damage detection in beams using frequency response function curvatures near resonating frequencies

Damage Detection in Beams UsingFrequency Response Function CurvaturesNear Resonating Frequencies

Subhajit Mondal, Bidyut Mondal, Anila Bhutiaand Sushanta Chakraborty

Abstract Structural damage detection from measured vibration responses has gainpopularity among the research community for a long time. Damage is identified instructures as reduction of stiffness and is determined from its sensitivity towards thechanges in modal properties such as frequency, mode shape or damping values withrespect to the corresponding undamaged state. Damage can also be detected directlyfrom observed changes in frequency response function (FRF) or its derivatives andhas become popular in recent time. A damage detection algorithm based on FRFcurvature is presented here which can identify both the existence of damage as wellas the location of damage very easily. The novelty of the present method is that thecurvatures of FRF at frequencies other than natural frequencies are used fordetecting damage. This paper tries to identify the most effective zone of frequencyranges to determine the FRF curvature for identifying damages. A numericalexample has been presented involving a beam in simply supported boundarycondition to prove the concept. The effect of random noise on the damage detectionusing the present algorithm has been verified.

Keywords Structural damage detection � Frequency response function curvature �Finite element analysis

1 Introduction

Damage detection, condition assessment and health monitoring of structures andmachines are always a concern to the engineers. For a long time, engineers havetried to devise methodology through which damage or deterioration of structurescan be detected at an earliest possible stage so that necessary repair and retrofitting

S. Mondal � B. Mondal � A. Bhutia � S. Chakraborty (&)Department of Civil Engineering, Indian Institute of Technology Kharagpur,Kharagpur, Indiae-mail: [email protected]

© Springer India 2015V. Matsagar (ed.), Advances in Structural Engineering,DOI 10.1007/978-81-322-2193-7_119

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can be carried out. Recently, due to the rapid expansion of infrastructural facilitiesas well as deterioration of the already existing infrastructures, the magnitude of theproblem has become enormous to the civil engineering community. Detection ofdamages using various local and global approaches has been explored in currentliterature. The measured dynamical properties have been used effectively fordetecting damages. The dynamical responses of structures can be very preciselymeasured using modern hardware and a large amount of data can be stored forfurther post processing to subsequently detect damages. The damage detectionproblem can be classified as identification or detection of damage, location ofdamage, severity of damage and at the last-estimation of the remaining service lifeof a structure and its possible ultimate failure modes. During the last three decadessignificant research has been conducted on damage detection using modal prop-erties (frequencies, mode shapes and damping etc.). The mostly referred paper ondamage detection using dynamical responses is due to Deobling et al. [1] whichgive a vivid account of all the methodologies of structural damage detection usingvibration signature until 90 s. Damage detection using changes in frequency hasbeen surveyed by Salawu and Williums [2]. The main drawback of detectingdamages using only frequency information is the lack of sensitivity for the smalldamage cases. The main advantage of this method is that, frequency being a globalquantity it can be measured by placing the response sensor such as an accelerometerat any position. Mode shapes can also be effectively used along with frequencyinformation to locate damage [3, 4], but the major drawback is that mode shape issusceptible to the environment noise much more than the frequency. Moreover,mode shapes being a normalized quantity is less sensitive to the localized changesin stiffness. The random noise can be averaged out to some extent but systematicnoise cannot be fully eliminated. Furthermore, in vibration based damage detectionmethodologies, depending upon the location the damage may or may not bedetected if it falls on the node point of that particular mode. Lower modes some-times remain less sensitive to localized damages and measurement of higher modesare almost always is necessary which is more difficult in practice.

In contrast with frequency and mode shape based damage detection, method-ologies using mode shape curvature, arising from the second order differentiation ofthe measured displacement mode shape is considered more effective for detectingcracks in beams [5]. Wahab and Roeck [6] showed that damage detection usingmodal curvature is more accurate in lower mode than the higher mode. Whalen [7]also used higher order mode shape derivatives for damage detection and showedthat damage produce global changes in the mode shapes, rendering them lesseffective at locating local damages. Curvature mode shapes also have a noticeabledrawback of susceptibility to noise, caused by these second order differentiation ofmode shapes. This differentiation process may amplify lower level of noise to suchan extent to produce noise-dominated curvature mode shapes [8] with obscureddamage signature. Most recently, Cao et al. [9] identified multiple damages ofbeams using a robust curvature mode shape based methodology.

In recent years, many methods of damage detection based on changes indynamical properties have been developed and implemented for various

1564 S. Mondal et al.

complicated structural forms. Wavelet transformation is one of the recent populartechniques for damage detection in local level, although its performance to detectsmall cracks is questioned [10, 11].

A structure vibrates on its own during resonance at high amplitude and thereforethe FRFs become very sensitive to noise. Ratcliffe [12] explored the frequencyresponse function sensitivities at all frequencies rather than at just the resonantfrequencies to define a suitable damage index which can be used in a robust mannerin presence of inevitable experimental noise. Sampaio et al. [13] have given anaccount of the frequency response function curvature methods for damage detec-tion. Pai and Young [14] detected small damages in beams employing the opera-tional defected shapes (ODS) using a boundary effect detection method. Scanninglaser vibrometer was used for measuring the mode shapes. Bhutia [15] and Mondal[16] have also investigated damage detection using operational deflected shapes andusing FRFs at frequencies other than natural frequencies respectively.

Therefore, it appears that at frequencies slightly away from the natural frequency(either above or below), it may be somewhat less affected by measurement noise.But, it is to be also remembered that the sensitivity of FRFs to damage will also falldown at the frequencies other than the natural frequencies. Hence, the FRFs atfrequencies other than natural frequencies, although less noise prone is less sen-sitive to damage as well. With all probability, there might exist an optimum locationin FRF curve nearer to the resonant peaks where the measured FRFs still haveenough sensitivity towards damage yet have substantial less sensitivity to noise. Itmust also be noted that most of the existing damage detection algorithm works wellwhen the damage is severe, because the level of stiffness changes will be substantialfor such damages and will be easily detectable. Such damages can be detectedeasily by other means such as direct visual observations. The real challenge in theresearch field of structural damage detection is to test the damage indicator’ssensitivity for small damages in presence of inevitable measurement noise. Mostalgorithms are observed to give spurious indications of damage when the noiselevel becomes somewhat higher.

In this paper FRF curvature is used at frequencies different than the naturalfrequencies to detect damage. Thus the fundamental principle behind this damagedetection methodology is to exploit the relative gain in terms of lower noise sen-sitivity, sacrificing a bit in terms of resonant response magnitude. Although theconcept appears to be attractive, the current literature does not provide enoughguidance in this regard. The present paper tries to explore the same through anexample beam in simply supported condition.

The present study concentrates on a forward problem of simulating damagescenarios, considering the FRF curvatures as the damage indicators to see if itperforms better than the methods employing FRFs at natural frequencies. The keyquestion is the robustness of the algorithm, i.e. whether the results obtained willremain unique in the presence of real experimental noise, especially under thecondition of modal and coordinate sparsity. Finite element analysis using ABAQUS[17] has been used to generate the required vibration responses for this simulatedstudy. Simulated noises into the data are added as a percentage of FRF magnitude.

Damage Detection in Beams Using Frequency … 1565

2 Theoretical Background of the Present Methodology

The mass, stiffness and damping properties of a linear vibrating structure are relatedto the time varying applied force by the second order differential force equilibriumequations involving the displacement, velocity and acceleration of a structure. Thecorresponding homogenized equation can be written in discretized form-

M½ �fx::ðtÞg þ ½C�f _xðtÞg þ ½K�fxðtÞg ¼ f0g ð1Þ

where ½M�; ½K� and ½C� are the mass, stiffness and viscous damping matrices withconstant coefficients and fx::ðtÞg; f _xðtÞg and fxðtÞg are the acceleration, velocity anddisplacement vectors respectively as functions of time. The eigensolution of theundamped homogenized equation gives the natural frequencies and mode shapes.

If the damping is small, the form of FRF can be expressed by the followingequation [18].

HjkðxÞ ¼ Xj

Fk¼

XN

r¼1

rAjk

k2r � x2ð2Þ

Here, Hjk(ω) is the frequency response functions, rAjk is the modal constant, λr isthe natural frequency at mode r and ω is the frequency.

The individual terms of the Frequency Response Functions (FRF) are summedtaking contribution from each mode [18]. At a particular natural frequency, one ofthe terms containing that particular frequency predominates and sum total of theothers form a small residue. But for a FRF at frequency just slightly away from thenatural frequency, the other terms also starts contributing somewhat significantly,thereby remaining sensitive to stiffness changes of the structure. For localizing thedamage, FRF curvature method can more effectively be used than the FRFsthemselves.

3 Numerical Investigation

In this current investigation a simply supported aluminum beam has been modeledusing the C3D20R element (20 noded solid brick element). Eigensolutions havebeen found out using the Block Lanchoz algorithm with appropriately convergedmess sizes for the modes under consideration. The material properties of beam areassumed to be E = 70 GPa and NEU = 0.33. Then, damage has been inflicted with adeep narrow cut of width 2.5 mm thick and 5 mm deep as shown in Fig. 1. TheFRFs are computed at 21 evenly spaced locations as shown in Fig. 2.

Figure 3 shows the natural frequencies and the corresponding mode shapes ofthe ‘undamaged’ beam and damaged beam.


The FRFs of the undamaged and the damaged beam has been overlaid in Fig. 4.The difference is noticeable in some modes, indicating more damage sensitivity.

Curvature of FRFs, i.e. the rate of change of FRFs measured at twenty onelocations and at different frequency (lies in between 90 and 110 % of the natural

Fig. 1 Dimension and damage location of simply supported beam

Fig. 2 Location of FRF measurement along the center line of beam

Fig. 3 Mode shape and frequency of undamaged and damaged beam


frequencies) for undamaged and damage cases are determined. The procedure tocompute the FRF curvature is a central difference scheme and is given below

H00ijðxÞ ¼

Hðiþ1ÞjðxÞ � 2HijðxÞ þ Hði�1ÞjðxÞðDhÞ2 ð3Þ

For example, as shown in the Fig. 5 FRF curvature was taken at 90, 95, and98 % of 1st natural frequency for both the undamaged and damaged cases and thisprocess was continued for twenty one different location of the beam. The differencewas taken as absolute difference of the curvature. Since there is a frequency ‘shift’due to damage, a mapping scheme has been adopted as shown. Many referencesjust directly compare FRFs without accounting for such frequency shifts and maynot truly represent the effect of FRF changes due to damage.

Figure 5 shows the absolute change in FRFs at different frequencies around thefirst natural frequency without considering the noise.

3.1 Damage Detection Using FRF Curvature Near the FirstFundamental Mode

Figure 6 shows the FRF curvatures at various locations along the beam length fordifferent values of frequencies away from the natural frequencies as a percentage ofthe resonant frequency. Thereafter, random noise is added to the FRFs and the samemethodology is applied to determine the sensitivities. Figure 7a–f shows that as thenoise level increases the FRF curvatures show pseudo peaks of much highermagnitudes to obscure the actual damages, however the effect is minimum aroundFRF curvature computed at 95 % of natural frequency.

0

1

2

3

4

5

6

7

8

0 500 1000 1500 2000 2500 3000

Mag

nitu

de

Frequency (Hz)

FRF for undamaged beam

FRF for damaged beam

Fig. 4 Comparisons of point FRFs (point 11) of undamaged and damaged beam


Figure 7a shows the effect of noise on the damage localization using the FRFcurvature at around 1st natural frequency. With 1 % noise, false peaks appear inaddition to peak at 14th number point, so the localization sensitivity reduces.Addition of 2 % noise gives pseudo peaks at other points having much highermagnitudes which are however actually not damaged. Thus addition of noise hascaused more and more false detection of damages as compared to the noise-freecase. Addition of 3 % noise gives an even more unacceptable result with substantialincrease in false detections apart from the actual damage at point 14.

Figure 7b which is the plot of FRF curvature at 98 % of 1st natural frequencyshows similar kind of result with very little improvement towards the noise resis-tance. However when curvature differences at 96 and 95 % of 1st natural frequencyare explored, they show substantial increase in resistance towards the added randomnoise as is evident from Fig. 7c, d. It can be easily observed that the small peaks are

Fig. 5 Mapping of FRF for undamaged and damaged beam at different frequency

0

0.00002

0.00004

0.00006

0.00008

0.0001

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

cha

nge

in F

RF

curv

atur

e

90%

93%

95%

96%

97%

98%

99%

Point number

Fig. 6 FRF curvature beam without noise


relatively suppressed, thereby locating the damage much more uniquely at thedesignated point number 14. Further downward movement along frequency scalehowever could not fetch any benefit and in fact shows reduction in damagedetection capacity. At 93 and 90 % of 1st natural frequency false peaks againstarted to predominate. Hence, Curvature difference away from 1st natural fre-quency shows very distinct damage localization capability, even with substantiallevel of added random noise. The peaks at the actual damage location are distinctenough to pin-point the actual damage location. Overall damage identificationcapability in presence of noise increases as we move away from resonant peak ofFRF and damage detection is most robust within certain range of frequency, veryclose to the natural frequency.

Similar phenomenon on other side of the FRF peak at natural frequency havebeen observed and are presented in Fig. 8.

From Fig. 9a–f it is clear that damage detection can be done better between 104and 105 % of natural frequency in noisy environment than the usual practice ofusing FRFs at resonant frequency.

Fig. 7 Difference between curvatures of FRFs of undamaged and damaged (point 14) Simplysupported beam for a 1st natural frequency b 98 %, c 96 %, d 95 %, e 93 % and f 90 % of 1stnatural frequency for different percentage of noise. Input force at mid point


0

0.00002

0.00004

0.00006

0.00008

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Abs

olut

e ch

ange

in c

urva

ture

101%102%103%104%105%107%110%

Point number

Fig. 8 Difference between curvatures of FRFs of undamaged and damaged (point 14) simplysupported beam for different percentage (100–110 %) 1st natural frequency. Input force at midpoint

Fig. 9 Difference between curvatures of FRFs of undamaged and damaged (point 14) simplysupported beam for a 1st natural frequency b 102 %, c 104 %, d 105 %, e 107 % and f 110 % of1st natural frequency for different percentage of noise. Input force at mid point


3.2 Damage Detection Using FRF Curvature Nearthe Second Mode

The investigation is extended further to include the second mode also and similarresults are obtained and are presented in Fig. 10.Most effective zone to detect damageis again found to be at 95–96 % of the resonant frequency (so also at 100–110 % ofthe second natural frequency) and is not presented here for brevity.

4 Conclusions

An attempt has been made to detect the location of damage in a simply supportedaluminum beam using FRF curvatures at frequencies other than natural frequenciesand is found to be more robust as compared to method using FRFs at resonantfrequencies when random noise are present in data. Upto 2–3 % of random noise inobserved FRF data are tried. An optimum frequency zone at around 95–96 % (or105–106 %) of the natural frequency has been identified as ideal to locate damageas they maintain the required sensitivity for damage detection yet being slightlyoffset from the peak value. Keeping all the above observations, we can concludethat damage detection using FRF curvature at other than natural frequency may be abetter option if considerable measurement noise is present into the data. The methodneeds to be further explored with appropriate model of noise actually present in realmodal testing of structures in various boundary conditions.

Fig. 10 Difference between curvatures of FRFs of undamaged and damaged (point 14) Simplysupported beam for a 2nd natural frequency, b 98 %, c 95 % and d 90 % of 2nd natural frequencyfor different percentage of noise. Input force at mid point


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