Substructure Vibration NARX Neural Network Approachfor Statistical Damage Inference

Linjun Yan, M.ASCE1; Ahmed Elgamal, M.ASCE2; and Garrison W. Cottrell3

Abstract: A damage detection approach is developed using nonlinear autoregressive with exogenous inputs (NARX) neural networks anda statistical inference technique. Within a large spatially extended dynamic system, an instrumented local substructure may be representedby a neural network, to predict the dynamic response of a given sensor from that of its neighbors. Without change in the system properties,the network prediction error will follow a stable statistical distribution. To infer damage, change in the prediction error variance as evaluatedby the statistical inference standard F test is utilized as a sensitive indicator. Validation of the described procedure is undertaken using twoexperimental data sets (from the Los Alamos National Laboratory in Los Alamos, NM). Reduced stiffness and nonlinear response of a mass-spring system is documented in the first set, while joint damage in a frame structure is explored in the second. Favorable results are obtained inboth cases with linear/nonlinear and single/multidamage patterns. Overall, the proposed framework may be particularly efficient for largespatially extended sensor network situations, where local condition assessment may be conducted based on the response of a few neighboringsensors. DOI: 10.1061/(ASCE)EM.1943-7889.0000363. © 2013 American Society of Civil Engineers.

CE Database subject headings: Structural Health Monitoring; Vibration; Damage; Neural networks; Statistics.

Author keywords: Structural health monitoring; Damage detection; Neural network; Damage indicator; Statistical inference; Sensornetwork.

Introduction

Health monitoring (damage identification) of structural systemscontinues to receive a growing level of interest with advances in therelated analytical and experimental techniques (Chang 1997, 2005,2007; Ou et al. 2005). Major efforts have been focused on exploringthe potential of identification, based on changes in the vibrationalsystem characteristics (e.g., Farrar and Doebling 1997; Doeblinget al. 1998; Sohn et al. 2003a; Overbey and Todd 2007; Overbeyet al. 2007).

For a civil engineering structure, actual dynamic response underdifferent damage scenarios of interest is typically unavailable orscarce. When such data are available, attention may be focused oninferring damage based on changes in the identified global systemproperties such as modal parameters and/or the derivatives thereof(e.g., Alampalli et al. 1997; Todd et al. 2004; Zonta et al. 2008;Nayeri et al. 2008). To build a knowledge base, a numerical modelmay be calibrated by available data, and then employed to explorethe impact of different potential damage states. This may necessitatean elaborate numerical model that requires labor-intensive fine-tuning and may include significant uncertainties from lack of in-formation and unavoidablemodeling limitations (Sohn et al. 2003a).

The important situation of spatially localized damage has beenshown to pose a particular challenge in view of the potential minimalimpact on the overall dynamic system properties (Farrar and Jaur-egui 1996; Farrar and Doebling 1997; Humar et al. 2003; Sohn et al.2003a). In terms of numerical simulation, such a damage patternimposes high demands on the geometric and material model fidelityand accuracy.

In view of the aforementioned challenges, significant researchefforts have been focused on development of damage detectionapproaches that do not depend on numerical modeling and/or globalsystem property identification. In this regard, different time series–prediction techniques have been employed, including neural networks(e.g., Masri et al. 1996; Nakamura et al. 1998; Conte et al. 1994; Kaoet al. 2003; Wu et al. 2002; Xu et al. 2003), autoregressive andautoregressivewith exogenous inputs (AR-ARX,withparameters thatcorrelate to the overall dynamic system properties)models (e.g., Sohnet al. 2001; Sohn and Farrar 2001; Sohn et al. 2003b; Lei et al. 2003;Nair et al. 2003), and response surface models (e.g., Iwasaki et al.2002, 2003; Casciati et al. 2003a,b,c). Among others, neural networksremain a popular techniquewith strong capabilities for approximatingresponse of linear and nonlinear systems (Bishop 1995).

Studies from the last 15 years show that the combination of timeseries–prediction techniques and statistical inference methods canprovide an effective solution (Masri et al. 1996; Nakamura et al.1998; Sohn et al. 2001; Sohn and Farrar 2001). The basic idea is thatfor each structure/substructure of interest, a time-series model is firstdeveloped to predict the measured baseline dynamic system re-sponse (intact state). Without significant change in the structure, theprediction error will remain at a stable statistical distribution. Assuch, variation in the prediction error distribution can be utilized asa sensitive and robust damage indicator (Sohn and Farrar 2001).

In this study, the nonlinear autoregressive with exogenous inputs(NARX) neural network approach (Leontaritis and Billings 1985;Chen and Billings 1989; Korbicz and Janczak 1996; Demuth andBeale 1998) was adopted to represent the dynamic system response.

1Project Engineer, TobolskiWatkins Engineering, 9246 Lightwave Ave.,Suite 140, San Diego, CA 92123. E-mail: james.linjun.yan@gmail.com

2Professor, Dept. of Structural Engineering, Univ. of California-SanDiego, La Jolla, CA 92093-0085 (corresponding author). E-mail: elgamal@ucsd.edu

3Professor, Dept. of Computer Science and Engineering, Univ. ofCalifornia-San Diego, La Jolla, CA 92093-0404. E-mail: gary@ucsd.edu

Note. This manuscript was submitted on September 28, 2010; approvedon December 5, 2011; published online on December 8, 2011. Discussionperiod open until November 1, 2013; separate discussionsmust be submittedfor individual papers. This paper is part of the Journal of EngineeringMechanics, Vol. 139, No. 6, June 1, 2013. ©ASCE, ISSN 0733-9399/2013/6-737–747/$25.00.

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On this basis, a statistical inference procedure was employed todetect structural damage based on change of the network predictionerror variance. Within this framework, local damage detection isfurther facilitated by developing a separate NARX network for eachlocal substructure of interest. Each network predicts dynamic re-sponse measured at a given sensor from that of its adjacent neigh-bors. As such, synchronized data would only be needed from a smallnumber of adjacent sensors.

In the following sections, the overall analysis framework is firstdescribed. NARX neural networks are then discussed, including net-work configuration and training. The statistical inference procedureemployed to compute the damage indicator is then presented. Finally,the developed framework is verified by two experimental data sets fromthe Los Alamos National Laboratory (LANL) in Los Alamos, NM.

Analysis Framework

The employed damage detection approach assumes that the in-vestigated structure is monitored by an array of neighboring sensors.Rather than addressing the structure as a single system, local sub-structures of interest are defined, and each is assessed independentlyusing the corresponding sensor network measurements. Such a de-centralization strategy, which also allows for distributed computingas needed, is desirable for practical implementation of large/densesensor networks. Within this context, the approach can be summa-rized in the following fashion (Fig. 1):1. Based on the spatial sensor layout, local substructures may be

defined. Each substructure is monitored by a set of sensors,with one designated as the master. Response of the master willbe predicted from that of its adjacent neighboring sensors(which provide input to the predictive network).

2. Usingstationaryambientvibration(e.g., fromtrafficonabridge)or a prescribed external excitation (e.g., shaker-induced white-noise excitation), the system dynamic response is recorded.For each substructure, an associated NARX neural network(Korbicz and Janczak 1996; Demuth and Beale 1998) is de-veloped to predict the measured response at the master senorfrom that of the neighboring sensors. Thus, the dynamic char-acteristics of the substructure are approximately implicitly iden-tified and represented by the associated network. Both thenetwork configurations including weights and biases (Bishop1995) and the time history of network prediction error arearchived as a reference baseline state.

3. The system response under the same excitation conditions isperiodically recorded. For each substructure, the developednetwork is employed to predict the dynamic response at themaster sensor and a new time history of prediction error iscalculated. Comparing this new time history with that of thebaseline state, a damage indicator is computed based onchanges in the error variance.

4. By assembling the spatial configuration of the computeddamage indicators, the occurrence, location, and relativeseverity of damage in the structural system may be inferred.

Note that it may be desirable to repeat Step 2 from the list fordifferent environmental and operational conditions (Sohn and Farrar2001) to build a baseline database consisting of a range of referencestates of interest. For each recorded data set, the reference state withthe closest environmental and operational conditions may be usedto perform the evaluation in Step 3. For a laboratory experiment(the applications shown in this paper), one reference state wasadequate as the environmental and operational conditions (in-cluding excitation pattern and mass spatial distribution) wererelatively stable.

NARX Neural Network

Network Configuration

As mentioned earlier, each substructure is monitored by a mastersensor andM neighboring sensors (M is the number of employedneighboring sensors). In this sense, the problem can be formulatedas a multiple-input, single-output nonlinear system. Given thediscrete-time nature of the data, the system can be defined bya NARX model (Leontaritis and Billings 1985; Chen and Billings1989)

yðtÞ ¼ f ½xðtÞ� ð1Þ

with xðtÞ a vector defined by

Fig. 1. Analysis framework

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xðtÞ ¼ ½xiðtÞ� ¼�y�t2 ny

�. . . yðt2 1Þ u1ðt2 nuÞ . . .

u1ðt2 1Þ u1ðtÞ . . . uMðt2 nuÞ . . . uMðt2 1Þ uM�t�� ð2Þ

inwhichyðtÞ denotes the output (master sensor response) at time stept; f 5 nonlinear function representing the substructure; umðtÞ5mthinput (measured by the mth neighboring sensor) at time-step t; andnu and ny 5 the input and output orders, respectively.

Herein, the function f is approximated by a multilayers per-ceptron as Bishop (1995) shows that such a neural network, withsigmoidal activation functions, provides strong capabilities forrepresenting continuous nonlinear systems (Bishop 1995 may beconsulted for a figure illustrating the multilayers perceptron networkstructure). The resulting framework is known as a NARX network(Korbicz and Janczak 1996; Demuth and Beale 1998), having oneinput layer of M3 ðnu 1 1Þ1 ny units, one hidden layer, and one

single unit output layer. For this single unit output case, it wasverified that setting the number of hidden units equal to the numberof input units ensures adequate representation capability whilecontrolling the network size (Bishop 1995).

The feed-forward network mapping from the input vectorxðtÞ5 ½xiðtÞ� to its output byðtÞ is defined by (Bishop 1995)

byðtÞ ¼ PM�ðnuþ1Þþnyj¼1 wð2Þ

j ghPM�ðnuþ1Þþny

i¼1 wð1Þji xiðtÞ þ wð1Þ

j0

i

þ wð2Þ0

ð3Þ

where wð1Þji denotes weight going from input unit i to hidden unit j;

wð1Þj0 denotes bias for hidden unit j; gð × Þ denotes activation function

for hidden units; wð2Þj denotes weight going from hidden unit j to the

single output unit; and wð2Þ0 denotes bias for the single output unit.

As such, the output layer is linear, and a hyperbolic tanh isadopted as the activation function gð × Þ of hidden units, defined by(LeCun et al. 1998)

Fig. 2. An 8-DOF mass-spring system attached to a shaker withaccelerometers mounted on each mass (Sohn and Farrar 2001; Sohn, H.,and Farrar, C. R., “Damage diagnosis using time series analysis ofvibration signals.” Smart Materials and Structures, Vol. 10, Issue 3,pp. 446–451, June 2001. © IOP Publishing. Reproduced by permissionof IOP Publishing. All rights reserved)

Fig. 3.A typical bumper used to simulate nonlinear damage (Sohn andFarrar 2001; Sohn, H., and Farrar, C. R., “Damage diagnosis using timeseries analysis of vibration signals.” Smart Materials and Structures,Vol. 10, Issue 3, pp. 446–451, June 2001. © IOP Publishing. Repro-duced by permission of IOP Publishing. All rights reserved)

Table 1. Features of the Ten 8-DOF System Experimental Scenarios

Scenarionumber

Occurrenceof damage

Damagelocationa

Damagetype

Stiffnessreduction

(damage level,percent)

Initial clearance(damage level,

mm)

1 No — — — —

2 No — — — —

3 Yes 1 Linear 24 —

4 Yes 5 Linear 24 —

5 Yes 7 Linear 24 —

6 Yes 1 Nonlinear — 0 (High)7 Yes 5 Nonlinear — 0 (High)8 Yes 7 Nonlinear — 0 (High)9 Yes 7 Nonlinear — 0.2 (Moderate)10 Yes 7 Nonlinear — 0.4 (Low)aDamage locations are shown in Fig. 4.

Fig. 4. Masses and sensors (shaded), damage locations (1–7), and the8-DOF system substructures labeled by the numbers in circles

Table 2. Definition of Substructures in the 8-DOF System

Substructure Master sensor Neighboring sensorsa

1 1 0, 22 2 1, 33 3 2, 44 4 3, 55 5 4, 66 6 5, 77 7 6, 88 8 7aSensor locations are shown in Fig. 4.

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g�aj� ¼ 1:7159 tanh

�2aj=3

� ð4Þ

where aj 5PM3 ðnu11Þ1ny

i51 wð1Þji xiðtÞ1wð1Þ

j0 5 the weighted sum ofinputs for hidden unit j.

Network Training

In this study, the minimum-to-maximum range of measured re-sponse at each sensor was normalized to ½21, 1�. For each sub-structure, a sequence of input-output patterns f xðtÞ yðtÞ g can then

be sampled from these normalized time histories. The patterns wererandomly partitioned into a training set (80% of the total patterns)and a validation set (20% of the total patterns). Based on the trainingset, the standard backpropagation algorithm (LeCun et al. 1998)was adopted for network training by sequentially minimizing thesquare error between the network outputbyðtÞ and the correspondingtarget yðtÞ

EðtÞ ¼ 12�yðtÞ2byðtÞ�2 ð5Þ

To best utilize the knowledge contained in the data, the training setwas repeatedly used,with each training loop over all patterns definedas an epoch. After each epoch, the average errors of the training andvalidation sets were computed. The optimal network configurationwas selected according to the minimum error of the validation set toavoid overfitting (Bishop 1995; Hastie et al. 2001).

Damage Indicator

After a NARX network is trained, the dynamic characteristics ofthe corresponding substructure are implicitly represented by thenetwork, which then may be employed to predict the structural re-sponse under different states. Herein, the prediction error of the

Table 3. Parameters of the 8-DOF System

Parameter Value

Number of time steps in each record 4,080Input order of NARX model, nu 20Output order of NARX model, ny 20Number of network training epochs 50Length of eðtÞ, Ne 4,060Length of ɛðtÞ, Nɛ 4,060Length of each divided segment, n 20Number of segments Np for eðtÞ 203Number of segments Nq for ɛðtÞ 203Confidence level, a (%) 99

Fig. 5. Sample comparison of network predictions and corresponding master sensor measurements of structural response of the 8-DOF mass-springsystem at the baseline state (Scenario 1), along with the residual errors (for clarity, only a 500-time-step segment is shown)

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baseline state at time-step t is denoted by eðtÞ, t5 1, . . .Ne, whereNe

is the baseline time history duration (number of data points).Similarly, the prediction error of any other investigated state at time-step t is denoted by ɛðtÞ, t5 1, . . .Nɛ , where Nɛ is the associatedtime history duration.Without significant change in the substructure(e.g., from occurrence of damage), the trained network will deliverconsistent prediction performance, with eðtÞ and ɛðtÞ expected toexhibit similar statistical distributions. In other words, change in thestatistical distribution of the prediction error may be utilized as thedamage indicator (Sohn and Farrar 2001).

As the prediction error generally follows a distribution witha mean value around zero, most efforts have been based on assess-ment of the changes in variance (or SD). For instance, Sohn andFarrar (2001) evaluated such changes by the F test, which assumesthat both eðtÞ and ɛðtÞ follow normal distributions. However, thisnormality condition may be hard to achieve in practice. In view ofthis fact, Sohn andFarrar (2001) suggested that the hypothesis test beperformed against a threshold value from a modified F distribution.

Herein, this issue was addressed in an equivalent way thatweakens the strong normality requirement, only assuming eðtÞ andɛðtÞ to exhibit stable distributions. Based on this approach, for eachsubstructure, a damage indicator can be computed as follows:1. Shuffle the time-step order of the eðtÞ and ɛðtÞ time histories

independently to eliminate the correlation between neighbor-ing prediction errors.

2. Divide the shuffled eðtÞ and ɛðtÞ time histories into segments ofequal length, where the number of time steps in each segmentis denoted by n, the number of eðtÞ segments is denoted byNp 5Ne=n, and the number of ɛðtÞ segments is denoted byNq 5Nɛ=n.

3. Thus, for each segment, the sequence of prediction errors [eðtÞor ɛðtÞ] is approximately independent and identically distrib-uted, and their sum can be computed as p5

Pnt51eðtÞ and

q5Pn

t51ɛðtÞ. According to the central limit theory (Laplace1812; Lyapunov 1954; Spiegel 1992), both p and q will ap-proach normal distributions as n increases. Based on a numberof conducted trials, it was verified that n5 20 yields satisfac-tory results as will be further discussed in the following text.

4. Compute the damage indicator D as the sample variance ratioof p and q

D ¼ s2ðqÞs2ðpÞ ð6Þ

Based onD, the similarity of eðtÞ and ɛðtÞ can be indirectly evaluatedby testing the null hypothesis H0:s2ðqÞ5s2ðpÞ against the one-sidealternative H1:s2ðqÞ.s2ðpÞ. It can be shown that D exhibits theF-distribution with Nq 2 1 and Np 2 1 degrees of freedom underthe null hypothesis (Miller 1997). Thus, given a confidence level a,the theoretical threshold value F12a

Nq 2 1,Np 2 1 can be found. If D ex-ceeds this threshold, the null hypothesis H0 is rejected, whichindicates a significant change in the substructure.

Note that the computation of D only requires local informationfrom sensor data within the substructure. This is desirable in prac-tical implementations as it allows for locally distributed comput-ing and facilitates large-scale, spatially extended monitoring efforts.

Fig. 6.Neighboring prediction errors essentially display no correlationafter the time-step order was randomly reshuffled (Substructure 1 at thebaseline state, Scenario 1; only a 500-time-step segment is shown forclarity)

Fig. 7. Comparison of the histogram of p and the theoretical normaldistribution with the same mean value and SD for Substructure 1 at thebaseline state (Scenario 1)

Fig. 8. Distributions of p for the baseline state (Scenario 1) and q forScenarios 2, 8, 9, and 10 (Substructure 7)

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While the computation of D is an independent process for eachsubstructure, the damage indicators can be compared with oneanother as long as the same a, Np, and Nq values are used. In thisregard, it would be expected to find that a larger value ofD indicatesthat the corresponding substructure: (1) has developed a higher levelof damage or (2) is closer to the damage location. Based on this logic,the location and relative severity of damage may be inferred byassembling the spatial configuration of the computed damageindicators.

Experimental Verification

Eight-DOF Mass-Spring System

Testing ConfigurationAn eight degree-of-freedom (8-DOF) experimental system wasdesigned and constructed to study damage identification problems at

the Los Alamos National Laboratory (Sohn and Farrar 2001). Thesystem (Fig. 2) consists of eight translating masses (each being analuminum disk 25.4-mm thick and 76.2 mm in diameter witha central hole) connected by springs. All masses slide on a highlypolished steel rod that allows translations along the rod axis only(Fig. 2).

The nominal value of mass 1 was 559.3 g, and the nominal valuefor masses 2–8 was 419.4 g each. Using a 215N peak force elec-trodynamic shaker, random excitation was applied to mass 1 andrecorded by a force transducer (sensor 0). The response of eachmasswas recorded by an accelerometer. In Sohn andFarrar (2001), furthermodeling details are presented and the data were employed to verify adamage diagnosis procedure combining autoregressive and autore-gressive with exogenous inputs (AR-ARX) techniques.

The baseline undamaged configuration of the system was thestate with an identical linear spring (stiffness k5 56:7 kN=m) con-necting the masses (Fig. 3). Linear damage was defined as a changein the system stiffness characteristics and simulated by replacing

Fig. 9. Damage indicators computed for Scenarios 2–10 of the 8-DOF mass-spring system, using Scenario 1 as the baseline state, where the actualseverity of damage in each substructure (Table 1 and Fig. 4) is indicated by circles (no damage); diamonds (low-level damage);triangles (moderate-leveldamage); or boxes (high-level damage)

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an original spring with another of a lower spring constant. Thus,the degree of linear damage was controlled by the percentage re-duction in the spring constant. Nonlinear damage denoted massesimpacting one another upon traversing a predefined initial clearance(Fig. 3), with a smaller initial clearance representing a higher level ofnonlinearity.

Using this setup (Figs. 2 and 3), a series of experiments (303 intotal) was conducted. Scenarios of linear and nonlinear damagestates at different locations between the adjacent masses weresimulated (Sohn and Farrar 2001).

Of this rich data set, 10 recorded representative experimentalscenarios (Table 1) were investigated herein. In these scenarios,random events were generated by the shaker with the RMS am-plitude level of excitation corresponding to a 5-V input. Eachemployed time-history was sampled at 500 Hz and consists of 4,080data points.

Data AnalysisFollowing the proposed analysis approach (Fig. 1), eight sub-structures were defined as shown in Fig. 4 and Table 2. Herein,Scenario 1 (Table 1) was employed as the baseline state to developa NARX network for each substructure. Thereafter, these networkswere applied to Scenarios 2–10 for damage inference purposes. Therelated parameters for this study are summarized in Table 3.

For each substructure, the associated network was employed topredict the response measured by the master sensor. For the baselinestate, Fig. 5 shows a representative time-history comparison ofnetwork predictions and corresponding sensor measurements, alongwith the residual errors. It is seen throughout that the prediction ofeach network closely matches the corresponding master sensormeasurement. Thus, it can be concluded that the dynamic properties

of each substructure in the baseline state are now adequately rep-resented by the associated NARX network.

The residual prediction error [eðtÞ or ɛðtÞ] time histories (Fig. 5)were systematically generated for the different experimental sce-narios of Table 1. Order of the time step in each history was thenrandomly shuffled such that the values of neighboring steps becomeessentially uncorrelated (e.g., Fig. 6). Thereafter, each shuffled eðtÞor ɛðtÞ time history was divided into segments of equal length(n5 20 as mentioned earlier), and the sum of each segment (p or q)was computed, resulting in the desired normal distribution as shown,for instance, in Fig. 7.

If the properties of any substructure do not significantly change, qwould be expected to exhibit a distribution that closely matches p.For instance, the results related to Substructure 7 for Scenarios 1, 2,and 8–10 (Table 1) are shown in Fig. 8. It is seen that for Scenario 2with no damage, q exhibits a very close distribution to p of the

Fig. 10. Experimental setup of the 3-story frame model [Sohn et al. 2003b; Sohn, H., Allen, D. W., Worden, K., and Farrar, C. R., Structural HealthMonitoring (Vol. 2, Issue 1), pp. 57–74, copyright © 2003 by Sage Publications, Reprinted by Permission of SAGE]

Table 4. Ten Experimental Scenarios of the 3-Story Frame

Scenario number Damage occurrence Damage locationa Damage level

1 No — —

2 No — —

3 No — —

4 No — —

5 Yes 1C D106 Yes 1C D057 Yes 1C DBB8 Yes 3A DBB9 Yes 1C/3A DBB/DBB10 Yes 1C/3A DBB/DBBaDamage location indicates floor number and corner per Fig. 10.

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baseline case (Scenario 1). As to Scenarios 8–10 with damage inSubstructure 7, the distributions of q are significantly differentfrom p. Such difference increases with the severity of damage,particularly in terms of the variance.

Based on p and q, the damage indicator D was computed foreach substructure using Eq. (6). Fig. 9 shows the results forScenarios 2–10. Log scale was employed for the vertical axis(Fig. 9) to clearly display changes within the relatively low valuerange, around the threshold value of 1.39, which corresponds to a99% confidence level.

It is seen that for Scenario 2 (no damage), the eight computeddamage indicators were all below the threshold (as expected). Asto Scenarios 3–10 with linear or nonlinear damage, each is seen todisplay certain damage indicators that exceed the threshold. Inother words, the occurrence of damage in these scenarios wassuccessfully detected.

Damage location may be inferred from the computed damageindicator relative values (Fig. 9). For instance, Scenario 3 (lineardamage at Location 1) appropriately shows damage indicators forSubstructures 1 and 2 (Fig. 4) that significantly exceed the threshold(Fig. 9). Similarly, damage locations for Scenarios 4–10 can all besuccessfully inferred based on the computed damage indicatorrelative values (Fig. 9).

In Fig. 9, it is seen that substructures with no damage may alsodisplay damage indicator values that exceed the threshold. However,the largest values were always associated with the actually damagedsubstructures. Finally, Scenarios 8–10 with high to low nonlineardamage at the same location (Table 1) displayed indicator values thatproportionately agree with these levels (Fig. 9). In other words, therelative severity of damagemay be inferred from the relative damageindicator value (note the Log scale in Fig. 9).

Three-Story Frame Structure

Experimental SetupA3-story framemodel (Fig. 10) was constructed of Unistrut columnsand aluminum floor plates at LANL for conducting structural healthmonitoring studies (AdamsandFarrar 2002;Worden et al. 2002; Sohnet al. 2003b). The floors were represented by 12.7 mm (0.5 in.) thickplates, and thebasewas a 38.1mm(1.5 in.) thick plate [Fig. 10(a)]. Foreach floor/base plate, the four corners were numbered from A to Das shown in Fig. 10(b). The structure was supported on four airmount isolators to allow free horizontal movement [Fig. 10(a)]. Ashaker was coupled to the structure by a stinger connected to atapped hole 95.25 mm (3.75 in.) away from corner D of the baseplate [Figs. 10(b and d)] so that both translational and torsionalmotions may be excited.

Table 5. Parameters of the 3-Story Frame Structure

Parameter Value

Number of time steps in each record 8,180Input order of NARX model, nu 20Output order of NARX model, ny 20Number of network training epochs 50Length of eðtÞ, Ne 8,160Length of ɛðtÞ, Nɛ 8,160Length of each divided segment, n 20Number of segments Np for eðtÞ 408Number of segments Nq for ɛðtÞ 408Confidence level, a (%) 99

Fig. 11. Sample comparison of network predictions and corresponding sensor measurements of structural response of the 3-story frame structure atthe baseline state (for clarity, only a 500-time-step segment is shown)

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In this experiment, damagewas simulated in thefloor-column jointsby loosening the bolts [Figs. 9(b and c)]. All bolted connectionswere tightened to a torque value of 6,779 N×mm (60 in.×lbs) in theundamaged state.Using this setup, a series of experiments (270 in total)was conducted, encompassing damage scenarios of different severitiesat single or multiple locations (Sohn et al. 2003b).

Ten recorded representative experimental scenarios were in-vestigated herein as summarized in Table 4. Three levels of damage(Table 4) were included in these scenarios: Level 1 (D10, a torquevalue of 1,130 N×mm (10 in.×lbs) was left on the bolt); Level 2 (D05,a torque value of 565N×mm(5 in.×lbs)was left on the bolt); andLevel3 (DBB, the bolt was removed).

For each scenario, a dynamic test of the structure was performed(Sohn et al. 2003b). The RMS voltage of the shaker was 8 V, andrandom excitation was generated. The structural response in theshaking direction was measured by 24 piezo-electric single-axisaccelerometers, each mounted on an aluminum block that wasattached to the structure by hot glue. Two accelerometers wereplaced at each joint with one attached to the floor and the otherattached to the column [Fig. 10(b)]. Each employed time-history

was sampled at 1,600 Hz and consisted of 8,180 points. Detailedinformation about this experimental setup is presented in Sohnet al. (2003b).

Data AnalysisFor this model, the substructures of interest are the 12 floor-columnjoints. As each joint was monitored by a pair of accelerometers,a NARX network was developed to simply predict the structuralresponse of the floor (master sensor) from the adjacent column re-sponse (neighboring sensor). Herein, Scenario 1 was employed asthe baseline state to train the networks for identification purposes.Thereafter, the networks were applied to Scenarios 2–10 for damageinference purposes. The parameters of this study are summarized inTable 5.

Fig. 11 displays a representative time-history comparison ofeach master sensor measurement and the corresponding networkprediction, showing the expected close match in this no-damagestate. Thus, it can be concluded that the properties of each intactjoint were now accurately represented by the correspondingNARXnetwork.

Fig. 12. Damage indicators computed for Scenarios 2–10 of the 3-story frame structure, using Scenario 1 as the baseline state

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Fig. 12 shows the damage detection results for Scenarios 2–10(Table 4). It is seen that for Scenarios 2–4 with no damage, thecomputed indicators were all below the threshold value of 1.26(corresponding to a 99% confidence level). As to Scenarios 5–10(Table 4), the occurrence of damage was always detected, and theindicator of the highest value (Fig. 12) clearly corresponded to theactual location (Table 4). Finally, by comparing the results ofScenarios 5–7 (Fig. 12), it is seen that the indicator value increasesin accordance with the level of damage at 1C (Table 4).

Summary and Conclusions

In this paper, a structural damage detection approach based onNARX neural networks and statistical inference techniques waspresented and investigated. This approach relies on data froma spatially localized substructure that is being monitored by a set ofneighboring sensors. In each substructure, a master sensor is se-lected and an associated NARX network is developed to predict theresponse at this sensor from that measured by its neighbors.Withoutsignificant change in a substructure, the network prediction errorfollows a stable statistical distribution. Change in the predictionerror variance as evaluated by the modified F test is utilized as thedamage indicator.

To verify applicability, this damage detection approach wasemployed to study data from two LANL experiments: an 8-DOFmass-spring system and a 3-story frame structure. Satisfactoryresults were achieved showing that:1. For each substructure, the master sensor response was

closely predicted using the associated trained baseline NARXnetwork;

2. The employed damage indicator performed satisfactorily indelineating the investigated damage/no-damage scenarios;

3. The damage locations (one or more) always corresponded tothe highest damage indicator values; and

4. The damage indicator value increased with the relative sever-ity of damage.

As a direction for future research, implementation of the dis-cussed NARX neural networks approach for full-scale structuralresponse scenarios is needed, to further assess the actual practicalranges of applicability.

Acknowledgments

Support for this research was provided by the U.S. National ScienceFoundation, under ITR Grant No. 0205720. This support is grate-fully acknowledged. The authors also wish to thank the CaliforniaInstitute for Telecommunications and Information Technology(UCSDCalit2, http://www.calit2.net) for providing fellowship sup-port to this project. Finally, the experimental data sets studied in thispaper were provided by Dr. Francois Hemez and Dr. Charles Farrarat Los Alamos National Laboratory and Professor Hoon Sohn atCarnegie Mellon University. The authors are most grateful for thisvaluable contribution.

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