellini and G M Revel ldquoExperimental technique forstructural diagnostic based on laser vibrometry and neuralnetworksrdquo Shock and Vibration vol 7 no 6 pp 381ndash397 2000

[14] C Zang and M Imregun ldquoStructural damage detection usingartificial neural networks and measured FRF data reducedvia principal component projectionrdquo Journal of Sound andVibration vol 242 no 5 pp 813ndash827 2001

[15] X FangH Luo and J Tang ldquoStructural damage detection usingneural network with learning rate improvementrdquo Computersand Structures vol 83 no 25-26 pp 2150ndash2161 2005

[16] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[17] A H Gandomi M G Sahab A Rahaei and M S GorjildquoDevelopment in mode shape-based structural fault identifica-tion techniquerdquoWorld Applied Sciences Journal vol 5 no 1 pp29ndash38 2008

[18] J L Zapico M P Gonzalez and K Worden ldquoDamage assess-ment using neural networksrdquo Mechanical Systems and SignalProcessing vol 17 no 1 pp 119ndash125 2003

[19] C B Yun J H Yi and E Y Bahng ldquoJoint damage assessmentof framed structures using a neural networks techniquerdquoEngineering Structures vol 23 no 5 pp 425ndash435 2001

[20] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[21] M Sahin andR A Shenoi ldquoVibration-based damage identifica-tion in beam-like composite laminates by using artificial neuralnetworksrdquo Proceedings of the Institution ofMechanical EngineersC vol 217 no 6 pp 661ndash676 2003

[22] W DrsquoAmbrogio and A Fregolent ldquoUse of antiresonances forrobust model updatingrdquo Journal of Sound and Vibration vol236 no 2 pp 227ndash243 2000

[23] E J Williams and A Messina ldquoApplications of the multipledamage location assurance criterionrdquo Key Engineering Materi-als vol 167 pp 256ndash264 1999

[24] M Dilena and A Morassi ldquoThe use of antiresonances for crackdetection in beamsrdquo Journal of Sound and Vibration vol 276no 1-2 pp 195ndash214 2004

[25] Y Bamnios EDouka andA Trochidis ldquoCrack identification inbeam structures usingmechanical impedancerdquo Journal of Soundand Vibration vol 256 no 2 pp 287ndash297 2002

[26] T Inada Y ShimamuraA Todoroki andHKobayashi ldquoDevel-opment of the two-step delamination identification method byresonant and anti-resonant frequency changesrdquoKeyEngineeringMaterials vol 270ndash273 no 3 pp 1852ndash1858 2004

[27] D Wang and H Zhu ldquoWave propagation based multi-crackidentification in beam structures through anti-resonancesinformationrdquo Key Engineering Materials vol 293-294 pp 557ndash564 2005

[28] V Meruane and W Heylen ldquoStructural damage assessmentwith antiresonances versus mode shapes using parallel geneticalgorithmsrdquo Structural Control and Health Monitoring vol 18no 8 pp 825ndash839 2011

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

[29] M A Arbib The Handbook of Brain Theory and NeuralNetworks Bradford Book 2003

[30] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[31] M Beale andHDemuthNeural Network Toolbox Version 703Release The MathWorks 2012

[32] D W Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo Journal of the Society for Industrial andApplied Mathematics vol 11 no 2 pp 431ndash441 1963

[33] M I Friswell J E T Penny and S D Garvey ldquoA combinedgenetic and eigensensitivity algorithm for the location of dam-age in structuresrdquo Computers and Structures vol 69 no 5 pp547ndash556 1998

[34] J J Lee J W Lee J H Yi C B Yun and H Y Jung ldquoNeu-ral networks-based damage detection for bridges consideringerrors in baseline finite element modelsrdquo Journal of Sound andVibration vol 280 no 3-5 pp 555ndash578 2005

[35] KDHjelmstad and S Shin ldquoDamage detection and assessmentof structures from static responserdquo Journal of EngineeringMechanics vol 123 no 6 pp 568ndash576 1997

[36] S V Barai and P C Pandey ldquoVibration signature analysisusing artificial neural networksrdquo Journal of Computing in CivilEngineering vol 9 no 4 pp 259ndash265 1995

[37] J E Mottershead ldquoOn the zeros of structural frequencyresponse functions and their sensitivitiesrdquo Mechanical Systemsand Signal Processing vol 12 no 5 pp 591ndash597 1998

[38] L Prechelt ldquoEarly stoppingmdashbut whenrdquo in Neural NetworksTricks of the Trade vol 1524 of Lecture Notes in ComputerScience pp 55ndash69 Springer New York NY USA 1998

[39] M H Richardson and D L Formenti ldquoParameter estimationfrom frequency response measurements using rational fractionpolynomialsrdquo in Proceedings of the 1st International ModalAnalysis Conference pp 167ndash181 1982

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of