Satpal et al. International Journal of Advanced Structural Engineering 2013, 5:2http://www.advancedstructeng.com/content/5/1/2

ORIGINAL RESEARCH Open Access

Structural health monitoring of a cantilever beamusing support vector machineSatish B Satpal1, Yogesh Khandare1, Anirban Guha1* and Sauvik Banerjee2

Abstract

In this article, the effectiveness of support vector machine (SVM) is examined for health monitoring of beam-likestructures using vibration-induced modal displacement data. The SVM is used to predict the intensity or location ofdamage in a simulated cantilever beam from displacements of the first mode shape. Twelve levels of damageintensities have been simulated at 12 locations, and six levels of white Gaussian noise have been added, therebyobtaining 1,008 simulations. About 90% of these are used for training the SVM, and the remaining are used fortesting. The trained SVM is able to predict damage intensity and location of all the training set data with nearly100% accuracy. The test set data reveal that SVM is able to predict damage intensity and damage location witherrors varying from 0.28% to 4.57% and 0% to 20.3%, respectively, when there is no noise in the data. Addition ofnoise degrades the performance of SVM, the degradation being significant for intensity prediction and less fordamage location prediction. The results demonstrate the use of SVM as a powerful tool for structural healthmonitoring without using the data of healthy state.

Keywords: Support vector machine, Structural health monitoring, FE simulation, noisy data

IntroductionThe process of implementing a damage identificationstrategy for aerospace, civil, and mechanical engineeringinfrastructure is referred to as structural health monitoring(SHM) (Farrar and Worden 2007). Any changes in thematerial and/or geometric properties of these structuresare considered as damage. The modal parameters (naturalfrequencies, mode shapes, and modal damping) arefunctions of the physical parameters (mass, stiffness, anddamping), and therefore, it is reasonable to assume thatthe existence of damage leads to changes in the modalproperties of the structure.Several researchers have studied vibration-based methods

and a detailed literature review has been reported by Zouet al. (1999). Though the changes in natural frequenciesgive useful information regarding the structure's status, i.e.,whether it is damaged or not, that is not sufficient to locatethe damage. As a result, methods based on mode shapedata and frequency response function of the structure havebeen subject of recent studies (Mal et al. 2005). It has been

* Correspondence: anirbanguha1@gmail.com1Department of Mechanical Engineering, Indian Institute of TechnologyBombay, Powai, Mumbai 400076, IndiaFull list of author information is available at the end of the article

© 2013 Satpal et al. This is an Open Access arti(http://creativecommons.org/licenses/by/2.0), wprovided the original work is properly cited.

demonstrated that the approximate location and severity ofan unknown defect can be determined using a damageindex, which is calculated by comparing the changes inextracted signal features at the healthy and damaged states.Both built-in piezoelectric patches and the laser vibrometerhave been used as response sensors in an effort to examinetheir sensitivity for health monitoring of metallic andcomposite structures (Banerjee et al. 2009).Traditional neural network approaches have suffered

difficulties with generalization, producing models thatover-fit the data. This is a consequence of theoptimization algorithms used for parameter selectionand the statistical measures used to select the ‘best’model (Gunn 1998). On the other hand, support vectormachine (SVM) is a novel machine learning methodbased on statistical learning theory introduced by Vapnik(1999) and is gaining popularity due to many features.SVM requires few training data and avoids over-fittingof data. Shan et al. (2012) used the multi-class patternclassification algorithm of C-support vector machineand the regression algorithm of ε-support vector machineto identify the damage location and damage extent,respectively, in a railway continuous girder bridge.SHM of aerospace structures based on dynamic strain

cle distributed under the terms of the Creative Commons Attribution Licensehich permits unrestricted use, distribution, and reproduction in any medium,

Satpal et al. International Journal of Advanced Structural Engineering 2013, 5:2 Page 2 of 7http://www.advancedstructeng.com/content/5/1/2

measurements using SVM classification has been studiedby Loutas et al. (2012). The use of SVM for prediction offault in power systems has been demonstrated by Kumaret al. (2011). They used support vector classification topredict the damage location.Bulut et al. (2005) demonstrate the damage detection

in civil structure using SVM classifier and wavelets.They found that the SVM was a robust classifier in thepresence of noise, whereas wavelet-based compressiongracefully degrades its classification accuracy. Kurekand Osowski (2010) used SVM classifier for faultdiagnosis of the broken rotor bars of squirrel-cageinduction motor. A new monitoring and diagnosticmethod using support vector data description (SVDD)was proposed by Chendong et al. (2012), which onlyneeds samples under healthy condition. They selectedseveral nodes of the monitored structure and decom-posed the signals from these nodes with waveletpacket transform. To monitor the structural healthefficiently, they assembled a combined feature usingwavelet packet energy distributions of these nodes.The feature was then applied as the input to a SVDDclassifier.In this work, we have considered normalized displace-

ment data of the first mode of a cantilever beam fortraining ε-SVM in regression problem. This approachneeds only damaged state data of the structure (beam).Finite element (FE) analysis of the beam is carried out inthe FE package ABAQUSW (ABAQUS Inc., Providence,RI, USA). The damages are simulated at 12 differentdamage locations, and 12 damage intensities at eachlocation are considered in this study. The efficiency ofSVM is examined by adding noise levels of 30 to 80 dBin steps of 10 dB to the original data.

MethodsFinite element modeling and analysisA cantilever beam of length 450 mm, width 40 mm, andthickness 3mm with Young's modulus 207 GPa, density7,850 kg/m3, and Poisson's ratio 0.33 is simulated. A 4-node rectangular shell element of size 5 × 10 mm is usedfor meshing. The damage at a given location is simulatedby reducing the thickness from 15% to 70% in steps of 5%over a length of 10 mm. The schematic of the damagelocations starting from 100 mm, separated by 25 mm upto 375 mm from fixed end, is shown in Figure 1a. Firstmodal displacement values are obtained at 91 points alongthe center line of the beam. There are 12 damage locationsand 12 damage intensities at each location, giving a totalof 144 cases for simulations, and data (displacement of the91 points at first mode shape) obtained from each case isused in SVM. Mode shape data is obtained from thepoints highlighted in Figure 1b along the length of beamat the spacing of 5 mm.

Overview of support vector machineSVMs are a set of related supervised learning methodsused for classification and regression (Gunn 1998) whichuse machine learning theory to maximize predictiveaccuracy while automatically avoiding over-fit to the data.In this section, we present a brief theoretical formulationof SVM for linear regression analysis.SVM was initially developed for solving classification

problems and successfully applied in regression problems.The general formulation of regression learning is carriedout as follows (Gunn 1998).Consider the problem of approximating the set of data,

y1; x1ð Þ; . . . ; y1; x1ð Þ; x∈Rn; y∈R; ð1Þwith a linear function

f xð Þ ¼ w⋅xð Þ þ b; ð2Þwhere

w ¼Xli¼1

ai � a∗i� �

xi ð3Þ

b ¼ � 12w xr þ xs½ � ð4Þ

The optimal regression function is given by the minimumof the functional,

φ w; ξ; ξ�ð Þ ¼ 12

wk k2 þ CXli¼1

ξ i þXli¼1

ξ∗i

!ð5Þ

where C is a pre-specified value, and ξ∗, ξ are slackvariables representing upper and lower constraints on theoutputs of the system. Using an ε-insensitive loss function,

Lε yð Þ ¼ 0for f xð Þ � yj j < εf xð Þ � yj j � ε otherwise

�ð6Þ

The solution is given by

maxα; α� W α; α�ð Þ ¼ max

α; α�

� 12

Xli¼1

αi � αi∗ð Þ αj � α�j� �

xi⋅xj� �

þXli¼1

αi yi � εð Þ � α�i yi þ εð Þ

8>>>><>>>>:

9>>>>=>>>>;

ð7ÞAlternatively,

�α;�α∗ ¼ arg minα; α�

� 12

Xli¼1

αi � αi∗ð Þ αj � α∗j

� �xi⋅xj� �

�Xli¼1

yi αi � αi∗ð Þ þ

Xli¼1

αi � αi∗ð Þε

8>>>><>>>>:

9>>>>=>>>>;

ð8Þ

(a) Damage locations from fixed end.

(b) Top view of beam with finite element mesh and data acquisition points (separated at 5mm) of mode shape.

Figure 1 Schematic of the damage locations and mode shape data. Damage locations from fixed end (a). Top view of the beam with finiteelement mesh and data acquisition points (b).

Satpal et al. International Journal of Advanced Structural Engineering 2013, 5:2 Page 3 of 7http://www.advancedstructeng.com/content/5/1/2

with the constraints,

0≤αi≤C; i ¼ 1; ::::; l ð9Þ

0≤α∗i ≤C; i ¼ 1; ::::; l ð10ÞXli¼1

αi � α∗i� � ¼ 0 ð11Þ

Solving Equation 7 along with constraints Equations9 and 10 determines the Lagrange multipliers (αi, αi

∗),and the regression function is given by Equation 1.The Karush-Kuhn-Tucker conditions that are satisfied

by the solution are as follows:

�αi�α∗i ¼ 0; i ¼ 1; . . . ; l ð12Þ

Therefore, the support vectors are points whereexactly one of the Lagrange multipliers is greater thanzero. When ε = 0, we get the L1loss function, and theoptimization problem is simplified to

minβ

12

Xli¼1

Xlj¼1

βiβj xi⋅xj� ��Xl

i¼1

βiyi ð13Þ

where β is the difference in Lagrange multipliers withconstraints

�C≤βi≤C; i ¼ 1; . . . :; l ð14Þ

and

Xli¼1

βi ¼ 0 ð15Þ

A MatlabW toolbox (MathWorks, Natick, MA, USA)developed by Gunn (1998) is used for support vectorregression analysis. The code requires ‘training’ and‘testing’. In our case, displacement at 91 points alongthe beam in the first mode shape obtained from a vibrationsimulation is used as the ‘input’, and the location orintensity of damage is the output to the SVM in thetraining stage. An unknown set of displacements at those91 points is then used as the input to the trained SVM,and the damage intensity (or location) ‘predicted’ bythe SVM is compared with the known damage intensity(or location) which would have caused those displacements.This is known as testing.Performance of the SVM model depends on the

proper selection of the SVM parameters (C, ε). Hence, itis one of the critical tasks in SVM analysis for the givendata set. There is no universal rule to select the bestparameters for SVM analysis. These need to be determinedby trial and error. However, Cherkassky and Ma (2004)proposed a simple yet practical analytical approach to SVMregression parameter setting directly from the trainingdata. However, in the present paper, the parameters areselected by trial and error.In order to simulate experimental uncertainties, white

Gaussian noise is added in the original mode shape

Satpal et al. International Journal of Advanced Structural Engineering 2013, 5:2 Page 4 of 7http://www.advancedstructeng.com/content/5/1/2

data using MatlabW function, awgn(x, snr). This functiongenerates a vector of white Gaussian noise of mentionedsignal-to-noise ratio (snr) value in decibel scale and addsto the signal vector x.

Results and discussionCase 1: prediction of damage intensityTwelve damage intensities have been simulated at eachof the 12 locations using ABAQUSW, thus giving 144simulations. The vibration-induced displacements of thebeam at 91 locations (evaluation points) for each ofthese simulations are used as input to the SVM, and thedamage intensity is used as the output. The format forthe training and test set data used to predict the damageintensity using the SVM is shown in the Figure 2.Damages of 12 different intensities are simulated at thelocation L1. The displacements of the beam at the firstmode shape for all the damage intensities except 15% arefed to the SVM as training input and the correspondingdamage intensities as training output. After training, themode shape displacements at 15% damage are fed to thetrained SVM, and the output (Damage intensitypredicted) iscompared to the value 15 (Damage intensityactual).The error is evaluated according to the formula given inEquation 16. The parameters for SVM are taken as follows:C = infinity, e = 0, exponential radial basis function (ERBF)kernel, and ε-insensitive loss function. The procedure isrepeated for all the damage locations (L1 to L12). To findthe damage location, mode shape data for all the locations

15%

20%

25%

X1

X2

.

.

.

.

X91

Damage Intensities

Mode shapes

Damage Location

Data seTest input set of SVM

X1

X2

.

.

.

.

X91

X1

X2

.

.

.

.

X91

Test o/p of SVM

Figure 2 Format of data used for SVM analysis for prediction of inten

(at particular damage intensity) are used. Mode shapes atone of the locations is used as test input data set whoselocation is to be predicted and the remaining are usedfor training.

% error ¼ jDamage intensitypredicted � Damage intensityactualjDamage intensityactual

� �

� 100

ð16Þ

The SVM analysis is carried out using the mode shapedata considering no noise at each location for all thedamage intensities. The results obtained in damageintensity prediction in terms of percentage error areplotted in Figure 3. Thereafter, noise levels of 30 to 80dB are added in steps of 10 dB to the original modeshape data, and similar exercises are carried out for eachnoise level. The errors have been averaged over damagelocations and plotted in Figures 4 and 5. Inspection ofthese figures shows that error in the damage intensityprediction by SVM is high for low damage intensity(15% to 25%) at increased noise level. If we accept thatSVM should only be used to predict the damage intensityabove 25%, then the error in the prediction for no noisecase varies from 1.55% to 2.45%.The plot for 80-dB noise level is seen to be nearly over-

lapping with no noise case, which indicates that the effectof this level of noise is very low. Noise in the level of 30 to50 dB however causes significant differences in the results.

L1

65%

70%

50%

ts for training input of SVM

X1

X2

.

.

.

.

X91

X1

X2

.

.

.

.

X91

X1

X2

.

.

.

.

X91

Training o/p of SVM

sity for a fixed location.

100

125

150

175

200

225

250

275

300

325

350

375

0.0

1.0

2.0

3.0

4.0

5.0

1520

2530

3540

4550

55

60

65

70

Err

or (

%)

Figure 3 Error in damage intensity prediction for noise-free data.

Satpal et al. International Journal of Advanced Structural Engineering 2013, 5:2 Page 5 of 7http://www.advancedstructeng.com/content/5/1/2

The capability of SVM to predict damage intensity is goodat high damage intensity and deteriorates with increase innoise. At low level of noise (80 dB), it is able to predict evenlow levels of damage. At the higher levels of noise (70 to 30dB), the prediction of high damage (beyond 25%) remainsacceptable, but prediction of low damage intensity deterio-rates significantly. At the highest level of noise (30 dB) andlowest level of damage (15%), the error in prediction is asbad as 95.59%. The overall trend of the plots indicates thathigh level of damage is better predicted than low damage,the only significant deviation being the range of 30% to50% damage for 30-dB noise.

Case 2: prediction of damage locationAn exercise similar to case 1 is carried out except that inthis case, for a given damage intensity, the damagelocation is unknown. The error in location prediction iscalculated using Equation 17 at each location consideringall damage intensities, and it is plotted in Figure 6. Thepercentage error in the location prediction is significant

0

2

4

6

8

10

12

15 20 25 30 35 40 45 50 55 60 65 70

Err

or (%

)

Damage intensity (%)

60 dB 70 dB

80 dB no noise

Figure 4 Variation of percentage error with noise levels 30, 40,and 50 dB. To find damage intensity averaged over damage location.

only near the fixed end (location, 100 mm) where it is20%. For the locations in the middle span of the beam(125 to 350 mm), the prediction of damage location isquite accurate, varying from 1.46% to 4.93%. The trainingparameters for SVM are taken as follows: C = infinity,e = 0, ERBF kernel, and ε-insensitive loss function.The percentage error is calculated as follows:

% error ¼ Damage locationpredicted � Damage locationactual

Length of beam

� �� 100

ð17Þ

Thereafter, a similar analysis is carried out by addingwhite Gaussian noise (30 to 80 dB) in steps of 10 dB inthe mode shape data. The error in location predictionfor all damage intensities with noise added in the data iscalculated. The error for all damage intensities is averagedover each location for different noise levels and plotted inFigures 7 and 8. The error in the prediction of damage

0

20

40

60

80

100

120

15 20 25 30 35 40 45 50 55 60 65 70

Err

or (

%)

Damage intensity (%)

no noise 50dB

40dB 30dB

Figure 5 Variation of percentage error with noise levels 60,70, and 80 dB. To find damage intensity averaged over damagelocation.

100

125

150

175

200

225

250

275

300

325

350

375

0

5

10

15

20

25

152025303540

45

50

55

60

65

70

Err

or (

%)

Figure 6 Error in damage location prediction for noise-free data.

Satpal et al. International Journal of Advanced Structural Engineering 2013, 5:2 Page 6 of 7http://www.advancedstructeng.com/content/5/1/2

locations near both fixed and free ends of the beam is highcompared to those in the middle span of the beam.Noise levels of 80, 70, and 60 dB produce almost the

same trend as that of no noise. These error graphsnearly overlap with that of no noise except near the freeend (where the error increases up to 5%). Thus, noise upto 60 dB has very little effect on the capability of SVMto predict damage location. As noise level increases from50 to 30 dB, the error increases considerably. However,all of them show the same trend of low error (below 6%)in the middle span and high error at near both fixed andfree ends. The increase in error with increase in noise isvery significant near the free end (0.31% to 30%), but notso significant near the fixed end (20% to 31%). Thus,SVM predicts damage locations in a cantilever beamquite well if the damage is in the middle span of thebeam. Damage locations near the free end continue tobe predicted accurately if the noise level is reasonablylow. However, damage locations near the fixed end aredifficult to predict even with no noise.

0

5

10

15

20

25

100

125

150

175

200

225

250

275

300

325

350

375

Err

or (

%)

Damage location (mm)

no noise

80 dB

70 dB

60 dB

Figure 7 Noise effect on percentage error averaged overintensity (no noise, 50, 40, and 30 dB).

ConclusionsThis work attempts to evaluate the capability of supportvector machines for predicting the intensity or locationof damage for health monitoring of a cantilever beam.The 1,008 simulations have been studied using 12 levelsof damage intensities, 12 locations, and 7 noise levels(including no noise). The SVM has been trained withvibration-induced displacements collected at 91 pointsfor the first mode shape as input and damage intensityor location as output using only 90% of the simulations,the rest being kept aside for testing. After training, theSVM is able to predict any damage intensity or locationof the training set data with almost zero error. Theerrors in the test set without noise are less than 2.5% fordamage intensity prediction and below 6% for damagelocation prediction in the central section of the beam.Location prediction degrades drastically near the fixedend of the beam for any level of noise and is inaccuratenear the free end only for high levels of noise. On thecontrary, damage intensity prediction is significantly

0

5

10

15

20

25

30

35

100

125

150

175

200

225

250

275

300

325

350

375

Err

or (

%)

Damage location (mm)

no noise50dB40dB30dB

Figure 8 Noise effect on percentage error averaged overintensity (no noise, 80, 70, and 60 dB).

Satpal et al. International Journal of Advanced Structural Engineering 2013, 5:2 Page 7 of 7http://www.advancedstructeng.com/content/5/1/2

affected by noise for low damage intensity. High noisecauses significant deterioration in performance atdamage intensities lower than 25%. The worst level ofnoise (30 dB) results in 11% to 22% error for damageintensities higher than 25% and a much higher error atlower damage intensities.

Competing interestsThe authors declare that they have no competing interests.

Authors' contributionsSBS carried out the finite element simulations. YK performed the supportvector machine training. AG optimized the performance of support vectormachine. SB prepared the finite element model. All authors read andapproved the final manuscript.

Author details1Department of Mechanical Engineering, Indian Institute of TechnologyBombay, Powai, Mumbai 400076, India. 2Department of Civil Engineering,Indian Institute of Technology Bombay, Powai, Mumbai 400076, India.

Received: 15 October 2012 Accepted: 22 December 2012Published: 16 January 2013

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doi:10.1186/2008-6695-5-2Cite this article as: Satpal et al.: Structural health monitoring of acantilever beam using support vector machine. International Journal ofAdvanced Structural Engineering 2013 5:2.

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