a novel method crack detection beam like structures measurements natural frequencies

Accepted 13 April 2014Handling Editor: L.G. Tham

Keywords:

parameters is derived as a modification of the Rayleigh quotient for the multiple crackedbeams that differ from the earlier ones by including nonlinear terms with respect to crack

ucturposedg the

sensitivity of thelso, the detectionto overcome the

tic equation thatn was conductedent forms for the

beam with multiple cracks [1620]. Though the characteristic equation has been obtained explicitly, the natural frequencies

frequencies causes a difficulty in solving the problems associated with the damage detection from only natural frequencies.

Contents lists available at ScienceDirect

Journal of Sound and Vibration

Journal of Sound and Vibration 333 (2014) 408441030022-460X/& 2014 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.jsv.2014.04.031

n Corresponding author.E-mail addresses: [email protected] (N.T. Khiem), [email protected] (L.K. Toan).could be computed just numerically as the implicit functions of damage parameters. This implicit representation of naturalThis is because of the fact that the natural frequencies are most easily and accurately measured in compdynamic characteristics of a structure. The major drawbacks of the frequency-based approach are the weakmeasured frequencies to damage, and the same change in frequencies might be caused by different damages. Aof unknown number of damages in a structure is in general an unsolved problem. Therefore, seeking the wayshortcomings of the frequency-based methods of damage detection is a promising subject.

The theoretical basis of the frequency-based methods for damage detection is the so-called characterisrelates the natural frequencies to damage parameters. The first compact form of the characteristic equatioin [1215] for a beam-like structure with a single crack. Then, the equation has been established in differdamage-induced change in natural frequencies have been most early engaged [48] and they have been used until now [911].arison with other1. Introduction

The condition assessment of strA number of methods have been prodynamic characteristics [13]. AmonMultiple cracked beamModal analysisFrequency-based methodCrack detectionRayleigh quotientthe beamwith arbitrary number of cracks instead of solving the complicated characteristicequation. The obtained nonlinear expression for natural frequencies in combination withthe so-called crack scanning method proposed recently by the authors allowed thedevelopment of a novel procedure for consistent identification of unknown amount ofcracks in the beamwith a limited number of measured natural frequencies. The developedtheory has been illustrated and validated by both numerical and experimental results.

& 2014 Elsevier Ltd. All rights reserved.

e and machinery is a vital concern in structural and mechanical engineering.to detect damage in structures and most of them are based on the change in theirnumerous procedures developed for damage detection, the techniques based onAvailable online 10 May 2014 severity. This expression provides a simple tool for calculating the natural frequencies ofA novel method for crack detection in beam-like structuresby measurements of natural frequencies

N.T. Khiem n, L.K. ToanInstitute of mechanics, VAST, 264, Doi Can, Ba Dinh, Hanoi, Vietnam

a r t i c l e i n f o

Article history:Received 12 June 2013Received in revised form20 January 2014

a b s t r a c t

A novel method is proposed for calculating the natural frequencies of a multiple crackedbeam and detecting unknown number of multiple cracks from the measured naturalfrequencies. First, an explicit expression of the natural frequencies through crack

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

N.T. Khiem, L.K. Toan / Journal of Sound and Vibration 333 (2014) 40844103 4085An explicit expression of natural frequencies in terms of crack magnitude was derived approximately in [21] and applied fordamage detection in [22] for the case of small cracks by using the perturbation method. A system of linear equations relatingthe shift of natural frequencies with a variation of both the crack magnitudes and positions has been conducted in [23] but itwas determined only numerically using the finite element method. By introducing the so-called element damage index, theauthors of references [2426] were able to express natural frequency shifts in terms of the damage indices in the form oflinear equations that provide a useful tool for damage localization by measured natural frequencies. Although theconventional Rayleigh method was earlier employed for determining the natural frequency of a cracked beam by Shenand Pierre [27], an explicit expression of natural frequency was obtained much later by Fernandez-Saez et al. in [28] withthe use of the Rayleigh method. Nevertheless, this appealing expression is applied only for calculating the fundamentalfrequency of a beam with a single crack. Later, Fernandez-Saez and Navarro [29] obtained a more accurate expression ofnatural frequencies for a singly cracked beam, but it was limited to applying for determining the upper and lower boundsof the fundamental frequencies only. Recently, an expansion of the Rayleigh quotient for calculating the natural frequenciesof a cracked beam has been developed in [30] but as with the former results, it has not been straightforward to use for thecrack detection problem.

Note here that though the comprehensive literature on the development of the frequency-based method for damagedetection in structures has been published, very few papers are devoted to investigating the case of previously unknownnumber of damages. This posed a more widespread problem to predict also the number of damages mutually with theirlocations and sizes by the measurement of modal parameters. Developing the idea that emerged in the paper [8], theso-called crack scanning method is proposed in [31] to detect unknown number of cracks in a beam based on the measurednatural mode shape. The main idea of the procedure is first to estimate unknown magnitudes of all the cracks assumed at achosen grid of positions in the structure using the given data and this process is performed iteratively by eliminating thepositions from the grid where the estimated magnitudes are zero or negative. The actual cracks would be acknowledged atthe locations of the grid that could not be reduced by the removing positions with zero and negative magnitude. Actually,the proposed procedure enables us to determine not only the location and magnitude but also the quantity of cracks.Certainly, this procedure can be applied not only for the case of measured mode shapes as performed in [31] but also for thecase of other modal parameters such as natural frequencies or frequency response functions.

The present paper aims to develop the scanning method for detecting unknown number of cracks in a beam by themeasurement of natural frequencies. Firstly, the Rayleigh quotient is derived for the multiple cracked beam that enables usto conduct an explicit expression of natural frequencies in terms of crack positions and sizes by choosing the shape functionfirst suggested in [28]. Such obtained frequency representation provides a simple and efficient tool for calculating everynatural frequency of the beam with arbitrary number of cracks. The most important difference of the constructed explicitexpression from those derived by the perturbation [21], sensitivity method [23] and the energy approach [2426] is that theobtained herein expression included additionally the nonlinear terms of the crack magnitudes. Then, the obtained explicitexpression is straightforward to apply the aforementioned scanning method for identification of the multiple cracks fromnatural frequencies. In this regard, the nonlinear terms taken into account could be helpful for us to overcome the non-uniqueness solution of damage detection problem in the beam with symmetrical boundary conditions. The theoreticaldevelopment is validated by both numerical and experimental examples.

2. The Rayleigh quotient for the multiple cracked beam

Let's consider a uniform EulerBernoulli beamwith clamped ends and the following material and geometrical constants:Young's modulus E, mass density , length L, cross section area Fbh and moment of inertia I. Suppose, moreover, that thebeam has been damaged to crack at a number of positions 0!e1!!en!1 with the depth (a1,, an). If the springmodel of the cracks is adopted the spring stiffness Kj is calculated from the crack depth aj by [16]

j EI=LKj 5:346h=LIcaj=h; (1)

Icz 1:8624z23:95z316:375z437:226z576:81z6126:9z7172z8143:97z966:56z10;where the parameter jEI/LKj has been introduced to represent severity of the crack and termed by crack magnitude.

For the beam, kth natural frequency and mode shape denoted by k, k(x) satisfy equation

d4kx=dx44kkx 0; xAej1; ej; j 1;;n1; e0 0; en1 1;4k L4F2k=EI (2)

and boundary conditions

k0 0k0 k1 0k1 0: (3)

Additionally, the mode shape k(x) should satisfy the following conditions at cracks:

kej kej ;kej kej ;kej kej ; 0kej kej jkej; j 1; :::;n: (4)

N.T. Khiem, L.K. Toan / Journal of Sound and Vibration 333 (2014) 4084410340860.995

0.996

0.997

0.998

0.999

1

1.001

of fi

rst e

igen

valu

eNow, first, multiplying both sides of Eq. (2) by k(x), then taking integration along the beam segment (ej1, ej) andsummation lead to

n1

j 1

Z ejej 1

d4kxdx4

kxdx( )

4k n1

j 1

Z ejej 1

2k xdx( )

: (5)

Note that for the functions x; 0x; x; 0x that are all continuous in the segment (a,b) it can be easily obtained

Z ba

d4xdx4

xdxZ ba

2xdx BbBa ; (6)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.991

0.992

0.993

0.994

Crack position along beam length

Rat

io

Fig. 1. Comparison of the fundamental frequency computed by using the characteristic equation method (solid and black lines) and the Rayleighapproximation with different values of the correcting factor (the dotted lines represent the first order approximation). The crack depth equals 10%, 20%,and 30%.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99

0.992

0.994

0.996

0.998

1

1.002


Rat

io o

f sec

ond

eige

nval

ue

Fig. 2. Comparison of the second frequency computed by using the characteristic equation method (solid black lines) and the Rayleigh approximation withdifferent values of the correcting factor (the dotted lines represent the first order approximation). The crack depth equals 10%, 20%, and 30%.

N.T. Khiem, L.K. Toan / Journal of Sound and Vibration 333 (2014) 40844103 4087where

Bx 0xxx0x:

Applying the equality (6) for the integrals on the left hand side of Eq. (5) we notice that e00, en11 gives

n1

j 1

Z ejej 1

d4xdx4

kxdxZ 10

2k xdx Bk1Bk0 n

j 1Bkej Bkej :

Taking into account the conditions (3) and (4) the latter equation becomes

n1 Z ej d4x

xdxZ 1

2xdx n

2e (7)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.988

0.99

0.992

0.994

0.996

0.998

1

1.002


Rat

io o

f Thi

rd e

igen

valu

e

Fig. 3. Comparison of the third frequency computed by using the characteristic equation method (solid and black lines) and the Rayleigh approximationwith different values of the correcting factor (the dotted lines represent the first order approximation). The crack depth equals 10%, 20%, and 30%.j 1 ej 1 dx4 k

0k

j 1j k j

and, consequently, Eq. (5) can be rewritten as

4k Z 10

2k xdx n

j 1j

2k ej

" # Z 10

2k xdx:,

(8)

This is the Rayleigh quotient extended for the multiple cracked beam. Note that the well-known Rayleigh quotient in thepast has been utilized for calculating the natural frequencies with a properly chosen mode shape as trial shape function. Inthe present study, the quotient (8) is employed for establishing an explicit expression of the eigenvalue in terms of crackparameters (positions and magnitude). For the purpose, following Fernandez-Saez et al. [28] the mode shape function isselected in the form

kx 0kxAkjx3Bkjx2CkjxDkj; xAej1; ej; j 1;;n1; (9)

where 0k(x) is kth mode shape of intact beamwith clamped ends and Akj, Bkj, Ckj, Dkj are constants determined as following.Substituting the shape function (9) into conditions (4) at the crack position ej yields

Akj Ak;j1;Bkj Bk;j1;Ck;j1 Ckjjkej; Dk;j1 Dkjjkejej (10)

or

Akj Ak1;Bkj Bk1;Ckj Ck1 j1

i 1i

kei;

Dkj Dk1 j1

i 1eii

kei; j 1;;n1: (11)

N.T. Khiem, L.K. Toan / Journal of Sound and Vibration 333 (2014) 4084410340880.995

1

0.98

0.99

1

0.98

0.99

1The latter equations show that it remains only four constants Ak1, Bk1, Ck1, Dk1 that can be easily determined fromboundary conditions (3) rewritten in the form

Ck1 Dk1 0;

Ak1Bk1 n

j 1j1ejkej 0; 3Ak12Bk1

n

j 1j

kej 0 (12)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.975

0.98

0.985

0.99

Crack depth

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.94

0.95

0.96

0.97

Crack depth

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.94

0.95

0.96

0.97

Crack depth

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

Crack depth

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Crack depth

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.94

0.95

0.96

0.97

0.98

0.99

1

Crack depth

Fig. 4. The six lowest natural frequencies versus equal crack depth (from 0% to 50%) of triple cracks at 0.2; 0.45 and 0.7 in different nonlinear factorscompared to solution of the characteristic equation. (a) 1st, (b) 2nd, (c) 3rd, (d) 4th, (e) 5th and (f) 6th frequencies.

Fig. 5. Experimental model for modal testing fixedfixed beam.

Table 1Comparison of calculated and measured natural frequencies for a beam with triple cracks at positions 0.2; 0.45; 0.7 in different scenarios of depth from 0%to 50%.

Crack scenarios Number of frequencies

1 2 3 4 5 6

(000)% (Uncracked) (a) 43.1596 118.971 233.2307 385.5422 575.9336 804.4030(b) 43.17 119.0 233.3 385.6 576.1 804.6(c) 43.1596 118.971 233.2307 385.5422 575.9336 804.4030

(0100)% (Single crack) (a) 43.1377 118.9588 233.1035 385.4218 575.7544 803.9720(b) 43.14 118.99 233.16 385.52 575.91 804.17(c) 43,1376 118,9588 233,1031 385,4214 575,7539 803.9700

(10100)% (Double crack) (a) 43,1371 118,9311 232,9609 385,2174 575,6749 803.9555(b) 43,14 118.96 233.0 385.29 575.82 804.15(c) 43,1370 118,9310 232,9603 385,2155 575,6744 803.9531

(101010)% (Triple crack) (a) 43,1323 118,8541 232,9065 385,1886 575,2884 803.5648(b) 43.14 118.87 232.93 385.2 575.38 803.7(c) 43,1323 118,8522 232,9047 385,1858 575,2675 803.5540

(102010)% (Triple crack) (a) 43,0714 118,8202 232,5525 384,8570 574,7953 802.3735(b) 43.07 118.82 232.52 384.87 574.8 802.29(c) 43,0709 118,8175 232,5444 384,8450 574,7522 802.3591

(202010)% (Triple crack) (a) 43,0696 118,7435 232,1551 384,2981 574,5729 802.3202(b) 43.07 118.74 232.14 384.26 574.55 802.29(c) 43,0692 118,7387 232,1450 384,2690 574,5226 802.3081

(202020)% (Triple crack) (a) 43,0567 118,5293 232,0036 384,2200 573,5113 801.2460(b) 43.05 118.52 231.99 384.18 573.46 801.19(c) 43,0557 118,5028 231,9776 384,1791 573,2092 801.0904

(203020)% (Triple crack) (a) 42,9502 118,4698 231,3852 383,65501 572,6729 799.1894(b) 42.94 118.46 231.35 383.58 572.56 799.04(c) 42,9474 118,4383 231,3258 383,5639 572,2382 799.0690

(303020)% (Triple crack) (a) 42,9469 118,3376 230,6894 382,7121 572,2812 799.1226(b) 42.9 118.32 230.64 382.55 572.17 798.95(c) 42,9442 118,2944 230.6249 382.5520 571.7976 798.9608

(303030)% (Triple crack) (a) 42,9247 117,9643 230,4249 382.5813 570,4717 797.2471(b) 42.92 117.93 230.36 382.41 570.22 796.99(c) 42.9198 117.8232 230.2858 382.3678 568.8627 796.4205

(303040)% (Triple crack) (a) 42,8883 117,3613 230.0024 382,3702 567,5830 794.3503(b) 42.88 117.29 229.91 382.17 566.98 793.74(c) 42.8752 116.9249 229.6113 381.9864 562.7077 790.7068

(304040)% (Triple crack) (a) 42,7170 117,2598 228,9992 381,4957 566,3634 790.9806(b) 42.69 117.19 228.84 381.16 565.48 790.12(c) 42,6991 116.7935 228.4308 380.8805 560.6276 788.1119

(404040)% (Triple crack) (a) 42,7113 117,0542 227,8633 380,0624 565,7112 790.9013(b) 42.69 116.96 227.65 379.45 564.81 789.98(c) 42.6932 116.5210 227.3346 379.2809 559.6245 787.7905

(504040)% (Triple crack) (a) 42,7019 116,7131 226,0229 377,8217 564,7007 790.7747(b) 42.68 116.56 225.58 376.5 563.68 789.74(c) 42.6830 116.0545 225.4519 376,5362 557.9088 787.2419

(505040)% (Triple crack) (a) 42,4167 116,5592 224,3570 376,5532 562,6580 785.5926(b) 42.37 116.39 223.73 374.62 561.09 783.53(c) 42.3807 115.8285 223.3967 374.6201 554,3245 782.7899

(505050)% (Triple crack) (a) 42,3618 115,5740 223,6569 376,2644 558,3072 780.8314(b) 42.3 115.27 222.9 374.34 555.47 777.87(c) 42.3028 113.7744 221.8603 373.7671 537.6546 770.4146

(a) Solution of characteristic equation; (b) measurement; (c) calculated by the Rayleigh quotient actual crack positions: (0.20.450.7).

N.T. Khiem, L.K. Toan / Journal of Sound and Vibration 333 (2014) 40844103 4089

Fig. 6. Iterative localization of single crack at position 0.45 with 10% depth.

Fig. 7. Iterative localization of double cracks at 0.2 and 0.45 with equal 10% depth.

N.T. Khiem, L.K. Toan / Journal of Sound and Vibration 333 (2014) 408441034090

Fig. 8. Iterative localization of triple cracks at 0.2, 0.45 and 0.7 with equal 10% depth.

Fig. 9. Iterative localization of triple cracks at 0.2, 0.45 and 0.7 with 10%20%10% depth.



Fig. 11. Iterative localization of triple cracks at 0.2, 0.45 and 0.7 with equal 20% depth.


or

Ak1 n

j 1j12ejkej; Bk1

n

j 1j3ej2kej: (13)

Thus, the shape function (9) is completely determined and it allows us to calculate the numerator and denominator ofthe quotient (8) as follows:Z 1

02k xdx

Z 10

0kx6Ak1x2Bk12dxZ 10

20kxdx4Z 10

0kx3Ak1xBk1dx

4Z 10

3Ak1xBk12dxZ 10

2k0xdx12A2k112Ak1Bk1B2k1;

n

j 1j

2k ej

n

j 1j

20kej4

n

j 1j

0kej3Ak1Bk1:

Therefore, the numerator of quotient (8) becomes

NumZ 10

20kxdx n

j 1j

20kej21;

21 n

i;j 1ijij

0keik0ej; ji 6eiej15eiej4; j; i 1;;n: (14)

On the other hand,Z 10

2k xdx n1

j 1

Z ejej 1

2k xdx( )

n1

j 1

Z ejej 1

20kxdx2Z ejej 1

0kxAk1x3Bk1x2CkjxDkjdx" #

n1

j 1

Z ejej 1

Ak1x3Bk1x2CkjxDkj2dx" #

Z 10

20kxdx2122; (15)

where the following notations have been introduced

1 Z 10

0kxAk1x3Bk1x2dx n1

j 1

Z ejej 1

0kxCkjxDkjdx 4k0 n

j 1j

20kej; (16)

22 Z 10

Ak1x3Bk1x22dx n1

j 1

Z ejej 1

CkjxDkj2dx2Ak1x3Bk1x2CkjxDkj

dx A2k1

7Ak1Bk1

3B

2k1

5C

2k;n13

Ck;n1Dk;n1D2k;n14Ak15Bk1

10Ck;n1

13n

j 12j

20keje3j

3Ak14Bk16

Dk;n1 n

j 1j

0kej

Ak110

e5j Bk16e4j

Ckj3e3j Dkje2j

n

j 1qjiji

k0ejk0ei; (17)

qij 17=21717ej=140153e2j =35e3j =3e4j =6e5j =10; j i;2=1543ejei=40ejei2eiej=212ej12ei=73ej23ei2=5

(

3ejei2e3j e3i =12ejei=51=3e4j e4i =2e5j e5i =5; ia j;

In the latter equations the terms of order 3 with respect to the crack magnitudes have been neglected by using thefollowing approximation of constants Ckj, Dkj, Ak1, Bk1 defined in (11) and (13)

Ckj j

i 1i

kei

j

i 1i

0kei; Ak1

n

j 1j12ejkej

n

j 1j12ej0kej;

Dkj j

i 1eii

kei

j

i 1eii

0kei; Bk1

n

j 1j3ej2kej

n

j 1j3ej20kej:

Finally, the quotient (8) in this case can be written as

4k Z 10

20kxdx n

j 1j

20kej21

" # Z 10

20kxdx24k0 n

j 1j

20kej22

" #,(18)


N.T. Khiem, L.K. Toan / Journal of Sound and Vibration 333 (2014) 408441034094with 21, 22 defined in (14) and (17) as bilinear forms of {1,,n}. If the mode shape of the intact beam is chosen to benormalized so that

R 10

20kxdx 1, consequently,

R 10

20kxdx 40k, then the quotient (18) becomes

k=0k4 14k0 n

j 1j

20kej4k0 21=124k0

n

j 1j

20kej22: (19)

This is an explicit expression of natural frequencies of the multiple cracked beam in terms of crack parameters.Representing the right hand side of Eq. (19) as the function

f 1

n

j 1ajej

n

i;j 1bijeij

12 n

j 1ajej

n

i;j 1dijeij

;

then expanding the function in Taylor series one obtains

f 1 n

j 1f 0j0j1=2

n

i;j 1n

j 1f ij0ijojj3

1 n

j 1f jej

n

i;j 1n

j 1f ijeij;

where f je aje; f ije bijedije and is a correcting factor that could be chosen to minimize the cut-off error ofthe Taylor series. So, the second order asymptotic approximation with respect to small crack magnitudes {1,,n} of Eq. (19)can be represented as

4k 4k0 n

j 1j

2k0ejk

n

i;j 1ijijei; ejk0eik0ej (20a)

with ijei; ej 4k0qijij. The linear approximation is obtained by setting k0 that gives rise to

4k 4k0 n

j 1j

2k0ej: (20b)



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5 x 10

Crack position

Cra

ck m

agni

tude

The First iteration - Linear model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5x 10

Crack position

Cra

ck m

agni

tude

The 6th iteration - Nonlinear mode

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8x 10

Crack position

Cra

ck m

agni

tude

The 9th iteration - Linear model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5 x 10

Crack position

Cra

ck m

agbn

itude

The 8th iteration - Nonlinear model



Fig. 15. Iterative localization of triple cracks at 0.2-0.45-0.7 with 30%30%40% depth.

Fig. 16. Iterative localization of triple cracks at 0.2-0.45-0.7 with 30%40%40% depth.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 x 10

Scanning crack position

Estim

ated

cra

ck m

agni

tude

The first iteration - Linear model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4x 10


Estim

ated

cra

ck m

agni

tude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6 x 10


Estim

ated

cra

ck m

agni

tude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015


Estim

ated

cra

ck m

agni

tude


Fig. 17. Iterative localization of triple cracks at 0.20.450.7 with 40%40%40% depth.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

x 10


Estim

ated

cra

ck m

agni

tude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014


Estim

ated

cra

ck m

agni

tude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7x 10


Estim

ated

cra

ck m

agni

tude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03


Estim

ated

cra

ck m

agni

tude




N.T. Khiem, L.K. Toan / Journal of Sound and Vibration 333 (2014) 408441034098The different forms of the linear Eq. (20b) have been obtained in [13,20,24] by using different approaches but thenonlinear terms in (20a) have not yet been previously taken into consideration. In a subsequent section the nonlinearexpression (20a) is used to develop a novel procedure for crack detection in the beam from measured natural frequencies.

3. A procedure for crack detection by natural frequencies

The problem investigated in this section is to find out the location, depth and number of cracks in a clamped end beamwith parameters E, , L, F, I from given m natural frequencies 1;;m. Since the number of potential cracks is unknownthe crack scanning method (CSM) proposed in [31] is used.

Procedure of the crack scanning method consists of the following tasks: (1) a number of cracks with unknown magnitude(1,, n) is assumed to occur at a mesh of positions 0re1!e2!!en!1 in the structure of interest; (2) a model ofthe structure with the presence of the cracks is constructed so that it could give rise to a straightforward correlationbetween measurable characteristics of the structure and unknown crack magnitudes (1,, n); (3) the crack magnitudeunknowns (1,,n) are evaluated from the available measured data by using the constructed model; (4) potential cracks areallocated at the positions e^1;; e^nc correspondingly to the definitely positive values ^1;; ^nc from the evaluated crackmagnitudes; (5) the crack positions detected e^1;; e^nc are used as a new crack position mesh for repeating the steps 24until the detected crack mesh is unchanged; (6) actual cracks position and depth are determined from the crack mesh andmagnitudes detected in step 5 and Eq. (1).

Thus, the focus of the CSM applied to the problem of multiple crack detection is to estimate the crack magnitude vector(1,,n)T from the given natural frequencies. The governing equations for the crack magnitude estimation are givenbelow.

As it is well known that the eigenvalues and normalized mode shapes of the intact beam with clamped ends are

10 4:7300408; 20 7:8532046; 30 10:9956078; 40 14:1371655; (21)

k0x sin kx sinh kxAk cos kx cosh kx;Ak sin k sinh k= cos k cosh k: (22)

For the damaged beam the eigenvalues k can be determined from the given natural frequencies 1;;m as4k FL42k=EI; k 1;2;3; and Eq. (19) can be rewritten in the form

AB b; (23)

A akj 2k0ej; B bkj 0kej n

i 1ij

0keii; diag1;; m;

b fb1;; bmgT ; bk k 4k04k ; k 1;;m; j 1;;n: (24)

This is a nonlinear equation for seeking the crack magnitude vector (1,,n) that can be solved by using the iterationmethod

Ai1i b;Ai1 ABi1; 0 0; i 1;2;3; (25)

The iteration process is stopped when ||(N)(N1)||rtolerance.It is necessary to make a note here that the problem for the solution of the linear equation system (25) is ill-conditioned

because of two reasons: first, the measured frequencies are usually contaminated with erroneousness that is able to causethe incorrectness of the solution of the problem; second, the number of measured frequencies is always much less than thenumber of the unknowns, m!!n that should be very large to more accurately capture the unknown crack positions overall the beam length. The ill-posed problem can be resolved in the following manner.

Firstly, the well-known Tikhonov regularization method that suggests replacement of Eq. (25) by the regularized one isemployed

ATi1Ai1I b; (26)with regularization factor determined from the equation

R

k 1

uTkbs2k

!2

n

k R1uTkb2 2; (27)

where R, sk, uk, vk are, respectively, rank, singular values and left and right singular vectors of the matrix Ai1, is the noise(erroneousness) level in the measurement data, the vector b [31]. The unique root of Eq. (27), i, allows one to calculateregularized solution of Eq. (26) as

i R

k 1

skuTkbis2k

!vk: (28)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

x 10


Estim

ated

cra

ck m

agni

tude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4x 10


Estim

ated

cra

ck m

agni

tude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03


Estim

ated

cra

ck m

agni

tude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03


Est

imat

ed c

rack

mag

nitu

de



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03


Estim

ated

cra

ck m

agni

tude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025


Estim

ated

cra

ck m

agni

tude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6x 10

-3


Estim

ated

cra

ck m

agni

tude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

x 10-3


Estim

ated

cra

ck m

agni

tude




N.T. Khiem, L.K. Toan / Journal of Sound and Vibration 333 (2014) 408441034100Secondly, among the crack magnitudes, denoted by ^ ^1;; ^nT , obtained as regularized solution (28) of Eq. (25), onlythe positive values are used to gather the corresponding locations e^1;; e^n^ that can be called herein the positivelydetected crack positions (PDCPs). Such detected crack positions are taken in use as a renewed crack position mesh to buildup the matrices

A^ a^kj 2k0e^j; B^ b^kj 0ke^j nc

i 1ijeî; e^j0keîî; k 1;;m; j 1;; n^: (29)

Using the matrices (29) and repeating the iteration process (25) result in a new solution with respect to crackmagnitudes ^1;; ^n^ that should be enhanced with correctness because the number of unknowns is reduced n^!n.This iteration process with respect to crack position mesh is sustained until the PDCPs get to be unchanged e1;; enc ,which are acknowledged as detected crack locations. Hence, the position and magnitude together with the number of actualcracks have been found and the depth of the cracks localized at the positions e1;; enc can be evaluated from the estimatedcrack magnitudes 1;; nc by solving the equation

5:346h=LIca=h j; j 1;;nc (30)with respect to a and function Ic(z) given in Eq. (1). So, either locations e1;; enc or depths a1;; anc together also withnumber nc of cracks have been identified and solution of the problem posed above is thus completed.

Note also here that to construct the matrix Ai1 in (25) additionally the matrix must be chosen representing thenonlinear terms included in the expression (20a). Assuming the matrix with a positive parameter and diagonalmatrix diag[1,,m] that led to the selection of the parameters and 1,,m for the matrix . The first one callednonlinear factor is selected to adjust the nonlinearity level in general and the subsequent parameters would be selectedindividually for every measured frequency dependent on the sensitivity of the frequency to the nonlinearity factor. Theselection 0 implies the use of linear approximation of the Rayleigh quotient and the positive value of the parameterwould increase the nonlinearity effect on natural frequencies. The mathematical aspect related to the selection of thenonlinearity parameters needs to be studied in future work. In the present paper the selection is accomplished intuitively onthe basis of the numerical analysis of frequencies dependent on the nonlinear terms.

4. Numerical and experimental validation

First, the three lowest eigenvalues of a beamwith single crack have been calculated by using the Rayleigh approximation,Eq. (20a), for different crack depths 10, 20, and 30 percent and correcting factor k r3k0 ; r 0;0:05;0:1;0:2;0:5;0:7;1:0.Note that the case when r0 corresponds to the linear or first order approximation, Eq. (20b). The ratios of the eigenvaluesto the ones of the undamaged beam versus crack position along the beam length are shown in Figs. 13 in comparison withthose computed from the characteristic equation. Obviously, natural frequencies calculated by using the Rayleighapproximations very well agreed with the solution of the characteristic equation for the crack of the depth within 20percent. The disagreement is visibly observed for crack depth approaching 30 percent and this discrepancy gets to be morenoticeable with increasing nonlinear correcting factor in comparison with the linear one. However, for the crack depthwithin 10 percent, see zoomed box in Fig. 1, the nonlinear approximation gives natural frequencies more close to solutionsof the characteristic equation with increasing the nonlinear factor. This means that for the small crack with 10 percent depththe nonlinear terms give better approximation in calculating the natural frequencies than the linear ones. Furthermore, itcan be seen from the figures that the frequencies computed from the characteristic equation and the linear Rayleighapproximation are a symmetrical function of crack position about the middle of the beam. This is surely the reason leadingto the nonunique solution of the crack localization in the beam with symmetrical boundary conditions based only on thecharacteristic equation or the first order Rayleigh approximation. Dissimilarly to the fact, it is apparent also in the figures,see Fig. 3 in the zoomed box, that the nonlinear approximation gives rise to unsymmetrical dependence of the frequencieson the crack position and the asymmetry becomes more significant with increasing the nonlinear factor. It must be notedhere that although the nonlinear terms make the frequency function asymmetrical, they do not modify the frequency nodeswhere the presence of a crack has no effect on the frequency. This might be important to solve the nonuniqueness problemin crack localization in the beam using only natural frequencies.

Secondly, a clamped end beam of the parameters: L104 cm; b2 cm; h0.9 cm; 7855 kg/m3; E2.01011 N/m2with triple cracks at positions 0.2, 0.45 and 0.7 is examined. Namely, the first six natural frequencies of the beam arecomputed by using Eq. (20) and the characteristic equation for equal crack depth varying from 0 percent to 50 percent.Results of computation are shown in Fig. 4 that demonstrates also the ratio of frequencies (damaged to undamaged) versusequal depth of the triple cracks. It can be observed from the later graphics that both the Rayleigh approximations agree wellwith the characteristic equation in calculating natural frequencies for multiple cracks of depth within 20 percent. For thecracks of depth larger than 20 percent, sensitivity of the frequencies calculated from the Rayleigh approximation gets to behigher than the solution of the characteristic equation. It is promising to use the Rayleigh quotient for multiple cracks in thebeam.

Finally, an experiment, as shown in Fig. 5, has been carried out in the Laboratory of Structural Dynamics, IMECH-VAST tomeasure natural frequencies of the beam with triple cracks that were studied above numerically. In this experiment, thetriple cracks are fabricated by saw cut with depth varying in the step of 10 percent from 0 percent to 50 percent. Scenarios of

N.T. Khiem, L.K. Toan / Journal of Sound and Vibration 333 (2014) 40844103 4101Table 2Numerical results of crack detection.

Actual crack scenarios Estimated crack parameters

0.4510% Number of detected cracks 2(Single crack) Position of detected cracks 0.45 0.5

Depth of detected cracks 0.1068 2.24e-50.20.45 Number of detected cracks 210%10% Position of detected cracks 0.2 0.45(Double crack) Depth of detected cracks 0.1 0.10.20.450.7 Number of detected cracks 3(101010)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.1 0.1 0.10.20.450.7 Number of detected cracks 3(102010)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.1 0.2001 0.10.20.450.7 Number of detected cracks 3(202010)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.2001 0.2001 0.10.20.450.7 Number of detected cracks 3(202020)% Position of detected cracks 0.2 0.45 0.7crack depth used for either computation or experimentation are given in Table 1 that include the case of single and doublecrack.The Multi Channel Signal Acquisition System PULSE 3060 is employed with one channel used for the impact hammer andanother channel used to measure the acceleration. In order to gather data, an accelerometer was roving through 9 points:6.25 cm, 10 cm, 12.5 cm, 16.7 cm, 25.1 cm, 30 cm, 31.25 cm, 37.5 cm and 50.2 cm from the right end, while the impactexcitation was applied at the point opposite to the accelerometer. The frequency response functions gathered from everymeasurement were collected and employed for extracting natural frequencies using the software ME'SCOPE 5.0. The firstsix measured natural frequencies are presented in Table 1 in comparison with those calculated by using the Rayleighquotient given by Eq. (20) and the characteristic Eq. [17]. The obtained results show good agreement of the lowest threefrequencies calculated by using the Rayleigh quotient and the characteristic equation with the experiment for the crackdepth to 50 percent beam thickness. Disagreement is observed for higher frequencies and this deviation increases withcrack depth that implies high sensitivity of frequencies calculated by using the Rayleigh approximation to crack depth.

The natural frequencies measured in the experiment are taken as input for multiple crack detection by using theprocedure proposed in the previous section. This is aimed to verify applicability of the Rayleigh quotient not only in modalanalysis of cracked beam but also for crack detection problem. For the purpose, the natural frequencies measured in theexperiment are taken in use for solving the diagnostic Eqs. (23)(30) solutions of which are shown in Figs. 620 for differentcases of crack depth scenarios. In each figure there are given typical four iterations leading to the final result given in the lastbox. To illustrate the difference between linear and nonlinear approximations, the chart given in the first box is the result of

(Triple crack) Depth of detected cracks 0.2001 0.2001 0.20010.20.450.7 Number of detected cracks 3(203020)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.2001 0.3 0.20010.20.450.7 Number of detected cracks 3(303020)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.3 0.3 0.20010.20.450.7 Number of detected cracks 3(303030)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.3 0.3 0.30.20.450.7 Number of detected cracks 3(303040)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.3 0.3 0.40030.20.450.7 Number of detected cracks 3(304040)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.3 0.4003 0.40030.20.450.7 Number of detected cracks 3(404040)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.4003 0.4003 0.40030.20.450.7 Number of detected cracks 3(504040)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.4997 0.4003 0.40030.20.450.7 Number of detected cracks 3(505040)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.4997 0.4997 0.40030.20.450.7 Number of detected cracks 3(505050)% Position of detected cracks 0.2 0.45 0.7(Triple crack) Depth of detected cracks 0.4997 0.4997 0.4997

[3] Q. Fan, P. Qiao, Vibration-based damage identification methods: a review and comparative study, Structural Health Monitoring 10 (10) (2011) 83111.

N.T. Khiem, L.K. Toan / Journal of Sound and Vibration 333 (2014) 408441034102[4] P. Cawley, R.D. Adams, The location of defects in structures from measurements of natural frequencies, Journal of Strain Analysis 14 (2) (1979) 4957.[5] O.S. Salawu, Detection of structural damage through changes in frequency: a review, Engineering Structures 19 (9) (1997) 718723.[6] R. Ruotolo, C. Surace, Damage assessment of multiple cracked beams: numerical results and experimental validation, Journal of Sound and Vibration

206 (4) (1997) 567588.[7] A. Morassi, M. Rollo, Identification of two cracks in a simply supported beam fromminimal frequency measurements, Journal of Vibration and Control 7

(5) (2001) 729740.[8] N.T. Khiem, T.V. Lien, Multi-crack detection for beam by the natural frequencies, Journal of Sound and Vibration 273 (2004) 175184.[9] X. Zhang, Q. Han, F. Li, Analytical approach for detection of multiple cracks in a beam, Journal of Engineering Mechanics 136 (3) (2010) 345357.[10] M. Dilena, A. Morassi, Reconstruction method for damage detection in beams based on natural frequency and antiresonant frequency measurements,

ASCE Journal of Engineering Mechanics 136 (3) (2010) 329344.[11] F.B. Sayyad, B. Kumar, Identification of crack location and size in a simply supported beam by measurement of natural frequencies, Journal of Vibration

and Control 18 (2) (2012) 183190.[12] R.D. Adams, P. Cawley, C.J. Pye, B.J. Stone, Vibration technique for non-destructively assessing the integrity of structures, Journal of Mechanical

Engineering Science 20 (2) (1978) 94101.[13] W.M. Ostanchowicz, M. Krawczuk, Analysis of the effect of cracks on the natural frequencies of a cantilever beam, Journal of Sound and Vibration 150

(2) (1991) 191201.[14] R.Y. Liang, J. Hu, F. Choy, Theoretical study of crack-induced eigenfrequency change on beam structures, Journal of Engineering Mechanics 118 (2) (1992)

384395.[15] Y. Narkis, Identification of crack location in vibrating simply supported beams, Journal of Sound and Vibration 172 (1994) 549558.[16] E.I. Shifrin, R. Ruotolo, Natural frequencies of a beam with an arbitrary number of cracks, Journal of Sound and Vibration 222 (3) (1999) 409423.[17] N.T. Khiem, T.V. Lien, A simplified method for natural frequency analysis of multiple cracked beam, Journal of Sound and Vibration 245 (4) (2001)

737751.[18] Q.S. Li, Vibratory characteristics of multi-step beams with an arbitrary number of cracks and concentrated masses, Applied Acoustics 62 (2001)

691706.[19] S. Caddemi, I. Cali, Exact closed-form solution for the vibration mode of the EulerBernoulli beam with multiple open cracks, Journal of Sound and

Vibration 327 (35) (2009) 473489.[20] K. Aydin, Vibratory characteristics of EulerBernoulli beams with an arbitrary number of cracks subjected to axial load, Journal of Vibration and Control

14 (4) (2008) 485510.[21] A. Morassi, Crack-induced changes in eigenparameters of beam structures, Journal of Engineering Mechanics 119 (9) (1993) 17981803.[22] L. Rubio, An efficient method for crack identification in simply supported EulerBernoulli beams, Journal of Vibration and Acoustics 131 (2009) 16.

(051001).[23] J. Lee, Identification of multiple cracks in beam using natural frequencies, Journal of Sound and Vibration 320 (2009) 482490.the first iteration obtained with initial 20 positions mesh and zero nonlinear factor (0), i.e., linear approximation.Therefore, symmetrical positions of cracks detected at the first iteration are apparent. However, the symmetry in the resultof crack localization in subsequent iterations with positive nonlinear factor starting from the second one is immediatelyeliminated. The number of frequencies used for the crack detection procedure needs to be not less than 6 to faithfullyestimate the crack magnitude. The numerical results including depth, location and number of detected cracks evaluatedfrom the chart in the last box of the figures and Eq. (30) are tabulated in Table 2. It is obviously that multiple cracks could betruthfully identified from only natural frequencies by using obtained in the present paper Rayleigh quotient.

5. Conclusion

In the present paper, the so-called Rayleigh quotient is derived for a clamped end beam with arbitrary number of cracks.Comparing natural frequencies calculated by using the Rayleigh quotient to those computed from the characteristicequation and measured in an experiment, it can be concluded that the obtained herein Rayleigh formulae provide a simpleand consistent tool for modal analysis of cracked structures. Moreover, since the Rayleigh quotient has been obtained in anexplicit expression of natural frequencies in terms of crack parameters, it is straightforward to develop a new procedure forcrack detection from only measured natural frequencies. The procedure consists of iteratively estimating unknownmagnitudes of cracks assumed to be at a grid of positions in the beam that is called crack scanning mesh. Since thenonlinear terms with respect to crack magnitude have been incorporated in the explicit expression of natural frequencies,the approach to the crack detection makes it possible to really overcome the non-uniqueness problem of crack localizationin the beam with symmetrical boundary conditions using only measured natural frequencies. Nevertheless, the questionhow the nonlinear regularization factor is chosen to obtain a unique solution of the crack detection should be furtherinvestigated. Usefulness of the developed herein technique in structural damage detection based on measurement of naturalfrequencies has been validated by an experimental study on a beam with fixed ends.

Acknowledgment

The authors are pleased to thank the NAFOSTED of Vietnam for support under Grant number 107.04.12.09 in completingthis paper.

References

[1] S.W. Doebling, C.R. Farrar, M.B. Prime, D.W. Shevitz, Damage identification and health monitoring of structural and mechanical systems from changesin their vibration characteristics: a literature review, Los Alamos National Laboratory Report, LA-13070-MS, 1996.

[2] H. Sohn, C.R. Farrar, F.M. Hemez, D.D. Shunk, D.W. Stinemates, B.R. Nadler, J.J. Czarnecki, A review of structural health monitoring literature: 19962001, Los Alamos National Laboratory Report, LA-1396-MS, 2004.

[24] R.Y. Liang, J. Hu, F. Choy, Quantitative NDE technique for assessing damages in beam structures, Journal of Engineering Mechanics 118 (7) (1992)14681487.

[25] D.P. Patil, S.K. Maiti, Detection of multiple cracks using frequency measurements, Engineering Fracture Mechanics 70 (12) (2003) 15531572.[26] A. Maghsoodi, A. Ghadami, H.R. Mirdamadi, Multiple crack damage detection in multi-step beams by a novel local flexibility-based damage index,

Journal of Sound and Vibration 332 (2013) 294305.[27] M.H.H. Shen, C. Pierre, Natural modes of BernoulliEuler beams with symmetric cracks, Journal of Sound and Vibration 138 (1) (1990) 115134.[28] J. Fernandez-Saez, L. Rubio, C. Navarro, Approximate calculation of the fundamental frequency for bending vibrations of cracked beam, Journal of

Sound and Vibration 225 (2) (1999) 345352.[29] J. Fernandez-Saez, C. Navarro, Fundamental frequency of cracked beams in bending vibrations: an analytical approach, Journal of Sound and Vibration

256 (1) (2002) 1731.[30] T. Zheng, T. Ji, An approximate method for determining the static deflection and natural frequency of a cracked beam, Journal of Sound and Vibration

331 (2012) 26542670.[31] N.T. Khiem, T.H. Tran, A procedure for multiple crack identification in beam-like structure from natural vibration mode, Journal of Vibration and Control

(2013), http://dx.doi.org/10.117/1077546312470478. (Online First Version, SAGE).


A novel method for crack detection in beam-like structures by measurements of natural frequenciesIntroductionThe Rayleigh quotient for the multiple cracked beamA procedure for crack detection by natural frequenciesNumerical and experimental validationConclusionAcknowledgmentReferences

a novel method crack detection beam like structures measurements natural frequencies

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