International Journal of Mechanical Sciences 47 (2005) 14771497

H. Nahvi, M. Jabbari

The proposed method is based on measured frequencies and mode shapes of the beam.r 2005 Elsevier Ltd. All rights reserved.

early detection of cracks has gained popularity over the years and in the last decade substantialprogress has been made in that direction.

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www.elsevier.com/locate/ijmecsci

0020-7403/$ - see front matter r 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijmecsci.2005.06.008

Corresponding author. Tel.: +98 311 3915242; fax: +98 311 3912628.

E-mail address: hnahvi@cc.iut.ac.ir (H. Nahvi).Keywords: Crack detection; Vibration analysis; Cantilever beam; Finite element method

1. Introduction

The purpose of the present work is to establish a method for predicting the location and depthof a crack in a cantilever beam using experimental vibration data. Using vibration analysis forMechanical Engineering Department, Isfahan University of Technology, Isfahan 84156, Iran

Received 3 September 2004; received in revised form 18 May 2005; accepted 12 June 2005

Abstract

In this paper, an analytical, as well as experimental approach to the crack detection in cantilever beamsby vibration analysis is established. An experimental setup is designed in which a cracked cantilever beam isexcited by a hammer and the response is obtained using an accelerometer attached to the beam. To avoidnon-linearity, it is assumed that the crack is always open. To identify the crack, contours of the normalizedfrequency in terms of the normalized crack depth and location are plotted. The intersection of contourswith the constant modal natural frequency planes is used to relate the crack location and depth.A minimization approach is employed for identifying the cracked element within the cantilever beam.Crack detection in beams using experimental modal data andnite element model

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 147714971478Nomenclature

a crack depthA crack cross-sectionb beam widthA crack in a structural member introduces local exibility that would affect vibration responseof the structure. This property may be used to detect existence of a crack together its location anddepth in the structural member. The effect of a crack on the deformation of a beam has beenconsidered as an elastic hinge by Chondros and Dimarogonas [1]. In Ref. [2] variations of thenatural frequencies were calculated by a perturbation method. A nite element model has beenproposed in Ref. [3], in which two different shape functions were adopted for two segments of thebeam, in order to consider the discontinuity of deformation due to the crack. Cawley and Adams

[C] exibility matrixCij elements of exibility matrix

C0ij exibility coefcients

C1ij additional exibility coefcient

E Youngs modulusEeff updated Youngs modulush beam heightI moment of inertiaK I; K II; K III stress intensity factorsK IM ; K IP; KIIP stress intensity factors for a rectangular cross-section crack[KC] cracked element stiffness matrixl length of the elementl1 distance of crack from the xed endL beam lengthm mass per unit lengthM bending moment on the elementMb total bending moments non-dimensional crack depthT shear force[T] transformation matrixU0 strain energy of uncracked elementU1 additional strain energy due to crack

X hi error functionfeij jth component of the ith experimental mode shapefmij jth component of the ith theoretical mode shapeli constant values depending on the boundary conditionsn Poissons ratiooi natural frequencies of intact beam

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 14771497 1479[4] showed that the stress distribution in a vibrating structure was non-uniform and was differentfor each mode of vibration. Therefore, any local crack would affect each mode differently,depending on the location of the crack. Stubbs [5] and Chondros and Dimarogonas [6] used theenergy method and the continuous cracked beam theory to analyze transverse vibration ofcracked beams.In the analytical study of this problem, two procedures have been used by researchers to

quantify local exibility due to the crack. In the rst procedure, a stiffness matrix is constructedfor the cracked section, in a similar way as an equivalent spring [7]. In the second procedure whichis more practical, a cracked nite element stiffness matrix is constructed and assembled with thenon-cracked elements of the structure. Dirr and Schmalhorst [8] used a three-dimensional niteelement model with an elastic material to model cracked beams. Qian et al. [9] formulated amethod of crack detection in beams based on the changes in the natural frequencies and modeshapes. Narkis [10] studied the dynamics of a cracked simply supported beam for bending andaxial vibrations. He has shown that for accurate crack identication the variations of the rst twonatural frequencies, caused by the crack, are needed. Shen and Pierre [11] investigated thevibration of cracked beams with single or symmetrical cracks using Galerkins method with manyterms. In Ref. [12] Nobile invoked equilibrium of elementary beam theory for evaluating theinternal forces at the crack tip. Recently, Saavedra and Cuitino [13] developed a nite elementmodel for a cracked element to evaluate the dynamic response of a crack-free beam underharmonic forces. Dilena and Morassi [14] detected a single crack in a beam when damage-inducedshifts in the mode shapes of the beam are known.In this work, a procedure is developed to detect the presence of crack in beams and determine

its location and size based on experimental modal analysis results. By applying fracture mechanicsmethods the nite element model was updated using experimental test data. Updated value of theYoungs modulus is calculated using modal analysis results. In the proposed method, a more exactmodel is developed in calculating inuence coefcients of the cracked beam using a high accuracyformulation with 11th order.

2. Crack model description

The presence of a crack in a structural member alters the local compliance that would affect thevibration response under external loads. This effect is related to the crack tip stress intensity factors.Consider a uniform EulerBernoulli beam as shown in Fig. 1. The length of the beam is L and it

has a crack at a distance l1 from the xed end. The beam is divided into elements. The beamstiffness matrix is derived using the energy method. The strain energy for the uncracked beamelement, subjected to a total bending moment Mb is given by

U0 1

2

Z lo

M2bEI

dx, (1)

where E is the Youngs modulus, I is the geometric moment of inertia and l is the length of theelement. The total bending moment may be written as

Mb Tl M, (2)

where T is the shear force and M is the bending moment on the element.

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 147714971480The additional strain energy due to the presence of crack is given by Tada et al. [15] as

U1 Z

A

1

AE0K2I K2II

1

1 nK2III

dA, (3)

where A is the crack cross-section, E0 E for plane stress and E0 E=1 u2 for plane strain, u isthe Poissons ratio and K I; K II and K III are stress intensity factors for opening, sliding and tearing-type cracks, respectively. To apply the linear fracture mechanics theory, it is necessary to considera plane strain state. Neglecting the effect of axial force, for a beam with cross-sectional area b hand a crack with depth a, Eq. (3) may be written as

U1 bZ a

O

fKlM KlP2 K2IIP=E0gda. (4)

The stress intensity factors are then given as [15]

KlM 6M

bh2pa

pF Is,

K IP 3TL

bh2pa

pF Is,

T p

Fig. 1. Schematic model of beam.K IIP bh

paF IIs, (5)

where s a=h and F Is and F IIs for a rectangular cross-section are expressed as follows:

F Is 2

pstan

ps2

r0:923 0:1991 sinps

24

cosps2 ,

F IIs 3s 2s21:122 0:561s 0:085s2 0:18s3

1 sp . (6)

From the denition of the compliance, the exibility coefcient for an element without crack isobtained as

C0ij

q2U0qTiqTj

; T1 T ; T2 M; i; j 1; 2. (7)

The additional exibility coefcient introduced due to the crack is

C1ij

q2U1qTiqTj

; T1 T ; T2 M; i; j 1; 2. (8)

3. Cracked beam nite element

In order to model the effect that the crack introduces to a structure, the stiffness matrix of a

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 14771497 1481cracked beam element is derived. The element is assumed to have a transverse crack under bendingand shearing forces. The equilibrium condition of the cracked element in Fig. 2 may be expressed as

Ti Mi Ti1 Mi1T T Ti1 Mi1T, (9)where T is the transformation matrix given as

T 1 l 1 00 1 0 1

T.

Applying the principle of virtual work, the stiffness matrix of the cracked beam element is dened as[16]

KC T TC1T . (10)From Eqs. (1), (2) and (4)(8) the elements of exibility matrix C may be obtained as

C11 l3

3EI 2B19l2B2 h2B3,

C12 l2

2EI 36lB1B2

C22 l

EI 72B1B2, (11)

where

B1 p1 v2

Ebh2; B2

Z :s:0

sF Isds; B3 Z :s

:0sF IIsds.Fig. 2. Cracked element diagram.

elements as a model. Using the nite element method and MATLAB program [17], the naturalfrequencies and mode shapes of the cracked beam are obtained for an edge crack located at a

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 147714971482normalized distance x=L from the xed end and with a normalized depth a=h in each location.The normalized natural frequencies are dened as the ratio of cracked beam natural frequency tothe uncracked beam natural frequency.Variations of the rst three normalized natural frequencies of the beam model in terms of crack

depth ratio and cracked element location are constructed and shown in Fig. 3. By intersecting thethree-dimensional plots at different crack element locations, variations of normalized naturalfrequencies of the beam model as a function of the crack depth ratio, for different crack elementlocations are obtained and shown in Fig. 4. Variations of the normalized natural frequencies ofthe beam model in terms of the cracked element number, for different crack depth ratios, areillustrated in Fig. 5.As can be seen from Fig. 4, natural frequencies of the cracked beam decrease as crack grows

deeper. The fundamental frequency is mostly affected when crack is located on the rst elementwhich is near the xed end. The reason is that the presence of a crack near the xed end reducesUsing Eq. (10), the cracked element stiffness matrix becomes

KC 1

C11C22 C12C21

C22 C22l C21 C22 C21C22l C21 C22l2 C21l C12l C11 C22l C12 C21l C11

C22 C22l C21 C22 C21C12 C12l C11 C12 C11

2666664

3777775. 12

The integrals B2 and B3 were evaluated using the MAPLE program [17] (see Appendix). Due tonegligible effect of crack on the mass of the beam element, mass matrix of the cracked element isassumed to be the same as that of uncracked element.

4. Crack detection

Detection of crack in a beam is performed in two steps. First, the nite element model of thecracked cantilever beam is established. The beam is discretized into a number of elements, and thecrack position is assumed to be in each of the elements. Next, for each position of the crack ineach element, depth of the crack is varied. Modal analysis for each position and depth is thenperformed to nd the natural frequencies of the beam. Using these results, a class of three-dimensional surfaces is constructed for the rst three modes of vibration, which indicate naturalfrequencies in terms of the dimensionless crack depth and crack position.

4.1. Cracked element behavior

To show the vibration behavior of a cracked beam, consider a cantilever beam with vethe stiffness near the support signicantly. The second and third natural frequencies change

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Fig. 3. Variations of normalized natural frequencies of the beam model versus crack depth ratio and crack element

location for different modes.

H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 14771497 1483

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Fig. 4. Variations of normalized natural frequencies versus crack depth ratio for different crack element locations:

(a) 1st frequency; (b) 2nd frequency; (c) 3rd frequency.

H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 147714971484

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 14771497 1485rapidly as the crack is located on the third element. It can be inferred that the changes in thefrequencies of higher modes depend on how close the crack location is to the nodes of thecorresponding mode shape. As a consequence, natural frequency of the cracked beam isunaffected for a crack located at the nodal points of the corresponding mode shape.From the results it can be seen that the decrease in the frequencies is greatest for a crack located

at the point of maximum bending moment. The trends of changes of the second and third naturalfrequencies, also, are not monotonic as in the rst natural frequency (see Fig. 5). It can beconcluded that both crack location and crack depth have inuence on the frequencies of the

Fig. 5. Variations of normalized natural frequencies versus cracked element number for different crack depth ratios:

(a) 1st frequency; (b) 2nd frequency; (c) 3rd frequency.

cracked beam. Also, a certain frequency may correspond to different crack depths and locations.Based on this, the contour line which has the same normalized frequency change can be plottedhaving crack location and crack depth as its axis. The location and depth corresponding to anypoint on this curve becomes a possible crack location and depth.

4.2. Identification of crack location

4.2.1. Numerical analysis

In order to show effectiveness of the proposed method, a steel cantilever beam is considered.Table 1 shows the geometrical and material properties of the beam. By modal analysis rst threenatural frequencies and corresponding mode shapes of the beam are obtained. As shown in Fig. 6,the beam is discretized into nine elements and crack is considered to be located on one element ateach modal analysis. For better convergence, the size of the cracked element is considered to besmaller than that of the other elements. For each crack location different depth ratios areconsidered. The stiffness matrix of the cracked beam is constructed using the method presented in

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Table 1

Geometrical and material properties of the beam

Beam length (mm) Cross section mm2 Density kg=m3 Poissons ratio Updated Youngs modulus (Mpa)

290 22:5 13 7800 0.3 175 000

H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 147714971486Fig. 6. Geometry of the beam with a crack; lengths in mm.

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 14771497 1487Section 3. Fig. 7 illustrates three-dimensional surfaces of the rst three natural frequencies of thebeam versus crack location and crack depth ratio.To overcome the inuence of unsatisfactory precision of published value for Youngs modulus

its updated value is calculated following Adams et al. [18]. Using natural frequencies of the intactbeam obtained from modal analysis, the updated Youngs modulus can be computed as [16]

Eeff mL4o2il2i I

, (13)

where m is the mass per unit length, L is the beam length, oi are natural frequencies of the intactbeam, I is the cross-sectional moment of inertia, and li are determined according to the boundaryconditions of the beam. For a cantilever beam l1 and l2 are given as 1.8751 and 4.6941,respectively [16]. Updated values of Youngs modulus are calculated for the rst and secondnatural vibrating modes of the uncracked beam and their average value is presented in Table 1.The natural frequencies of the cracked beam are then computed using the updated value of theYoungs modulus which is determined for the uncracked beam.

4.2.2. Experimental analysis

The experimental setup, shown in Fig. 8, is designed to measure natural frequencies and modeshapes of the beam. An impact hammer (type B&K 8202) generates excitation on the nodes of thenite element beam model. The excitation is applied parallel to the neutral plane of the beam. Anaccelerometer (type PCB Triaxial ICP) is attached to the right node of element 8 to capture thevibration signals. A power amplier (type 2707 B&K) is also utilized in the experiment.Since the frequencies could be measured accurately for low modes of vibration, only the

contour lines for the rst and second modes are plotted. Generally, the rst two measuredfrequencies and modes of vibration are sufcient to identify the crack in the beam. Initially, modaltests are conducted with the uncracked beam to determine rst two natural frequencies and modeshapes of the beam. In order to get best results, excitation is applied to different nodes of thebeam. The test was repeated 8 times. The locations of excitation and the position of accelerometerare shown in Fig. 9. Fig. 10 shows FRF diagrams of the intact and cracked beam. The softwareICATS is utilized for signal analysis. The natural frequencies of the uncracked beam are presentedin Table 2. The corresponding mode shapes are shown in Fig. 11.The crack was introduced by a saw perpendicular to the longitudinal axis of the beam. For this

case the crack is assumed to be always open during vibrations. The true crack position and depthratio were 135mm and 0.5, respectively. In the procedure of identifying the crack location, rst,the natural frequencies of the cantilever beam are measured experimentally. The rst and secondnatural frequencies of cracked beam are shown in Table 2. The corresponding mode shapes areshown in Fig. 12.The three-dimensional surfaces shown in Fig. 7 are then intersected by the constant planes

corresponding to the experimental values of natural frequencies of the cracked beam. On thisbasis, a contour line which has the same frequency of a particular mode could be plotted withcrack location and crack depth ratio as its axes. The contour lines for the rst and second modesare shown in Fig. 13. These plane curves show couples of crack depth ratio and crack locationsthat result the natural frequencies of the cracked beam. Hence, the natural frequencies of the

beam could correspond to different crack locations and crack depths.

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Fig. 7. Variations of natural frequencies of the beam versus crack location and crack depth ratio: (a) 1st mode; (b) 2nd

mode; (c) 3rd mode.

H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 147714971488

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 14771497 14894.2.3. Error functionTo nd the element containing the crack, the experimentally measured mode shapes are

compared with the modes generated from the nite element analysis of the cracked beam. Anerror function is dened for each mode shape i, for the crack located at a specic position X h. Theerror function is expressed as [17]

X hi XJj1

feij fmij

2" #1=2

, (14)

where fij denotes the jth component of the ith mode shape and superscripts e and m refer to theexperimental and nite element analysis, respectively. For i 1 and 2, the values of error functionare computed for the location of crack on different elements. In this stage, the mode shapes of thebeam for the location of crack on different elements, obtained by nite element analysis, areneeded. The values of error function for the rst and second modes are shown in Table 3. It can beobserved that for both modes the error function takes its minimum value when crack occurred inelement 5. This indicates that mode shapes obtained from experiment and nite element analysis

Fig. 8. Experimental setup.

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 147714971490are closer to each other when crack location is in element 5. Hence, for the rst approximation,crack is located in any place on element 5; i.e., in the interval between 130 and 140mm from thexed end of the beam.

Fig. 9. Schematic representation of the beam, points of excitation and point of accelerometer attachment.

Fig. 10. FRF diagrams of the beam: (a) uncracked beam; (b) cracked beam.

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Table 2

Natural frequencies of uncracked and cracked beams

Uncracked beam (Hz) Cracked beam (Hz)

First mode Second mode First mode Second mode

179.66 1208.27 171.18 994.42

Fig. 11. Mode shapes of the uncracked beam: (a) 1st mode; (b) 2nd mode.

H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 14771497 1491

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 1477149714924.3. Crack depth determination

In the previous stage only the cracked element has been detected. Using the rst and secondmodal graphs in Fig. 13, the crack depth ratio is found to be 0.506 and 0.553, respectively. Inorder to nd the location of the crack more precisely, the detected element (number 5) issubdivided into four elements. The same procedure performed in the previous section must beconducted for the new discretized beam to achieve the contour lines illustrating variations of crackdepth ratio in terms of crack location, more accurately. These contours are obtained and depictedin Fig. 14.It should be noted that discretizing the beam into more elements has noticeable effect only in

the crack position. Therefore, using the crack depth ratios determined before, the more accurate

Fig. 12. Mode shapes of the cracked beam: (a) 1st mode; (b) 2nd mode.

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 14771497 1493crack position is obtained from Fig. 14. The results are shown in Table 4. The true crack positionand depth ratio were 135mm and 0.5, respectively.As can be seen from Table 4, the relative errors of the crack position in both modes are less than

1%. For the tested beam the relative error of the crack depth ratio in the rst mode is much lessthan that in the second mode. It should be noted that some errors are inevitable in determinationof experimental mode shapes. For the rst and second modes errors of the experimental naturalfrequencies of the intact beam were calculated to be 0.02% and 4%, respectively, compared to thetheoretical results of the nite element analysis, that are considered as exact. This error may bedue to the fact that xed boundary condition is very difcult to achieve experimentally, and the

Fig. 13. Frequency contour plots: (a) 1st mode (171.18Hz), (b) 2nd mode (994.42Hz).

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Table 3

Values of error function of different cracked elements

Cracked element Error function values in the rst mode Error function values in the second mode

1 0.094 0.244

2 0.105 0.243

3 0.0848 1.078

4 0.057 0.24

5 0.023 0.15

6 0.033 0.86

7 0.2085 0.886

8 0.659 0.85

Fig. 14. Frequency contour plots: (a) 1st mode (171.18Hz); (b) 2nd mode (994.42Hz).

H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 147714971494

different modes, which can be obtained from experimental modal analysis. The contour lines ofthe cracked beam frequencies were plotted for the rst and second modes. In a situation that the

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Depth ratio 0.5 0.506 1.2 0.553 10crack position coincides with a vibration node, the frequency remains unchanged, and the contourlines tend to disappear. In this case, contours of the higher modes should be obtained to detect thecrack correctly.The identication procedure presented in this study is believed to provide a useful tool for

detection of medium size cracks in a beam. For such size of cracks, the method gives reliableand accurate results (within 1%) for crack depth. The relative error at crack location is alsoeffect of damping, which is ignored in the nite element analysis. The assumption that was madeon opening of the crack may not also be completely true during the test. This creates somedamping which has inuence on the results.Due to insignicant change in the stiffness of the cracked beam, the proposed method cannot

detect cracks with small relative depth of about 0.25 or less. This connes the applicability of themethod to moderate size cracks.

5. Conclusions

A method for identifying the crack location and depth of the uniform cantilever beam wasdeveloped by using the linear fracture mechanics theory. The nite element model of the crackedbeam is constructed and used to determine its natural frequencies and mode shapes. From thetheoretical analysis and experimental measurements, it is found that the crack location, as well ascrack size, has noticeable effects in the rst and second natural frequencies of the cantileverbeam. Natural frequencies decrease signicantly as the crack location moves towards the xedend of the beam. In fact, the crack near the xed ends would modify the boundary constraints ofthe beam.In practice, one needs natural frequency ratios of the cracked beam to the intact one in twoTable 4

Crack depth ratio and crack location obtained from Fig. 14

Exact First mode Error (%) Second mode Error (%)

Location (mm) 135 134.4 0.4 134.6 0.3

H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 14771497 1495acceptable.

Appendix A

Using MAPLE program [17] the integrals B2 and B3 are evaluated as

B2 0:1215854204 107

18b

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 147714971496

139124880 sin3ps 6259043520 sinps cos2ps6259043520 logsecps tanps cos2ps398772120 tan2ps cos2ps6259043520 logcosps cos2ps 709379280 sin5ps2039972880 sin3ps cos2ps 842641620 sin6ps1365374820 sin4ps cos2ps2730749640 sin2ps cos2ps798356160 sin5ps cos2ps 33264800 sin8ps348488800 sin6ps cos2ps 95042400 sin9ps95042400 sin7ps cos2ps 11880300 sin10ps11880300 sin8ps cos2ps 2545670725456707 cos2ps

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

=cos2ps,

B3 1

b

0:01178181818 s11 0:01368 s10 0:1101488889 s9 0:356885 s80:082612 s7

1:483734833 s6 3:1577768 s5 2:66192 s4 0:2274253333 s3 0:341138 s2

0:682276 s 0:682276 logabs1 s

0BB@

1CCA.

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[9] Qian GL, Gu SN, Jiang JS. The dynamic behavior and crack detection of a beam with a crack. Journal of Soundand Vibration 1990;138(2):23343.

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H. Nahvi, M. Jabbari / International Journal of Mechanical Sciences 47 (2005) 14771497 1497

Crack detection in beams using experimental modal data and finite element modelIntroductionCrack model descriptionCracked beam finite elementCrack detectionCracked element behaviorIdentification of crack locationNumerical analysisExperimental analysisError function

Crack depth determination

ConclusionsReferences