A closed-form solution to the problem of crack

International Journal of Solids and Structures 150 (2018) 154–165

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International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr

A closed-form solution to the problem of crack identification for a multistep beam based on Rayleigh quotient N.T. Khiem a,b,∗, T.H. Tran b, V.T.A. Ninh c a

Center for Interdisciplinary Research, Ho Chi Minh City University of Technology (HUTECH), 475A Dien Bien Phu Street, Binh Thanh District, Ho Chi Minh City, Vietnam b Institute of Mechanics, VAST, 264 Doi Can, Ba Dinh, Hanoi, Vietnam c University of Transport and Communication, Hanoi, Vietnam

a r t i c l e

i n f o

Article history: Received 4 July 2017 Revised 27 May 2018 Available online 19 June 2018 Keywords: Multistep beam Crack detection Natural frequencies Rayleigh quotient

a b s t r a c t In the present paper, the problem of detecting unknown number of cracks is solved in a closed form for a multistep beam. First, general Rayleigh quotient is established for a multistep beam with an arbitrary number of cracks and boundary conditions. Then, it is used for obtaining an explicit expression of natural frequencies through crack parameters for cantilever multistep beams. The obtained expression is straightforward for calculating natural frequencies of a multistep beam with an arbitrary number of transverse cracks and enables to develop an efficient procedure for detecting unknown number of cracks in the beam structure. Numerical examples show that the Rayleigh quotient is really a useful tool for modal analysis and identification of cracked multistep beams. An experimental study is accomplished for a two-step cracked cantilever beam to validate the theoretical development. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction A stepped beam structure is frequently used in the practice of robotics, mechanical engineering, and structural engineering. It is also an appropriate model for studying beams of nonuniform profiles. Vibrations of such structures and benefits of the structure model were investigated in the widespread literature, some examples of which are studies by Jang and Bert (1989), Jaworski and Dowell (2008), Subramanian and Balasubramanian (1987), Koplow et al. (2006), and Yang (2010). Numerous methods were also developed for the vibration analysis of multistep beams such as transfer matrix method (TMM) (Sato 1983; Patil and Maiti 2003; Attar 2012), finite difference method (Sarigul and Aksu 1986), finite element method (FEM) (Ju et al., 1994), Green’s function method (Kukla and Zamojska 2007), composite element method (CEM) (Lu et al., 2009), adomian decomposition method (Mao 2012), and differential quadrature element method (Wang and Wang 2013). The problem of vibration in stepped beams becomes more appealing when a damage such as crack is present in the structures and need to be detected. Namely, using the Green’s function method, Kukla (2009) demonstrated that the nonuniformity of a column and the location and depth of a crack can signifi-

∗ Corresponding author at: Institute of Mechanics, VAST, 264 Doi Can, Ba Dinh, Hanoi, Vietnam. E-mail address: [email protected] (N.T. Khiem).

https://doi.org/10.1016/j.ijsolstr.2018.06.010 0020-7683/© 2018 Elsevier Ltd. All rights reserved.

cantly change the eigenfrequencies and critical forces of the column. Tsai and Wang (1996) showed that a crack leads to a sharp change in the mode shape of stepped shaft at the crack position that is a useful indication for crack localization. Moreover, Zhang et al. (2009) revealed that the crack-induced change in the mode shape of a stepped beam becomes more apparent in its wavelet coefficient plotted along the beam span. By using the conventional approach, Nandwana and Maiti (1997) established the frequency equation for a three-step beam with a single crack in the form of a 16-dimensional determinant that allows natural frequencies of the cracked multistep beam to be calculated as functions of the crack position and depth. This provides a constructive tool for crack detection in stepped beams by measurements of natural frequencies. Chaudhari and Maiti (20 0 0) extended the above study and obtained similar results for a segmented beam with a crack. A remarkable achievement in the vibration of cracked multistep beam was given by Li (2001) who obtained almost explicit expressions for frequency equation and mode shapes of a multistep beam with an arbitrary number of cracks and concentrated masses. This result provides a straightforward tool for exactly calculating natural frequencies and mode shapes of a multistep beam with multiple cracks. Skrinar (2013) accomplished the finite element formulation for a multistep cracked beam element that could be used for the structural analysis of more complicate frames with multiple cracks.

N.T. Khiem et al. / International Journal of Solids and Structures 150 (2018) 154–165

The problem of crack identification in multistep beams, as usually, can be solved by the vibration-based technique that consists of using given modal parameters of a structure such as natural frequencies and/or mode shapes for detecting potential cracks (location and size) in the structure. Indeed, Tsai and Wang (1996) used, first, the change in mode shape for locating a single crack in a three-step beam, and then, the depth of the crack is determined graphically from measured natural frequencies. These findings demonstrated that a single crack could be detected from changes in natural frequencies and mode shapes. By the same way, Zhang et al., (2009) exhibited that the crack location can be more easily detected from wavelet coefficients of mode shape extended by a smooth extrapolation. However, the wavelet transform is usually sensitive to errors vitally accompanied with the extrapolated or measured mode shape. Nandwana and Maiti (1997) proposed a procedure for detecting a single crack in a three-step beam based only on natural frequencies that are more easily and accurately measured in practice. According to the procedure, the crack position and size are determined from the interaction of the so-called frequency contours drawn up numerically as an implicit function of the crack parameters for every given natural frequency. This is really a useful procedure for identifying a single crack in stepped beam, but it would be difficult to extend the procedure for detecting multiple cracks in the beam. Attar (2012) developed a general procedure for multiple crack identification in a multistep beam by solving a system of 2 N nonlinear equations to find the unknown position and size of N cracks using the Newton-Raphson method. One of the shortcomings of the procedure is the need to calculate the Jacobian matrix of an implicit vector function that may be time consuming and erroneous, especially, in the case of a large number of cracks. This is why the author could examine the proposed procedure only in the case of a three-step beam with double cracks. The energy approach (Liang et al., 1992) was used by Maghsoodi et al. (2013) to obtain diagnostic equations like the Rayleigh quotients (Fernandez-Saez et al., 1999) that express crackinduced changes in natural frequencies in terms of crack parameters. The obtained approximate diagnostic equations have been employed for detecting single, double, and triple cracks in various stepped beams. The main disadvantage of the approach is the need to find the crack position belonging to an assumed segment and crack depth be greater than 5% of beam thickness. It is difficult for diagnostic equations that are usually ill-posed due to the errors in modeling and measurement. The above results lead us to the following: Natural frequencies are good indicators for multiple crack identification in stepped beams; however, diagnostic equations established for determining crack parameters from given natural frequencies should be most exact but sufficiently simplified to find a consistent solution regarding both measurement and modeling inaccuracy. This motivates the authors of the present paper to use the exact Rayleigh quotient (Khiem and Toan, 2014) for cracked multistep beams and Tikhonov’s regularization method for solving the inverse problem of multiple crack identification in stepped beams by measurements of natural frequencies. Thus, the objective of the present study is to develop a fruitful procedure for detecting unknown number of cracks in multistep beams by measurements of natural frequencies. First, the Rayleigh quotient is accurately obtained from the governing vibration equation of a multistep beam with an arbitrary number of open transverse cracks for every vibration mode. Then, the obtained quotient is employed for deriving an explicit expression of natural frequencies through crack parameters and mode shapes of an intact multistep cantilever beam. These expressions are straightforward to construct a system of diagnostic equations simply relating changes in natural frequencies with the crack positions and depths. The diagnostic equations established in a standard

155

form can directly apply the crack scanning procedure proposed by Khiem and Tran (2014) and Khiem and Toan (2014) in combination with Tikhonov’s regularization method for detecting unknown number of cracks in the multistep beam by measurements of limited number of natural frequencies. The theoretical development is illustrated and validated in both numerical and experimental examples. 2. Rayleigh quotient for cracked multistep beam Let’s consider a multistep beam consisting of n segments x ∈ (xj − 1 , xj ),j = 1, ..., n with x0 = 0, xn = 1. Denoting natural frequencies and mode shapes of the multistep beam by {ωk2 , φk (x ), k = 1, 2, ...}, one obtains φ k (x) = {φ kj (x),j = 1, ..., n}, where the function φ kj (x) satisfies the following equation (Khiem and Toan, 2014) ) −ωk2 m j φk j (x ) + S j φk(IV (x ) = 0, x ∈ (x j−1 , x j ) j

(2.1)

Sj = Ej Ij ; mj = ρ j Aj are mechanical constants (the bending stiffness and mass per unit length, respectively), and the coordinate x is normalized by the total length of the beam. Multiplying both sides of Eq. (2.1) by φ kj (x) and taking integration along the segment (xj − 1 , xj ) yield

bk j ωk2 = S j ak j ,

(2.2)

where



ak j =

xj x j−1

φk(IVj ) (x )φk j (x )dx; bk j = m j



xj x j−1

φk2j (x )dx.

(2.3)

Taking the sum of both sides in Eq. (2.2) along j leads to n 

S j ak j = bk ωk2 ; k = 1, 2, 3, ....

(2.4)

j=1

where

bk =

n 

bk j =

j=1

n  j=1

 mj

xj x j−1

φk2j (x )dx.

(2.5)

If the functions φ kj , φ  kj , φ   kj , φ    kj are all continuous in the segment (xj − 1 , xj ), one can calculate



ak j =

xj x j−1

φk(IVj ) (x )φk j (x )dx = Uk j + [Bk j (x−j ) − Bk j (x+j−1 )],

(2.6)

where the superscripts + and − of xj − 1 , xj denote the right and left limits respectively, for the points and

 Uk j =

xj x j−1

φkj2 (x )dx; Bk j (x ) = [φkj (x )φk j (x ) − φkj (x )φk j (x )]. (2.7)

In the case if the functions φ kj , φ  kj , φ   kj , φ    kj are discontinuous at a single point ej ∈ (xj − 1 , xj ) only, the integral (2.6) could be rewritten as follows:



ak j =

x j−1

 =

xj

ej

x j−1

φk(IVj ) (x )φk j (x )dx φk(IVj ) (x )φk j (x )dx +



xj ej

φk(IVj ) (x )φk j (x )dx

= Uk−j + [Bk j (e−j ) − Bk j (x+j−1 )] + Uk+j + [Bk j (x−j ) − Bk j (e+j )] = (Uk−j + Uk+j ) + [Bk j (x−j ) − Bk j (x+j−1 )] + [Bk j (e−j ) − Bk j (e+j )] or

ak j = Uk j + [Bk j (x−j ) − Bk j (x+j−1 ) + Bk j (e−j ) − Bk j (e+j )]. Substituting (2.8) into (2.4) yields

(2.8)

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N.T. Khiem et al. / International Journal of Solids and Structures 150 (2018) 154–165

n 

S j ak j =

n 

j=1

where φk0j (x ) is the mode shape of the beam in the segment (xj − 1 , xj ). Note, Eqs. (2.14)–(2.16) can be used to analyze the boundary effect on natural frequencies of beams including those with elastic supports. For instance, if two ends of a beam are supported on translational and rotational springs of stiffness T0 ,R0 ,T1 ,R1 , when elastic boundary conditions are expressed as

S jUk j + Bkn (1 )Sn − Bk1 (0 )S1

j=1 n−1 

+

[Bk j (x−j )S j − Bk, j+1 (x+j )S j+1 ]

j=1 n 

+

S j [Bk j (e−j ) − Bk j (e+j )].

(2.9)

j=1

It is well-known that at step joint xj ,j = 1,…, n − 1, the continuity conditions must be satisfied

φk j (x−j ) = φk, j+1 (x+j ); φk j (x−j ) = φk, j+1 (x+j );  +  −  + S j φkj (x−j ) = S j+1 φk, j+1 (x j ); S j φk j (x j ) = S j+1 φk, j+1 (x j ),

 (1 ) = −R φ  (1 ) S1 φk1 (0 ) = R0 φk 1 (0 ); Sn φkn 1 kn

one would have got

Sn Bkn (1 ) − S1 Bk1 (0 )

2  2 ( 1 ) + T φ 2 ( 0 ) + R φ  2 ( 0 )] . = [T1 φkn (1 ) + R1 φkn 0 k1 0 k1

(2.10)

Therefore, the Rayleigh quotient for cracked multistep beams with elastic boundary supports is



that allow one to calculate n−1 

 (1 ) = T φ (1 ); S1 φk1 (0 ) = −T0 φk1 (0 ); Sn φkn 1 kn

ωk2 =

[Bk j (x−j )S j − Bk, j+1 (x+j )S j+1 ] = 0.

n 



Sj

xj

x j−1

j=1

 2 2 ( 1 ) φkj2 (x )dx + γ j φkj2 (e j ) + T1 φkn (1 ) + R1 φkn 

j=1

+ T0 φk21 (0 ) + R0 φk21 (0 )

Furthermore, if the beam segment (xj − 1 , xj ) is cracked at position ej , the following condition must be satisfied:

φk j (e−j ) = φk j (e+j ); φkj (e−j ) = φkj (e+j ); φkj (e−j ) = φkj (e+j ); φk j (e−j ) = φk j (e+j ) − γ j φkj (e+j ),

(2.11)

where γ j = EIj /Lj Kj , called the crack magnitude and determined from the crack depth, see Appendix. According to Eq. (2.11), the last term in (2.9) can be given as follows:

[Bk j (e−j ) − Bk j (e+j )] = γ j φkj2 (e j ).

(2.12)

S j ak j =

j=1

n 

2

Finally, using Eqs. (2.4) and (2.13), Eq. (2.4) is rewritten in the form



ω =

n 



Sj

xj

φ 2 (x )dx + γ

j

kj

x j−1

j=1

φ 2 (e



+ Bkn (1 )Sn − Bk1 (0 )S1

kj

n 

 j) 

mj

j=1

xj

x j−1

 φ (x )dx , 2 kj

(2.14)

that is the Rayleigh quotient expanded for a cracked multistep beam with arbitrary boundary conditions. It is also easy to verify that Eq. (2.14) leads to the Rayleigh quotient obtained in (Khiem and Toan, 2014) for a multiple cracked uniform beam, when S1 = S2 = ... = Sn = S0 and m1 = m2 = ... = mn = m0 , 2 ωuk = ( S0 /m0 )



1 0

φkj2 (x )dx +

+ Bkn (1 ) − Bk1 (0 )

n 



ω = 2 k



1 0

φ (x )dx 2 k

(2.15)

Furthermore, from Eq. (2.14), the Rayleigh quotient for an uncracked multistep beam can be obtained



ωk20 = 

n 

 Sj

j=1 n  j=1

x j−1

 mj

xj





2

x j−1

2



[φk0j (x )] dx ,

n 



Sj

x j−1

j=1 n 

 mj

xj

xj x j−1

φ 2 (x )dx + γ kj

j

φ 2 (e kj

 j)



φ (x )dx . 2 kj

(2.18)

 1 = φ  1 = 0. φk1 (0 ) = φk (0 ) = 0; φkn ( ) k ( )

(2.16)

(3.1)

In this case, one selects the trial function as

φk j (x ) = φk0j (x ) + Ck j x + Dk j + γ j φk0j (e j )L(x − e j ), where φk0j (x ) is the k-th mode shape of tistep beam without cracks satisfying the φ k0 (0) = φ  k0 (0) = φ   k0 (1) = φ    k0 (1) = 0 and functions



0, x ≤ 0  ; L (x ) = x :x≥0

(3.2) the mulconditions



0 , x ≤ 0  ; L (x ) = L (x ) = 0. 1 :x>0

(3.3)

Note that φkj (x ) = φk0j (x ); φkj (x ) = φ 0k  (x ) and in consequence j function (3.2) satisfies condition (2.11) at crack ej . Namely,

φk j (e+j ) = φk0j (e+j ) + Ck j e+j + Dk j =

φk0j (e−j ) + Ck j e−j + Dk j = φk j (e−j );

φ  k j (e+j ) = φk0j (e+j )

[φ 0k j (x )] dx + Bkn (1 )Sn − Bk1 (0 )S1

xj

(2.17)

The above obtained Rayleigh quotient could be employed in the calculation of the arbitrary natural frequency of a cracked multistep beam by choosing an appropriate mode shape as trial functions corresponding to given boundary conditions. In this subsection, a multistep cantilever beam is investigated. For the beam, the boundary conditions are φ k (0) = φ  k (0) = φ   k (1) = φ    k (1) = 0 that lead to

L (x ) =

γ j φkj2 (e j )

j=1

 

x j−1

 φk2j (x )dx .

3. Calculation of natural frequencies of cracked multistep cantilever beam

j=1

(2.13)

2 k

xj

In this study, only beams with the conventional boundary conditions, such as simply supported, clamped, or cantilevered beams are investigated, so that Bk1 (0) = Bkn (1) = 0 and Eq. (2.14) is thus simplified to

j=1

S j [Uk j + γ j φ  k j (e j )] + Bkn (1 )Sn − Bk1 (0 )S1 .

 mj

j=1

Hence, we can rewrite Eq. (2.9) as n 

n 

=

φk0j (e−j )

=

φ  k j (e−j )

=

φk0j (e j ); φ  k j (e+j )

N.T. Khiem et al. / International Journal of Solids and Structures 150 (2018) 154–165

=

+ 0 − φk0 j ( e j ) = φk j ( e j )

=

φ  k j (e−j ) = φk0 j ( e j );

where Pjm = (xm − xm )/m; T jm = (xmj − emj )/m; m = 1, 2, 3 and j j−1 Q j ( ei , er ) =

φ  k j (e+j ) = φk0j (e+j ) + Ck j + γ j φk0j (e j ) =

φk0j (e−j ) + Ck j + γ j φk0j (e j )

=

φ  k j (e−j ) + γ j φk0j (e j ).

j−1 

Thus, natural frequencies of a cracked multistep beam are represented as

j−1 

ei γi φk0j (ei ), j = 1, . . . , n

(3.4)

n  xj j=1

n   j=1

and

2

x j−1

m j [φk0j (x )] dx + 2

n  j=1

xj x j−1



2

[φ 0k j (x )] dx +

n 



j=1

 −2 γ j S j ωks [φ 0k j (e j )] + 2

2

S j γ j [φ 0k j (e j )]

j n  



j=1 i,r=1



m j Q j (ei , er )γi γr φ 0ki (ei )φ 0kr (er )

xj x j−1

2

m j [φk0j (x )] dx = 1,

(3.8)

natural frequencies of the uncracked beam would be

n−1 

0 kj

γi φ (ei );

ωk20 =

i=1

Dkn = Dk1 −

ω =

Sj

j=1

If the mode shapes of an uncracked multistep beam are normalized by

i=1

Ckn = Ck1 +

n  2 k

(3.7)

γi φk0j (ei );

i=1

Dk j = Dk 1 −



T j3 + 2e j T j2 + e2j T j1 r = i = j; [T j3 − (e j + ei )T j2 + e j ei T j1 ] : i < r = j; [P j3 − (er + ei )P j2 + ei er P j1 ] : r, i < j; [T j3 − (e j + er )T j2 + e j er T j1 ] : r < i = j.

(3.6)

Furthermore, function (3.2) would also satisfy condition (2.10) if the constants Ckj ,Dkj are chosen so that Ck, j+1 = Ck, j + γ j φk0j (e j ); Dk, j+1 = Dk, j − γ j φk0j (e j )e j . The latter relationships lead to

Ck j = Ck1 +

157

n−1 

n 

 Sj

j=1

ei γi φk0j (ei ).

xj



x j−1

2

[φ 0k j (x )] dx.

(3.9)

Substituting (3.8) and (3.9) into (3.7) we obtain finally

i=1

By substituting the function

φk1 (x ) = φk0j (x ) + Ck1 x + Dk1 ;

ωk2 /ωk20 =

φkn (x ) = φ (x ) + Ck1 x + Dk1 + 0 kj

n 

0 kj

1 + ωk−2 0 1 + 2ωk−2 0

γ j φ ( e j )L ( x − e j )

n 

γ j S j [φ 0k j  (e j )] + 2

j=1

n  j=1



2

S j γ j [φ 0k j (e j )]

n 

mj

j=1

j  i,r=1



(3.10)

j=1

into boundary condition (3.1), we have Ck1 = Dk1 = 0; therefore, one obtains

φk j (x ) = φk0j (x ) +

j 



γi φ 0k j (ei )L(x − ei ).

(3.5)

i=1

By using expression (3.5), the numerator and denominator in (2.15) can be calculated as

Num =

n 



Sj

j=1

=

n 

x j−1

 Sj

Denom =

n 

 mj

j=1

=

n 

xj

x j−1

j=1

x j−1

mj



+ +

xj

x j−1 n 

φ 2 (x )dx + γ kj

[φ

0 kj

xj

=

+

x j−1

) 2

ωk2 /ωk20 = 1 − ωk−2 0



2





ej

x j−1 2



xj

−

x j−1

2φ (x )(Ck j x + Dk j )dx 0 kj



ej

x j−1

[φk0j (x ) + Ck j x + Dk j ](e j − x )dx



(e j − x )2 dx n 

2

−2 γ j S j ωks [φk0j (e j )]

j=1

j=1 i,r=1



2

(3.11)

n 

S j γ j [φ 0k j (e j )]

ϑ

n 





R(ei , e j )γi γ j φ 0ki (ei )φ 0k j (e j ),

(3.12)

R ( ei , e j ) =

n 

[mr Qr (ei , e j ) + mr Qr (e j , ei )]/2; i, j = 1, ...n.

r=max(i, j )

(3.13)

m j [φk0j (x )] dx + 2



2

where ϑ is a constant that represents the truncation error and



j



S j γ j [φ 0k j (e j )] ;

j=1

(Ck j x + Dk j )2 dx

2

n 

j

0 kj

[φ (x )] dx +

γ j2 [φk0j (e j )]

j=1

n  j=1

(x )] dx + γ j [φ (e j )] ;

m j 2γ j φk0j (e j )

n  xj 

kj

2

0 kj

j=1

+

j

φ 2 (e

φk2j (x )dx

x j−1

j=1

ωk2 /ωk20 = 1 − ωk−2 0

i, j=1

xj



This is the Rayleigh quotient for the cracked multistep beam that gives natural frequencies of the beam explicitly expressed through crack parameters. Note, expression (3.10) allows one to obtain asymptotic approximations of the first and second orders with respect to crack magnitudes, respectively, as



xj



Q j (ei , er )γi γr φ 0ki (ei )φ 0kr (er )

0 0 m j Q j (ei , er )γi γr φki (ei )φkr ( er ),

Expressions similar to Eq. (3.11) were obtained in a previous study (Maghsoodi et al., 2013) by the energy approach and hypothesis that cracks do not change the mode shape of beams. However, as shown above, crack-induced change in mode shape is important to establish the Rayleigh quotient (3.10). Moreover, the quotient enables to obtain a novel expression (3.12), the asymptotic approximation of second order that has not been previously published for a multistep beam. Obviously, natural frequencies of a cracked multistep beam could be calculated with given frequencies and mode shapes of an uncracked one. Natural frequencies and mode shapes of an uncracked multistep beam can be computed by different methods developed in the aforementioned studies. In contrary to a previous study (Maghsoodi et al., 2013), where

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the uncracked natural frequencies and mode shapes were computed by the Rayleigh-Ritz approximate method, we proposed to compute the modal characteristics using the TMM that was proved to be the most exact method for computing the frequencies and mode shapes. Chan et al. (2011) declared that the use of trial functions more close to the exact mode shapes gives natural frequencies more accurately calculated by the Rayleigh quotient. Finally, a procedure is described below for computing natural frequencies and mode shapes of an uncracked multistep beam using the TMM. As well known, the general solution of Eq. (2.1) is represented by

φ j (x ) = A j cosh λ j x + B j sinh λ j x + C j cos λ j x + D j sin λ j x

Thus, all the mode shapes of subsequent segments of the beam would be determined. Obviously, the k-th mode shape of the beam contains an arbitrary constant Ck0 that is usually selected from the condition n 



mj

j=1

xj x j−1

2

[φk0j (x )] dx = 1.

(3.20)

Thus, the natural frequencies and mode shapes of the uncracked multistep beam are calculated. 4. Crack detection procedure

φ  j (x ) = λ j [A j sinh λ j x + B j cosh λ j x − C j sin λ j x + D j cos λ j x]; M j (x ) = λ2j S j [A j cosh λ j x + B j sinh λ j x − C j cos λ j x − D j sin λ j x]; Q j (x ) = λ3j S j [A j sinh λ j x + B j cosh λ j x + C j sin λ j x − D j cos λ j x].

This section describes the problem of crack identification for a multistep beam by measurement of natural frequencies. Herein, we distinguish two cases of the problem: first is the case when the number of cracks is defined, and in the other, the number of cracks is unknown. In the first case, the problem is solved by an analytical method for the number of cracks being not very large. For solving the second problem, the crack scanning method is employed.

Therefore, one can express the state vector Vj (x) = {φ j (x),φ  j (x),Mj (x),Qj (x)}T in the matrix form as follows:

4.1. Detection of a known number of cracks

where λj = (ω2 mj /Sj )1/4 and Aj ,Bj ,Cj ,Dj are constants. Therefore, modal slope, bending moment, and shear force are calculated as

{ V j ( x ) } = [ H j ( x )] { C ( j ) }

(3.14)

where {C(j)} = {Aj ,Bj ,Cj ,Dj }T and

⎡

cosh λ j x ⎢ λ j sinh λ j x [ H j ( x )] = ⎢ 2 ⎣λ j S j cosh λ j x λ3j S j sinh λ j x

sinh λ j x λ j cosh λ j x λ2j S j sinh λ j x λ3j S j cosh λ j x

cos λ j x −λ j sin λ j x −λ2j S j cos λ j x

λ3j S j sin λ j x

⎤ sin λ j x λ j cos λ j x ⎥ ⎥. −λ2j S j sin λ j x ⎦ 3 −λ j S j cos λ j x

(3.15) Using (3.14), one can get {V j (x j−1 )} = [H j (x j−1 )]{C( j )} ≡ V−j , {V j (x j )} = [H j (x j )]{C( j )} ≡ V+j that directly leads to

V+j = [T j ]V−j ; [T j ] = [H j (x j )][H j (x j−1 )]−1 .

(3.16)

Furthermore, continuity condition (2.10) can be rewritten as V+j = V−j+1 , which in combination with (3.16) yields V−j+1 = [T j ]V−j ; consequently, − − V+ n = [Tn ][Tn−1 ]...[T1 ]V1 = [T]V1 .



(3.17)

Substituting boundary condition (3.1) into (3.17), we have

      φn ( 1 ) T13 (ω ) T14 (ω ) M1 ( 0 ) T (ω ) = · ; 33 φ  n (1 ) T23 (ω ) T24 (ω ) Q1 ( 0 ) T43 (ω )   M1 ( 0 ) · = 0. Q1 ( 0 )



T34 (ω ) T44 (ω )

(3.18)

The second equation in (3.18) allows one to get the frequency equation for a multiple stepped cantilever as

F (ω ) ≡ T33 (ω )T44 (ω ) − T43 (ω )T34 (ω ) = 0,

(3.19)

the solution of which gives natural frequencies ωk0 , k = 1, 2, 3,…, together with vector {M1 (0 ), Q1 (0 )} = Ck0 {α , β}, where Ck0 is a constant. Therefore, the mode shape φk0j (x ), associated with the natural frequency ωk0 , is determined as follows: First, the constant vector for the first segment of the beam is calculated as {A1 , B1 , C1 , D1 }T = Ck0 [H1 (0 )]−1 {0, 0, α , β}T ; then, the k-th mode shape of the first segment is

φk1 (x ) = A1 cosh λ1 x + B1 sinh λ1 x + C1 cos λ1 x + D1 sin λ1 x;  2 m /S . x ∈ (0, x1 ); λ1 = 4 ω10 1 1 Next, the constant vector for the second segment is calculated

Suppose that there are r (1 ≤ r ≤ n) cracked segments in an n-step beam, the beam segments are numbered by subscripts j1 ,..., jr . The crack locations and magnitudes are denoted by e j1 , ..., e jr , γ j1 , ..., γ jr , respectively. Furthermore, assume that m ∗ of the cracked beam are (m > r) natural frequencies ω1∗ , ..., ωm measured, the corresponding undamaged natural frequencies and mode shapes are given as ωk0 , φk0j (x ), k = 1, ..., m; j = j1 , ..., jr . As well known, the discrepancy between measured and computed data is unavoidable, and it is not only caused by the measurement error but also due to the difference between the computational model and testing structure. Therefore, model updating is mandatory before accomplishing any model-based damage identification procedure. Because the proposed above procedure assumes natural frequencies and mode shapes of an undamaged multistep beam to be given together with measured natural frequencies, we suggest herein a model updating procedure for an intact multistep beam using its measured natural frequencies. Namely, for an intact multistep beam with mass normalized mode shapes {φ¯ k0j (x )}, Eq. (3.9) is now rewritten as

ωk20 =

n 

 Uk j S j ; Uk j =

j=1

xj

x j−1



2

[φ 0k j (x )] dx.

Hence, by using the measured natural frequencies for undamaged beam ωk∗0 , k = 1, ..., m, the system of linear equations n ∗2 j=1 Uk j S j = ωk0 can be obtained for determining the parameters Sj ,j = 1, ..., n. Thus, the model of the beam is updated by the structural parameters Sˆ = {S∗ , ..., S∗ } determined from the measured natural frequencies as

1

1

∗2 ∗2 T {Sˆ } = [UT U]−1 UT {∗0 }, ∗0 = {ω10 , ..., ωm 0} .

(4.1)

By model updating, the bending stiffness along the beam span is corrected, so that both the material and geometry of the beam are updated by the measured natural frequencies. By using the updated model, one can construct the following system of equations r 

akr (e ji )γ ji = bk ; k = 1, ..., m; akr = S∗jr [φk0ji (e ji )]2 ;

i=1

bk = (ωk0 )2 − (ωk∗ )2 .

(4.2)

as

r first equations in (4.1) enable to express r crack magnitudes γ j1 , . . . , γ jr through crack locations e = (e j1 , . . . , e jr ) as

{A2 , B2 , C2 , D2 }T = [H2 (x1 )]−1 [H1 (x1 )]{A1 , B1 , C1 , D1 }T

γ ji = hi (e j1 , . . . , e jr ), i = 1, . . . , r.

(4.3)

N.T. Khiem et al. / International Journal of Solids and Structures 150 (2018) 154–165

r 

159

Substituting (4.3) into unused m − r equations in (4.2), we get

4.2. Detection of unknown number of cracks

akr (e ji )hi (e j1 , ..., e jr ) − bk = 0; k = r + 1, . . . , m.

The crack scanning method is proposed specifically for detecting unknown number of cracks in structures by measurements of modal characteristics (Khiem and Tran, 2014; Khiem and Toan 2014). The method consists mainly of the following tasks:

(4.4)

i=1

If m = 2r, the system of nonlinear equations is closed to find crack locations (e¯ j1 , ..., e¯ jr ) by using the well-known methods. In case if m < 2r, we could find crack locations by solving the problem

(e j1 , ..., e jr ) =

m r   [ akr (e ji )hi (e j1 , ..., e jr ) − bk ]2 ⇒ k=r+1 i=1

min

(e j1 ,...,e jr )

(4.5) in the domain R = {(e j1 , ..., e jr ) : x j1 −1 < e j1 < x j1 , ..., x jr −1 < e jr < x jr }. The solution of this problem gives rise the demanded crack locations (eˆ j1 , ..., eˆ jr ) that allow the crack magnitudes to be calculated using (4.3). Namely

γˆ ji = hi (eˆ j1 , ..., eˆ jr ), i = 1, ..., r.

(4.6)

To evaluate the depth of the detected cracks, one needs to solve the equation

γˆ ji = g(z ), i = 1, ..., r,

(4.7)

g(z) = π (1 − ν 2 )(h/L)I

where c (z) with function Ic (z) as given in Appendix. Thus, the problem of identifying a known number of cracks in a multistep beam is solved by measuring natural frequencies. For example, consider a two-step beam with both segments cracked (n = 2, r = 2). Assuming first that m = 4, Eq. (4.3) is rewritten as

a11 (e1 )γ1 + a12 (e2 )γ2 = b1 ; a21 (e1 )γ1 + a22 (e2 )γ2 = b2 ; a31 (e1 )γ1 + a32 (e2 )γ2 = b3 ; a41 (e1 )γ1 + a42 (e2 )γ2 = b4 ,

(4.8)

where 0 a11 (e1 ) = S1∗ [φ11 ( e 1 )] 2 ;

0 a12 (e2 ) = S2∗ [φ12 ( e 2 )] 2 ;

0 ( e 1 )] 2 ; a21 (e1 ) = S1∗ [φ21

0 ( e 2 )] 2 ; a22 (e2 ) = S2∗ φ22

0 a31 (e1 ) = S1∗ [φ31 ( e 1 )] 2 ;

0 ( e 2 )] 2 ; a32 (e2 ) = S2∗ [φ32 0 S2∗ [ 42 e2 ]2 ; 2 ∗2 10 − 1 ; 2 ∗2 20 − 2 ; 2 ∗2 30 − 3 ; 2 ∗2 40 − 4 .

ω b2 = ω b3 = ω b4 = ω b1 =

φ ( ) ω ω ω ω

A = [ak j = φk0j 2 (e j )S j ; k = 1, ..., m; j = 1, ..., n]; 02 b = {bk = (ωks − ωk∗2 ), k = 1, ..., m};



(4.9)

G1 (e1 , e2 ) = det[A1 ] = 0; G2 (e1 , e2 ) = det[A2 ] = 0,



[A1 ] =

(4.13)

where

The latter system of linear equations with respect to crack magnitudes (γ 1 , γ 2 ) would be resolvable under the condition where matrix

Obviously, crucial points of the procedure are steps 2 and 3 where diagnostic equations are established and solved. Note first that though the Rayleigh quotient was established above for the case when each beam segment contains only one crack, it could be easily obtained for the case of multiple cracked segments. This is simply to divide a segment with p cracks into p sub-segments of the same material and geometry, and each of them contains only one crack. Therefore, the Rayleigh quotient is obtained in the same form as given above but includes a larger number of cracks. Thus, Eq. (3.12) can be used now for modeling a multistep beam with a given crack mesh, and it is rewritten in the form

[ A + ϑ B ( γ )] γ = b ,

0 a41 (e1 ) = S1∗ [φ41 ( e 1 )] 2 ;

a42 (e2 ) =

(1) A mesh of locations (e1 ,..., en ) of assumed cracks with unknown magnitudes (γ 1 ,.., γ n ) is introduced in the structure of interest; (2) A model of the cracked structure is constructed so that a system of equations relating the unknowns (γ 1 ,.., γ n ) with given measured data and structural parameters, including the crack location mesh; (3) The constructed diagnostic equations are solved, giving the crack magnitude estimated for every crack location in the assumed mesh: (γˆ1 , .., γˆn ); (4) Elimination of positions of the mesh where crack magnitudes are negative or very small results in a new mesh of possible crack locations (eˆ1 , ..., eˆnc ); (5) Model of the structure is reconstructed with the mesh obtained above, and go to step (3) for estimating the crack magnitudes again. This task is continued until no new mesh of crack locations is resulted; (6) The locations in the finally obtained mesh and associated magnitudes provide desired crack positions and magnitudes. These estimated crack magnitudes are employed for evaluating the crack depth accordingly to the adopted model of cracks.



a11 (e1 ) a12 (e2 ) b1 a21 (e1 ) a22 (e2 ) b2 ; [A2 ] = a31 (e1 ) a32 (e2 ) b3



(4.10)

a11 (e1 ) a12 (e2 ) b1 a21 (e1 ) a22 (e2 ) b2 a41 (e1 ) a42 (e2 ) b4



(4.11) The system of Eqs. (4.10) is closed to find out crack locations eˆ1 , eˆ2 . Substituting crack locations determined above into the two first equations in (4.6) allows the estimation of crack magnitudes

b1 a22 (eˆ2 ) − b2 a12 (eˆ2 ) ; a11 (eˆ1 )a22 (eˆ2 ) − a21 (eˆ1 )a12 (eˆ2 ) b2 a11 (eˆ1 ) − b1 a21 (eˆ1 ) γˆ2 = . a11 (eˆ1 )a22 (eˆ2 ) − a21 (eˆ1 )a12 (eˆ2 )

γˆ1 =

(4.12)

B (γ ) =

bk j =

n 

0 R(ei , e j )γi φki (ei )φk0j (e j );

i



k = 1, ..., m; j = 1, ..., n .

(4.14)

Solution of Eq. (4.13) is obtained by the iteration technique that can be described as follows:

[ A ( i )] γ ( i ) = b ;

(4.15)

[A(i )] = [A + ϑ B(γ (i−1) )]; i = 1, 2, 3, ...; γ (0) = 0. The iteration process is stopped as the condition γ (i + 1) − γ (i) ≤ δ with a given tolerance δ is fulfilled, and the correction of truncation error coefficient ϑ now will be employed as a regularization factor. Once the crack magnitudes γˆ j are estimated, the corresponding crack depths are obtained from the equation

F (a/h ) ≡ Ic (a/h ) − γ¯ = 0,

(4.16)

160

N.T. Khiem et al. / International Journal of Solids and Structures 150 (2018) 154–165

Fig. 1. Experimental setup: (a) stepped beam model; (b) measurement equipment.

where γ¯ = γˆ L/π h(1 − ν 2 )and function Ic (a/h) is given in Appendix. For instance, the matrices A, B in (4.14) are specified for a three-step beam with crack mesh consisting of n = n1 + n2 + n3 equidistant crack locations, where n1 ,n2 ,n3 are the number of cracks in the first, second, and third segments of the beam, respectively. For this case, one has

S j = {S1 : j = 1, ..., n1 ; S2 : j = n1 + 1, ..., n1 + n2 ;

φk0j (e j ) = {φk01 (e j ) : j = 1, ..., n1 ; φk02 (e j ) : j = n1 + 1, ..., n1 + n2 ; (4.17)

Therefore,

ak j =



φk012 (e j )S1 : j = 1, ..., n1 ;

φk022 (e j )S2 : j = n1 + 1, ..., n1 + n2 ;  φk032 (e j )S3 : j = n1 + n2 + 1, ..., n ;

bk j =

{φk012 (e j )k j : j = 1, ..., n1 ; φk022 (e j )k j : j = n1 + 1, ..., n1 + n2 ; φk032 (e j )k j : j = n1 + n2 + 1, ..., n},

where k = 1, 2,…, m and

n1 

R(ei , e j )γi φk01 (ei ) +

i=1

+

n2 

R(ei , e j )γi φk02 (ei )

i=n1 +1 n1 

R(ei , e j )γi φk03 (ei ).

(4.20)

i=n1 +n2 +1

This particular case of a three-step cantilever beam is numerically examined below for illustrating the crack scanning method. 5. Methodology validation and numerical examples

S3 : j = n1 + n2 + 1, ..., n};

φk03 (e j ) : j = n1 + n2 + 1, ..., n}.

k j =

(4.18)

5.1. Experimental validation A two-step cantilever beam with single and double cracks is tested by the modal testing technique as shown in Fig. 1. First, natural frequencies of the beam are measured and compared to those computed by using the Rayleigh quotient and TMM. The beam properties and results of both the test and computation are presented in Table 1 that shows a good agreement of frequencies computed by the TMM with experimental ones. This is because both the TMM and modal testing technique are the most exact methods for determining natural frequencies of beam-like structures. Obviously, the Rayleigh quotient, as usually, overestimates the natural frequencies (except some particularities), and this disagreement increases with the mode number (higher modes) but limited to less than 5% of discrepancy. The measured natural frequencies are employed as inputs for crack detection below using the proposed procedure. 5.2. Numerical comparative study

(4.19)

The obtained Rayleigh quotient is numerically validated in subsection by comparison of the frequencies computed by ing the Rayleigh quotient with those computed by the FEM TMM. The comparison is carried out for two-step (Nandwana Maiti, 1997; Table 2) and three-step beams (Attar, 2012; Table

this usand and 3).

N.T. Khiem et al. / International Journal of Solids and Structures 150 (2018) 154–165

161

Table 1 Comparison of computed natural frequencies with experimental results for a two-step cantilever. Crack scenarios

Natural frequencies (Hz)

No crack One crack of 12% depthat e2 (second step)

One crack of 31% depthat e2 (second step)

One crack of 50% depthat e2 (second step)

Two cracks at e1 (13%−1st step) and e2 (50% - 2nd step)

Two cracks at e1 (25% −1st step) and e2 (50%- 2nd step)

Two cracks at e1 (42% −1st step) and e2 (50%- 2nd step)

Measured TMM Rayleigh Measured TMM Rayleigh Measured TMM Rayleigh Measured TMM Rayleigh Measured TMM Rayleigh Measured TMM Rayleigh Measured TMM

first

second

third

fourth

fifth

sixth

13.00 13.0 0 01 12.9927 12.44 12.9890 12.9495 12.38 12.9240 12.8416 12.29 12.7498 12.8277 12.29 12.7220 12.7905 12.23 12.6474 12.6772 12.15 12.4227

53.69 53.7776 53.7198 53.56 53.6913 53.3825 53.19 53.1962 52.5354 51.81 51.9460 52.5331 51.69 51.9383 52.5267 51.61 51.9176 52.5074 51.13 51.8557

139.3 139.5445 139.5096 139.4 139.4924 139.3068 139.0 139.1958 138.8017 138.4 138.4601 138.6964 138.0 138.2257 138.4140 136.8 137.6004 137.5555 135.4 135.7389

289.25 289.6203 289.3426 288.94 289.2068 287.7235 287.38 286.8662 283.6615 281.44 281.2123 283.1060 280.19 280.1977 281.6130 277.06 277.5498 277.0437 269.69 270.1743

449.63 449.9872 449.6960 449.44 449.5551 448.0 0 01 446.88 447.1676 443.7591 440.63 441.7878 443.4379 440.38 440.9707 442.5765 436.88 438.8968 439.9583 432.75 433.5308

675.81 677.7374 677.4342 675.75 677.2888 675.6699 674.06 674.7977 671.2673 669.19 669.1519 671.1484 668.44 668.9736 670.8301 663.69 668.5051 669.8652 665.13 667.1685

Beam parameters: E = 200 Gpa;ρ = 7855 kg/m3 ; L1 = 0.4 m; L2 = 0.6 m; b1 = b2 = 0.02 m; h1 = 0.015 m; h2 = 0.008 m; Crack positions: e1 = 0.3 m at the first step; e2 = 0.6 at the second step. Table 2 Comparison of natural frequencies computed by the Rayleigh quotient with those computed by the FEM (Nandwana and Maiti, 1997) for a two-step cantilever beam (single crack at first step). Methods of computation

Crack in

Natural frequencies (Rad/s)

first step

First

(Err. %)

Second

(Err. %)

Third

(Err. %)

Rayleigh-PresentFEM-[Nandwana]

non

0.12

Rayleigh-PresentFEM-[Nandwana]

0.20 (10%)

Rayleigh-PresentFEM-[Nandwana]

0.40 (10%)

Rayleigh-PresentFEM-[Nandwana]

0.45 (10%)

Rayleigh-PresentFEM-[Nandwana]

0.20 (20%)

Rayleigh-PresentFEM-[Nandwana]

0.20 (30%)

Rayleigh-PresentFEM-[Nandwana]

0.20 (40%)

Rayleigh-PresentFEM-[Nandwana]

0.20 (50%)

6623.7859 6506.7 6613.5130 6483.7 6603.9375 6498.4 6599.6488 6488.3 6576.7442 6499.4 6546.7057 6480.9 6442.7901 6448.3 6269.7788 6398.3 5995.5085 6323.1

1.79

0.05 (10%)

2367.5910 2345.9 2366.1463 2334.0 2358.1826 2345.7 2363.5362 2341.6 2365.8514 2340.1 2331.0016 2344.6 2281.4427 2342.7 2198.2958 2339.7 2064.6620 2335.5

0.92

Rayleigh-PresentFEM-[Nandwana]

455.5595 455.0 455.4928 451.5 454.7097 453.0 453.1491 454.2 452.7806 454.4 452.2647 447.6 447.8463 438.3 440.5518 423.8 429.1556 402.2

0.88 0.37 −0.23 −0.35 1.04 2.17 3.95 6.7

1.37 0.53 0.93 1.10 −0.58 −2.61 −6.04 −11.59

2.00 1.62 1.71 1,19 1.01 −0.08 −2.00 −5.18

E = 210 Gpa; ρ = 7860 kg/m3 ; L1 = L2 = 0.25; b1 = b2 = 0.012; h1 = 0.02; h2 = 0.016 (m).

Results presented in the tables show that natural frequencies computed by the Rayleigh quotient excellently agree with those obtained by the FEM and TMM for a crack depth up to 40%. Particularly, in some cases, natural frequencies calculated by the Rayleigh quotient are more close to those computed by the FEM than to those computed by the TMM. A discrepancy between the frequencies is observed for the case of crack depth 60%, and it may reach 16–17% for the fourth frequency. Consequently, it can be noted that the Rayleigh quotient method proposed in this study is useful for calculating natural frequencies of cracked multistep beam with cracks of depth up to 40% beam thickness. 5.3. Numerical examples of crack detection for a multistep beam by natural frequencies Because the mode shape and modal curvature of an undamaged multistep beam are essential for solving both the forward and inverse problems of a cracked multistep beam using the Rayleigh quotient, the modal characteristics of the two-step beam used in

the experimental study are computed and compared to those of a uniform beam (see Fig. 2). Obviously, a stepwise change in crosssection significantly deviates the beam mode shape and modal curvature, especially for higher modes of vibration. The curvature graphs are visibly non-smooth at the beam step that may cause an abnormal difference between curvatures in both sides of the step. Furthermore, in this section, numerical results of crack identification for two-step and three-step cantilever beams by using the procedure proposed above are presented. For the two-step beam, the crack identification is accomplished in both cases of single and double cracks from natural frequencies measured in the experiment (see Table 1). It is assumed that only second step is cracked in the case of single crack, and each beam segment contains one crack for the case of double cracks. The crack identification problem for the three-step beam is examined without knowing how many and which segments of beam are cracked using the natural frequencies computed by FEM (Attar, 2012). In the case of a single crack appeared at the free end segment of the two-step cantilever beam, the equations for model updating

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N.T. Khiem et al. / International Journal of Solids and Structures 150 (2018) 154–165 Table 3 Comparison of natural frequencies computed by Rayleigh quotient with the TMM and FEM [Attar, 2012] for a three-step cantilever beam. Methods of computation

Crack depth (a/h) 1st step

2nd step

Natural frequencies (Error, %) 3rd step

First

Second

Third

446.4974 446.497(0.0) 445.805(0.15)

3075.8333 3075.83(0.0) 3054.13(0.71)

8004.6817 8004.68(0.0) 7903.62(1.27)

Rayleigh-PresentTMM – [Attar]FEM – [Attar]

No crack

Rayleigh-PresentTMM – [Attar]FEM – [Attar]

0.2

0

0

Rayleigh-PresentTMM – [Attar]FEM – [Attar]

0.2

0.2

0

Rayleigh-PresentTMM – [Attar]FEM – [Attar]

0.2

0.2

0.2

440.9822 440.664(0.07) 439.880(0.25)

Rayleigh-PresentTMM – [Attar]FEM – [Attar]

0.4

0.4

0.4

Rayleigh-PresentTMM – [Attar]FEM – [Attar]

0.6

0.6

0.6

420.6294 421.910(−0.3) 420.414(0.05) 350.7282 375.719(−6.6) 373.604(−6.1)

441.6481 441.353(0.02) 440.571(0.24) 441.0154 440.698(0.07) 439.911(0.25)

fourth

16,216.3119 16,216.30 (0.0) 15,951.10(1.66) 3075.8228 7975.5192 16,093.3269 3075.61(0.0) 7973.79(0.02) 16,088.1(0.03) 3053.88(0.72) 7873.46(1.29) 15,824.8(1.69) 15,883.0386 3044.0356 7975.3666 3042.17(0.06) 7973.54(0.02) 15,874.5 3020.52(0.77) 7872.83(1.30) (0.05) 15,615.0 (1.71) 3036.3942 7909.2825 15,711.0504 3034.30(0.07) 7904.25(0.06) 15,704.0 3012.60(0.79) 7804.34(1.34) (0.04) 15,446.6 (1.71) 2890.6793 7557.8108 13,749.2292 2902.82(−0.4) 7587.65(−0.4) 14,257.3(−3.56) 2878.26(0.43) 7483.90(0.98) 13,993.3(−1.74) 2387.5732 6355.5685 2588.12(−7.7) 6839.14(−7.0) 13,366.8083 2562.91(−6.8) 6739.97(−5.7) 11,613.9(15.09) 11,380.1(17.45)

E = 210 Gpa; ρ = 7860 kg/m3 ;L1 = L2 = L3 = 0.1;b1 = b2 = b3 = 0.02;h1 = h3 = 0.0075;h2 = 0.009375 (m);e1 = 0.06;e2 = 0.15;e3 = 0.24.

Fig. 2. Comparison of mode shapes (a) and curvatures (b) of uniform and stepped beams: solid lines – uniform beam and dash lines – stepped beam.

are ∗ ∗ ∗2 ∗ ∗ ∗2 U11 S1 + U12 S2 = ω10 ; U21 S1 + U22 S2 = ω20 ,

∗ ω10

(5.1)

∗ 13.00, ω20

where = = 53.69 are the measured natural frequencies of the intact beam given in Table 1 and

Uk∗1 =

 0

L1



2

[φ 0k1 (x )] dx; Uk∗2 =



L1 + L2

L1



2

[φ 0k2 (x )] dx, k = 1, 2. (5.2)

=

∗ ∗ U11 U22

∗ ∗ − U21 U12 .

a12 (e2 )γ2 = b∗1 ; a22 (e2 )γ2 = b∗2 .

∗2 ∗ ω10 U21 )/;

(5.3)

Using the updated stiffness and three pairs of measured natural frequencies(ω1∗ = 12.44, ω2∗ = 53.56 ); (ω1∗ = 12.38, ω2∗ = 53.19 ),

(5.4)

that allow the determination of crack location as solution of the equation

f (e ) = b∗1 a22 (e ) − b•2 a12 (e ) = 0, e ∈ (0.4 − 1.0 ).

Therefore, the updated stiffness parameters are ∗2 ∗ ∗2 ∗ ∗2 ∗ S1∗ = (ω10 U22 − ω20 U12 )/; S2∗ = (ω20 U11 −

and(ω1∗ = 12.29, ω2∗ = 51.81 ) as given in Table 1 corresponding to the three cases of crack depth (12%, 31%, and 50%), the diagnostic Eqs. (4.2) can be obtained in the following form:

(5.5)

For the first crack scenario as shown in Table 1 (where the actual crack depth equals to 0.12, 0.31, and 0.50), Eq. (5.5) gives crack locations detected as eˆ2 = 0.5897; 0.5913; 0.5945, respectively; these values are close to the actual position e2 = 0.6. The associated depths of the crack are estimated as 0.117, 0.303, and

N.T. Khiem et al. / International Journal of Solids and Structures 150 (2018) 154–165

163

Fig. 3. Results of crack localization for a three-step beam with (five) various scenarios of cracks as shown in Table 3 (single, double, and triple cracks) by the crack scanning method.

0.490, respectively. Obviously, the crack of a larger depth is more accurately detected; this well agrees with the practical application where small cracks are more difficult to be detected than large cracks. Moreover, the latter fact allows one to note that although the Rayleigh quotient is useful to calculate natural frequencies of a beam with cracks of depth only up to 40%, it is still efficient also for detecting cracks of depth up to 50%. In the case of double cracks, four measured natural frequencies given in Table 1 (three last scenarios) (a) 12.29–51.69–138.0– 280.19; (b) 12.23–51.61–136.8–277.06, and (c) 12.15–51.13–135.4–

269.69 are used as inputs for solving problem (4.11) that allows one to detect the crack locations by using MATLAB function fsolve.m as follows:

(a ) : eˆ1 = 0.3515, eˆ2 = 0.5989; (b ) : eˆ1 = 0.3372, eˆ2 = 0.5919; (c ) : eˆ1 = 0.3281, eˆ2 = 0.5912. Using the detected crack positions, the corresponding depths are estimated as

(a ) : aˆ1 /h1 = 0.105, aˆ2 /h2 = 0.510;

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N.T. Khiem et al. / International Journal of Solids and Structures 150 (2018) 154–165 Table 4 Results of crack identification in a three-step cantilever beam by the crack scanning procedure. Crack scenarios Single crack Double cracks

Triple cracks

Actual Detected Actual Detected Actual Detected Actual Detected Actual Detected

0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

Crack position

Crack depth

0.06 0.06 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15

0.2 0.2041 0.2 0.1985 0.2 0.1980 0.4 0.4009 0.6 0.6038

(b ) : aˆ1 /h1 = 0.271, aˆ2 /h2 = 0.498; (c ) : aˆ1 /h1 = 0.425, aˆ2 /h2 = 0.502. Comparing the results with the actual crack positions and depths as shown in Table 1 demonstrates that the detection of small cracks (13% and 25%) in this case is still more erroneous while the larger cracks could be detected almost accurately. The inaccuracy in crack detection is caused by that only the model was updated, but the noise accompanied with the measured natural frequencies was treated. Both the modeling and measurement errors would be instantaneously corrected when the crack scanning method is applied in combination with Tikhonov’s regularization technique for detecting unknown number of cracks in a three-step beam. The crack detection is performed herein for a three-step cantilever beam studied by Attar 2012 using the scanning method with four natural frequencies computed by the FEM (see Table 3). Suppose that the beam contains n = n1 + n2 + n3 cracks at positions 0 < e1 < e2 < ... < en < L with unknown depths a1 ,..., an . For correcting the deficiency and error in the given input data, the regularization technique proposes the replacement of the diagnostic Eq. (4.15) by

[AT ( i )A ( i ) + β I ] · {γ ( i ) } = b,

(5.6)

where I is the unique matrix of dimension n and β is the regularization factor. Using the singular value decomposition, the solution of Eq. (5.6) is

γˆ (i) =

R 



k=1

 σk uTk b v , β + σk2 k

(5.7)

where σ k ,uk ,vk ,k = 1, ..., R are singular values and left and right singular vectors, respectively, and R is the rank of the matrix A(i). The regularization factor β is determined from the equation

g( β ) =

R  k=1



β uTk b β + σk2

2

−

n 



2

uTk b

= δ2,

(5.8)

k=R+1

where δ = b − b¯ represents the erroneousness level of vector in the left hand side of Eq. (5.6) compared to the exact one b¯ . In this example, the noise level δ is determined as discrepancy between natural frequencies computed by FEM and TMM. Results of crack localization for the crack scenarios given in Table 3 are shown in Fig. 3. Crack magnitudes estimated subsequently in the first, an intermediate, and last iterations for one of the crack scenarios given in Table 3 are presented in one row of Fig. 3. Obviously, a single crack at the first step of a three-step cantilever beam is exactly detected (at position 0.06) by four natural frequencies. The double cracks at 0.06 and 0.15 were accurately predicted, but in this case, a false crack is also detected at position 0.26 but with very small depth. In the case of triple cracks, the results of the detection well agree with the actual scenarios. In particular, three cracks are immediately detected at the first iteration

– (0.26) 0.24 0.24 0.24 0.24 0.24 0.24

0.2 0.1944 0.2 0.1901 0.4 0.3906 0.6 0.6045

– (0.1081) 0.2 0.1917 0.4 0.3964 0.6 0.6060

for the case of 60% crack depth. Although the detected cracks have the peak values of magnitude, cracks at the positions neighboring to the actual ones but having less magnitudes are also detected. This is probably typical for the stepped beam in comparison with the uniform one (Khiem and Toan, 2014). Estimated crack depth and detected crack positions are tabulated in Table 4 that show exactness in crack locating and acceptable accuracy in crack depth estimation. Evidently, the crack at the middle step is more exactly predicted because its actual depth is greater than that at other two steps although their relative depths are the same and, as usually, cracks of more severity are more accurately detected. Finally, the number of cracks, in general, could be truly predicted by the crack scanning method, except in one case when a false crack was detected with a small depth. Convergence of the crack scanning method is adjusted by the parameter ϑ in Eq. (4.13), and the iteration process of mesh renewal actually converges after a maximum of six iterations if the truncation parameter is properly selected. However, the question how to select the truncation parameter to obtain fast convergence remains unanswered for the authors. 6. Conclusion In the present study, a closed-form solution was obtained to the problem of detecting multiple cracks in multistep cantilever beams by measurements of natural frequencies. First, the well-known Rayleigh quotient for calculating natural frequencies of beam-like structures was extended to a multiple cracked beam with stepwise change in cross-section. The established Rayleigh quotient enables to obtain an explicit expression of natural frequencies of a cracked multistep cantilever beam through crack parameters that provides an efficient tool for modal analysis and crack detection for the beam. Numerical and experimental results demonstrate that natural frequencies of a multistep beam with an arbitrary number of cracks can be easily calculated with adequate accuracy by using the proposed Rayleigh quotient. The obtained natural frequency expression can be directly used to derive basic diagnostic equations for crack detection from measured natural frequencies. General problem of multiple crack detection is solved in both cases of known and unknown number of cracks in a multistep beam. In the former case, the crack detection is accomplished by an analytical method, and in the latter one, the problem is solved by the application of the crack scanning method. The solution of the crack detection problem validated in both experimental and numerical examples shows that the procedure based on the Rayleigh quotient is efficient for multiple crack identification in a multistep beam. The important advantage of the crack identification procedure proposed in the present study is that it allows crack localization and crack depth estimation to be subsequently accomplished with the well-known technique for incorrectness regularization. Nevertheless, the crack detection procedure proposed in this study has not been validated for the case when cracks occur at steps of the beam; this is a subject of further study for the authors.

N.T. Khiem et al. / International Journal of Solids and Structures 150 (2018) 154–165

Although the problem was solved only for multistep cantilever beam, the procedure can be directly applied for beams of other boundary conditions also including the elastic ones. Acknowledgment This study is supported by Vietnam NAFOSTED, Grant No. 107.01–2015.20. Appendix Formula for calculating magnitude from the depth of a crack

γ j = E I j /L j K j = 6π (1 − ν 2 )(h j /L j )Ic (a j /h j ), j = 1, ..., n,

(A.1)

where aj /hj is the relative crack depth, ν is the Poisson’s coefficient, Lj = xj − xj − 1 and function Ic (z) is (Chondros et al., 1998) given as follows:

Ic (z ) = 0.6272z2 − 1.04533z3 + 4.5948z4 − 9.9736z5 + 20.2948z6 − 33.0351z7 + 47.1063z8 − 40.7556z9 + 19.6z10 .

(A.2)

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T.H. Tran, Principle researcher of IMECH, VAST; Lecturer of GUST and VNU Degrees: Engineer of Mechanics (20 0 0); Master of Science (2005); PhD (2012); Research fields: Engineering Vibrations; Structural Dynamics; Technical Diagnostics.

V.T.A. Ninh, Lecturer of University of Transport and Communication, Hanoi, Vietnam; Degrees: Bachelor (2003); Master of Science (2010); PhD student (from 2015) Research fields: Engineering Mechanics; Structural Dynamics; Technical Diagnostics.