development of residual operational deflection shape for crack detection in structures

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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 43 (2014) 113–123

0888-32http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/ymssp

Development of residual operational deflection shapefor crack detection in structures

Erfan Asnaashari, Jyoti K. Sinha n

School of Mechanical, Aerospace and Civil Engineering, University of Manchester, Sackville Street, Manchester M13 9PL, UK

a r t i c l e i n f o

Article history:Received 2 May 2013Received in revised form27 September 2013Accepted 6 October 2013Available online 26 October 2013

Keywords:Crack detectionCrack breathingModal analysisOperational deflection shape

70/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.ymssp.2013.10.003

esponding author. Tel.: þ44 161 3064639; fail addresses: [email protected]

a b s t r a c t

Excitation of a cracked structure at a frequency always generates higher harmoniccomponents of the exciting frequency due to the breathing of the crack. In this paper,the deflection of cracked structures at the exciting frequency and the second harmoniccomponent is mapped by a new method based on the operational deflection shape (ODS)for the purpose of crack detection. While the ODS is helpful in understanding dynamicbehaviour of structures and machines, it is not always possible to determine the locationof cracks in structures or machines based on the ODS itself. Therefore, a new conceptcalled residual ODS (R-ODS) has been defined for crack detection in beam-like structures.This paper presents the details of the proposed method and its results when applied tonumerical and experimental examples.

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1. Introduction

One of the important tasks of structural health monitoring systems is the detection of cracks in structures such asbuildings and bridges. A number of vibration-based methods are available in the literature [1] that can identify the presenceand location of cracks globally in any structures.

The presence of crack in a structure changes flexibility in the vicinity of the crack. This change causes the dynamicbehaviour for the whole cracked structure to be different from the healthy one [2]. Vibration-based crack detection methodsgenerally consider either open or breathing crack in their approaches. Gentile and Messina [3] identified open cracks inbeam structures by minimising measurement data and baseline of the structures and using continuous wavelet transform(CWT). Although the ability of this method to efficiently identify locations of open cracks was shown, the method was notverified experimentally. Khaji et al. [4] presented an analytical approach for identifying cracks in uniform beams with openedge cracks based on vibration measurements. The method was able to predict the crack location and depth with errors lessthan 8% and 25% respectively. Xiang et al. [5] proposed a model-based crack identification method to estimate the cracklocation and size in a static shaft with an open crack. The method investigated the effects of rotary inertia on the naturalfrequencies of the rotor system to construct B-spline wavelet on the interval and was verified experimentally.

While some approaches use open cracks in their analysis, the existence of breathing fatigue crack in a structure can berepresented accurately by considering its non-linear behaviour. Yan et al. [6] identified breathing-fatigue cracks in beams byemploying the difference between natural frequencies of each stiffness region of a beam according to the stiffness interface.Surace et al. [7] used higher order frequency response functions (FRFs) based on the Volterra series for the purpose offatigue crack detection in a cantilever beam. Tsyfansky and Beresnevich [8] used the higher harmonic components of

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ax: þ44 161 3064601.nchester.ac.uk (E. Asnaashari), [email protected] (J.K. Sinha).

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E. Asnaashari, J.K. Sinha / Mechanical Systems and Signal Processing 43 (2014) 113–123114

external harmonic excitation to identify fatigue cracks in flexible geometrically bars. Same approach was utilised bySemperlotti et al. [9] to locate a breathing fatigue crack in an isotropic rod.

In this paper, vibration responses at the higher harmonics of exciting frequency for the structure, generated due to thebreathing of fatigue crack (opening and closing), have again been considered. These responses can be analysed byconsidering either amplitude of deflection (AOD) or operational deflection shape (ODS). Ullah and Sinha [10] utilised theAOD method to map the deflection of E-glass fibre and epoxy resins composite plates with centre and off-centredelaminations. It was shown based on the experimental results that the normalised summation of higher harmonics(NSH) of the exciting frequency at each mode and its cumulative NSH (CNSH) were able to detect and locate thedelamination respectively. The disadvantage of the AOD method at higher harmonics is that only absolute values ofvibration amplitude at a specific mode are considered. Therefore, a true representation of the deflection shape of thestructure which requires both the amplitude and phase information cannot be constructed. The other drawback of the AODmethod at higher harmonics is that it may not be successful in detecting cracks in all types of structures [11]. Therefore,a new method for identifying and locating the crack based on the ODS at higher harmonics, which considers both amplitudeand phase, is presented here.

Vibration amplitudes need to be measured at a number of locations along a structure during external excitation in orderto construct the ODS for that structure. This can be done practically using a scanning laser Doppler vibrometer (SLVD).The key advantage of the ODS is that it uses both the amplitude and phase of vibration responses at a frequency to map thedeflection pattern of structures. This results in realistic vibration patterns of structures and/or machines at a specificfrequency. The ODS method is well-known in the vibration analysis [12,13]. Applications of the ODS analysis to differentstructures e.g., machines, bridges, gearboxes, and wind turbines are presented in the literature for different purposes.Pai and Young [14] investigated the use of a boundary effect detection (BED) method for identifying small damages in beamsbased on ODSs. Boundary layer solutions were extracted from experimental ODSs using a sliding-window least-squarescurve-fitting method. It was showed both numerically and experimentally that the BED method is reliable for locating smalldamages; however, some difficulties and limitations were observed during the application of this method. Sundaresan et al.[15] used a scanning laser vibrometer to compute and then compare the ODSs of healthy and damaged turbine blades.Changes in curvature of the ODSs were used to locate the damage. Zhang et al. [16] proposed a new damage detectionalgorithm called global filtering method (GFM) for beam and plate like structures based on ODS curvature (ODSC). A vehiclewas used to move along a line on damaged structure as an exciter. The ODSCs were then constructed from dynamicresponses of the vehicle and not the structure. The GFM which is based on wavelet decomposition was employed to makethe experimental ODSC smoother so that it can be used as baseline data. Then, the effect of damage could be detectedthrough comparing the ODSC and the filtered one. The method does not require a number of preinstalled sensors on thestructure which can be considered as an advantage; however, optimum velocity of the vehicle needs to be discovered sincethe detection quality of the GFM depends on it. Moreover, the results from the GFM were sensitive to the chosendecomposition level of filtering and the method may not be helpful in detecting small damages.

The ODS method is generally helpful in identifying vibration related problems in structures and machines; however, itmay not be straightforward to locate cracks in structures by using only the ODS. Recently, the residual ODS (R-ODS) has beendefined for the purpose of crack detection [11]. The preliminary study on simulated numerical examples showedencouraging results for identifying cracks in beam-like structures. In this paper, details of the R-ODS method as well asthe results of applying it to both numerical and experimental examples are presented.

2. Proposed method

A breathing crack opens and closes alternatively during every cycle of loading and consequently produces non-lineardynamics of the cracked structure (Fig. 1). Due to this phenomenon, excitation of the cracked structure at a frequency alwaysgenerates higher harmonic components of the exciting frequency. A typical spectrum for the beam with a breathing crack isshown in Fig. 2 where the exciting frequency (1� ) as well as its higher harmonics (2� , 3� , 4�…) can be seen.

The ODS of the cracked beam can be generated at the exciting frequency and the second harmonic components to mapthe deflection of the cracked structure. However, it has been observed from numerical simulations [11] that the ODSs at the

D

L

x

ydc

lc

Fig. 1. (a) A typical cracked beam, (b) open crack and (c) closed crack.

0 50 100 150 200 250 300 35010-3

10-2

10-1

100

101

Frequency, Hz

Acc

eler

atio

n, m

/s2

4x

1x

2x

3x

Fig. 2. A typical acceleration spectrum for the free–free beam.

E. Asnaashari, J.K. Sinha / Mechanical Systems and Signal Processing 43 (2014) 113–123 115

higher harmonic components are greatly affected by the ODS at the frequency of excitation. As a result, the ODSs at 2� andother higher harmonics cannot always distinguish the crack location because these data may contain the effect of the firstharmonic component (1� ; representing the mode shape) in addition to non-linearity due to the crack breathing. Based onthis assumption, a simple approach is proposed here to eliminate the effect of the first harmonic component from the ODSsat higher harmonics. In order to make such elimination possible, all the ODS values at 1� and higher harmonics arenormalised first using the following equation:

ðNODSÞi;m ¼ ðODSkÞi;mðMODSÞi;m

; i¼ 1;2;3;4;… ð1Þ

where ðNODSÞi;m is the normalised ODS at i-th harmonic for mode m. The subscript k refers to the node number, andðMODSÞi;m is the maximum value of ODS data at i-th harmonic for modem. Either of displacement, velocity or accelerationcan be used to construct the ODS of the structure. In this paper, measured and simulated acceleration responses are used.

The residual operational deflection shape (R-ODS) is then defined as the NODS at the p-th harmonic with respect to thefirst harmonic as expressed in Eq. (2). It removes the effect of the first harmonic from the ODS at p-th harmonic so that onlythe non-linear effect remains which is useful for the purpose of crack detection:

ðR� ODSÞp;m ¼ ðNODSÞp;m�ðNODSÞ1;m; p¼ 2;3;4;… ð2Þ

Therefore, when a cracked beam is excited at its first natural frequency and the R-ODS is calculated for 2� data, p andm areequal to 2 and 1 respectively.

3. Numerical example

An example of a steel beamwith the geometrical properties – length, 1000 mm and square cross-section 15 mm�15 mmand material properties – Young's modulus 210 GPa and density 7800 kg/m3 was considered. A finite element model of thebeam under free–free boundary condition was constructed using two node Euler–Bernoulli beam elements. The beam wasdivided into 10 elements each has a length of 100 mm. Each node has two degrees of freedom, the translationaldisplacement and bending rotation. Therefore, the beam consists of 11 nodes and 22 degrees of freedom in total. Fig. 3shows the finite element model of the free–free beam and the location of the external excitation. It is important to note thatthe excitation location must be away from nodal point(s) of the exciting mode. In this model, the approach proposed bySinha et al. [17] is used to introduce crack to the finite element model of the beam. Two crack depths (ðdc=DÞ ¼ 0:2; 0:3 ) andlocations (ðlc=LÞ ¼ 0:25; 0:35) are utilised in each analysis, where dc and lc are the depth and location of the crack and D and Lare the depth and length of the beam respectively as shown in Fig. 1(a). Natural frequencies of the cracked beam under free–free boundary condition were computed by carrying out modal analysis while the crack was assumed to be open. Table 1shows the results of the modal analysis. The modelled cracked beam is then excited by a sinusoidal input at its first naturalfrequency according to Table 1. The location of the excitation force was at node 7 as shown in Fig. 3. Time steps of 0.0002 swere used in the Newmark-beta numerical integration method to solve the equation of motion of the excited beam.

Breathing of the crack is also simulated causing the generation of higher harmonic components of the exciting frequencyin the frequency spectrum. The crack was assumed to be closed when the displacement of the nodes of the cracked elementin y-direction is less than zero for the applied excitation and to be open when that displacement is greater than zero. Thesampling frequency of 5000 Hz was used to convert the time-domain signals into frequency-domain. It is assumed here that

-Node point Excitation Force

111098764 52 31

Fig. 3. . Finite element model of the beam and the location of excitation force.

Table 1Computed natural frequencies for the numerical healthy and cracked beams.

Case Crack Frequency (Hz)

Size dcD

� �Location lc

L

� �Mode 1 Mode 2 Mode 3 Mode 4

Free–free No crack No crack 79.72 208.54 377.68 642.670.2 0.25 79.36 205.89 373.84 641.670.2 0.35 78.90 206.55 377.64 637.180.3 0.25 79.22 204.85 372.41 641.310.3 0.35 78.58 205.78 377.62 635.09

0 0.2 0.4 0.6 0.8 1-20

-10

0

10

Distance, m

OD

S, m

/s2

0 0.2 0.4 0.6 0.8 1-20

-10

0

10

Distance, m

0 0.2 0.4 0.6 0.8 1-0.2

-0.1

0

0.1

0.2

Distance, m0 0.2 0.4 0.6 0.8 1

-0.2

0

0.2

0.4

Distance, m

Depth=0.2DDepth=0.3D




OD

S, m

/s2

OD

S, m

/s2

OD

S, m

/s2

Fig. 4. ODSs at 1� (a, b) and 2� (c, d) for the free–free beam with crack depths of 0.2D and 0.3D at the location of 0.25 m (a,c) and 0.35 m (b,d) fromone end.


the vibration of entire structure can be scanned through a laser vibrometer or a number of accelerometers attached alongthe beam. Therefore, taking vibration measurements in terms of velocity or acceleration at a large number of locations ispossible. Fig. 2 shows a typical frequency spectrum of the velocity response of the cracked beam at a specific node where thehigher harmonics of the exciting frequency can be observed clearly.

Simulated data were initially analysed using the ODS at higher harmonic components of the first resonant frequency forthe cracked beam under free–free boundary condition at all crack depths and locations (Fig. 4). The obtained ODSs frommeasured data are generally complex quantities; therefore, the method explained in [18] is used to convert them intonormal ODSs. It can be observed that the ODSs at 1� data (Fig. 4(a) and (b)) correspond to the mode shape of the beamwhile the results acquired from the ODS analysis at 2� data (Fig. 4(c) and (d)) present some indications regarding thelocation of the crack but they are not clear enough. The proposed R-ODS method is now applied to the numerical examplesand the results are shown in Fig. 5. It can be observed that the first and the second highest values of (R-ODS)2,1 have beenoccurred at two consecutive nodes. In fact, the nodes corresponding to those two highest values determine the location ofthe cracks. As shown in Fig. 5, the intersections of the dashed lines provide accurate identifications of the crack location.

The influence of noise on the numerical results is also investigated. Calculated acceleration responses are polluted withnormalised random noise with signal-to-noise ratio (SNR) of 20 dB.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Distance, m

(R-O

DS)

2,1

(R-O

DS)

2,1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Distance, m



Fig. 5. The (R-ODS)2,1 for calculated vibration responses without noise at mode 1 for the free–free beam with crack depths of 0.2D and 0.3D and cracklocations of (a) 0.25 m and (b) 0.35 m from one end.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

Distance, m

(R-O

DS)

2,1

(R-O

DS)

2,1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

Distance, m



Fig. 6. The (R-ODS)2,1 for calculated vibration responses with noise at mode 1 for the free–free beamwith crack depths of 0.2D and 0.3D and crack locationsof (a) 0.25 m and (b) 0.35 m from one end.


Fig. 6 presents the application of the R-ODS method to noisy simulated data. The R-ODS results locate the cracksuccessfully which verify that the detection is not affected by the noisy signal.

3.1. Off-resonance excitation

In the previous section, the proposed method is tested when the beams are excited at their first natural frequency.However, depending on the situation, excitation at resonant frequency may not be possible in practice. Hence, off-resonanceexcitation is used here to examine the usefulness of the proposed method. Sinusoidal input with the frequency of 70 Hz,which is below the natural frequency (Table 1), is used to excite the beams. The (R-ODS)2,0 is then computed for 2�components. Subscript “0” indicates off-resonance excitation. Once again, acceleration responses are polluted with SNR of20 dB. Fig. 7 illustrates the obtained results which confirm the accuracy of the proposed method in locating cracks usingoff-resonance excitation.

4. Experimental verification of proposed method

Experimental tests were conducted on healthy and cracked solid aluminium beams having dimensions of 25�100�3000 mm3 to show the effectiveness of the proposed R-ODS method. Laser cutting method was used to create cracks in the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

Distance, m

(R-O

DS)

2,0

(R-O

DS)

2,0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Distance, m



Fig. 7. The (R-ODS)2,0 for calculated vibration responses with noise from off-resonance excitation for the free–free beam with crack depths of 0.2D and0.3D and crack locations of (a) 0.25 m and (b) 0.35 m from one end.

Table 2Crack sizes and locations used in the experimental beams.

Case Crack location (cm) Crack depth (mm) Crack width (mm)

Beam 1 (healthy) No crack No crack No crackBeam 2 150 6 0.2Beam 3 100 6 0.2Beam 4 37.5 6 0.2

Fig. 8. Test equipments.


form of small slots at different locations along the beams. Table 2 shows the crack sizes and locations for all the beams usedin the experiment.

4.1. Test setup

The aluminium beams were suspended from a hollow steel beam through two bicycle brake cables to provide free–freeboundary condition. The cables were connected to the beams using two clamps. Since the beams were excited at their firstnatural frequencies, the clamps were placed at nodal points of the first mode (free–free) where there are no displacementsduring external excitation. Laptop 1 (Fig. 8) was used to send the input signals in the form of simple sinusoidal waves to theshaker. Fifteen accelerometers were installed at equal intervals of 21 cm along the beams to acquire vibration responses(Fig. 9). A force gauge was also used to measure the input force applied by the shaker to the beams. Amplifiers wereemployed to amplify all the output signals and send them into 16 bit data acquisition card which was connected to laptop 2(Fig.8). All the response channels of the data acquisition card were acquired simultaneously so that valid ODSs with correct

Fig. 9. Test setup.

Amplifiers

Data Acquisition Card Laptop 2

Shaker Laptop 1

Fig. 10. Schematic view of the experiment.


magnitudes and phases can be guaranteed. Figs. 9 and 10 show the experimental setup and the schematic view of itrespectively for vibration data acquisition.

4.2. Experiment

Modal testing was carried out using laptop 1 in order to obtain the frequency response function (FRF) of each beam. Thealuminium beams were then excited at their first natural frequencies according to the FRF results. The excitations were inthe form of continuous sinusoidal waves applied to the beams through the shaker. The results obtained from the modalanalysis are summarised in Table 3.

It should be noted that the dimensions of each experimental beam slightly differ from the others. Also, the mountingplace of the accelerometers might not be exactly the same in all the cases causing to have slightly different mass distributionalong each beam. Therefore, minor inconsistency can be seen in the results for natural frequencies.

Vibration responses from all the sensors were recorded using the data acquisition card for each beam. Samplingfrequency of 5000 Hz was used to convert time domain responses into frequency domain. Fig. 11 demonstrates typicalacceleration spectra obtained from the 4th accelerometer (distance of 63 cm from one end) for Beams 1–4 when excited attheir first mode. Higher harmonic components can be seen clearly for Beams 2–4. However, the spectrum of Beam 1(healthy) does not show most of the harmonics. This confirms the breathing of the cracks during conducting theexperiments for Beams 2–4.

4.3. Analysis and results

The experimental ODSs at 1� and 2� data for all the three cracked beams are plotted in Figs. 12–14. As it can be seen,only Fig. 12(b) shows a negative peak at the crack location. All the other ODSs display the first mode shape of the free–freebeams. The reason is that in Beam 2, the crack is located at the centre of the beam where maximum crack breathing at thefirst mode is occurred. However, Beams 3 and 4 have their cracks at locations where crack breathing has not been prominent

Table 3Measured natural frequencies of the experimental beams.

Case Mode 1 (Hz) Mode 2 (Hz) Mode 3 (Hz)

Beam 1 (healthy) 14.95 39.98 78.43Beam 2 14.95 39.98 78.13Beam 3 15.26 40.26 79.65Beam 4 15.26 39.98 78.74

0 10 20 30 40 50 60 70 80 9010-4

10-3

10-2

10-1

100

Frequency, Hz

Acc

eler

atio

n, m

/s2

Acc

eler

atio

n, m

/s2

Acc

eler

atio

n, m

/s2

Acc

eler

atio

n, m

/s2

0 10 20 30 40 50 60 70 80 9010-5

10-4

10-3

10-2

10-1

Frequency, Hz

0 10 20 30 40 50 60 70 80 9010-6

10-5

10-4

10-3

10-2

Frequency, Hz0 10 20 30 40 50 60 70 80 90

10-6

10-5

10-4

10-3

10-2

Frequency, Hz

1x2x

3x

4x

1x

2x

3x

4x

2x

1x

3x

4x

a b

c d

Fig. 11. Typical spectra for (a) Beam 1, (b) Beam 2, (c) Beam 3, and (d) Beam 4 from the accelerometer located at 630 mm from the left end.

0 0.5 1 1.5 2 2.5 3-15

-10

-5

0

5

10

Distance, m

OD

S, m

/s2

OD

S, m

/s2

0 0.5 1 1.5 2 2.5 3-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

Distance, m

Fig. 12. Experimental ODSs at (a) 1� and (b) 2� data for Beam 2.


0 0.5 1 1.5 2 2.5 3-8

-6

-4

-2

0

2

4

6

Distance, m

OD

S, m

/s2

OD

S, m

/s2

0 0.5 1 1.5 2 2.5 3-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Distance, m


0 0.5 1 1.5 2 2.5 3-10

-8

-6

-4

-2

0

2

4

6

Distance, m

OD

S, m

/s2

OD

S, m

/s2

0 0.5 1 1.5 2 2.5 3-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Distance, m



enough to affect the overall trend of the ODS plots. As a result, it is not always possible to determine the location of cracksbased on the ODS itself.

The R-ODS method, explained in Section 2, is applied to all the experimental frequency-domain responses of the crackedaluminium beams. Fig. 15 presents these results. As it can be observed, positive peaks are appeared after removing the effectof mode 1 from the ODSs at second higher harmonics (R-ODS). These peaks show the location of sensors which had theclosest distance to the crack location. This means that the R-ODS method is capable of locating the cracks even in situationswhere the crack breathing is not very high. Therefore, the method can be employed as a trustworthy tool for crack detectionin beam-like structures.

5. Finite element simulation of the experiments

Finite element models of cracked aluminium beams which were used in the experiments are simulated in this section.The beams are divided into 15 elements and two degrees of freedom are considered for each node. The same approachesmentioned in Section 3 for modelling and simulating the crack and its non-linear breathing behaviour are utilised oncemore. The masses of all the mounted accelerometers in the experiment as well as the two clamps are considered in thissimulation.

The first natural frequencies of the cracked aluminium beams under free–free boundary condition were found to be14.926, 14.781, 14.837, and 14.921 Hz for Beams 1–4 respectively. Open crack was assumed during the modal analysis. Themodelled cracked beams are then excited by sinusoidal inputs at their first natural frequencies.

Fig. 16 demonstrates the results obtained from the application of the R-ODS method to numerical data. As it is shown, thenumerical (R-ODSs)2,1 show proper correlations with the experimental ones in Fig. 15. Good indications of the crack locationwere also observed when the R-ODS method applied to higher modes of vibration. Fig. 17 illustrates typical results obtained

0 0.5 1 1.5 2 2.5 3-0.4

-0.2

0

0.2

0.4

0.6

Distance, m

(R-O

DS)

2,1

(R-O

DS)

2,1

(R-O

DS)

2,1

0 0.5 1 1.5 2 2.5 3-0.1

-0.05

0

0.05

0.1

Distance, m

0 0.5 1 1.5 2 2.5 3-5

0

5

10x 10-3

Distance, m

Crack LocationCrack Location

Crack Location

Fig. 15. Experimental (R-ODS)2,1 at mode 1 for (a) Beam 2, (b) Beam3 and (c) Beam 4.

0 0.5 1 1.5 2 2.5 3 3.5-0.2

-0.1

0

0.1

0.2

0.3

0.4

Distance, m

(R-O

DS)

2,1

(R-O

DS)

2,1

(R-O

DS)

2,1

0 0.5 1 1.5 2 2.5 3 3.5-1

-0.5

0

0.5

1

1.5

Distance, m

0 0.5 1 1.5 2 2.5 3 3.5-0.4

-0.2

0

0.2

0.4

0.6

Distance, m

Crack location

Crack location

Crack location

Fig. 16. Numerical (R-ODS)2,1 at mode 1 for (a) Beam 2, (b) Beam 3 and (c) Beam 4.


0 0.5 1 1.5 2 2.5 3-6

-4

-2

0

2

4

6

8 x 10-3 x 10-3

Distance, m

(R-O

DS)

2,2

(R-O

DS)

2,2

0 0.5 1 1.5 2 2.5 3-4

-2

0

2

4

6

Distance, m

Crack location Crack location

Fig. 17. Numerical (R-ODS)2,2 at mode 2 for (a) Beam 2 and (b) Beam 3.


for mode 2 of the aluminium beams. This confirms that the proposed R-ODS method is also applicable to higher modes ofvibration for the purpose of crack detection provided that the crack is not located at or close to the nodal point(s) of theexciting mode. Beam 4 is not included in Fig. 17 since its crack is located near one of the nodal points at mode 2.

6. Concluding remarks

It is known that the breathing of crack in a structure which is excited at a frequency generates vibration responses at thefrequency of excitation and its higher harmonics. The proposed method utilises the concept of the ODS analysis at theexciting frequency and its higher harmonics to identify the location of the crack. The external excitation can be at resonanceor off-resonance frequency as per field convenience. A new term called R-ODS is defined which eliminates the effect of theODS at exciting frequency from those at higher harmonic components for clear and accurate identification of crack location.The method has been successfully validated by applying to both numerical and experimental beam-like examples. Theadvantage of the R-ODS method is its simplicity as well as its ability to identify the location of crack accurately.

References

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development of residual operational deflection shape for crack detection in structures

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