Crack detection of a double-beam carrying a concentrated mass

Mechanics Research Communications 75 (2016) 20–28

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Crack detection of a double-beam carrying a concentrated mass Khoa Viet Nguyen Institute of Mechanics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Ha Noi, Viet Nam

a r t i c l e

i n f o

Article history: Received 16 January 2016 Received in revised form 16 May 2016 Accepted 22 May 2016 Available online 24 May 2016 Keywords: Natural frequency Concentrated mass Crack Double-beam

a b s t r a c t This paper presents the influence of a concentrated mass location on the natural frequencies of a cracked double-beam. The double-beam consists of two different beams connected by an elastic medium. The concentrated mass is located on the main beam. The relationship between the natural frequency and the location of concentrated mass is established and called “Frequency–Mass Location” (FML). The numerical simulations show that when there is a crack, the frequency of the double-beam changes irregularly when the concentrated mass is attacked at the crack position. This irregular change can be amplified by the wavelet transform and this is useful for crack detection: the crack location can be detected by the location of peaks in the wavelet transform of the FML. Finite element model for the cracked double-beam carrying a concentrated mass is presented and numerical simulations are also provided. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Cracks may cause serious failure of mechanical system and civil engineering structures. Therefore, early crack detection is extremely important and this issue has attracted many researchers in the last three decades. Since the dynamic characteristics of structures may be influenced seriously by the existence of cracks, the dynamic characteristics of cracked structures such as natural frequencies and mode shapes have been investigated and applied wildly for the crack detection of structures. Chondros and Dimarogonas [1], Liang et al. [2], Lee and Chung [3] investigated the influence of a crack on natural frequencies of a beam. Orhan [4] and Zheng and Kessissoglou [5] used finite element method to investigate the relationship between natural frequency of a cracked beam to the depth and location of the crack. Khaji et al. [6] presented closed-form solutions for crack detection problem using natural frequencies of Timoshenko beams with various boundary conditions. Gudmundson [7] used a flexibility matrix approach to study the influence of cracks on the natural frequencies of slender structures. VakilBaghmisheh et al. [8] used genetic algorithms for crack detection of beam-like structures using natural frequencies. In other works [9,10] cracks were modelled as massless rotational springs, whose stiffness was obtained by using fracture mechanics to investigate the change in natural frequencies caused by cracks. However, the change in natural frequency is generally not sensitive to small cracks, thus it is difficult to use directly the change in natural frequency to detect small cracks.

E-mail addresses: [email protected], nguyenvietkhoa [email protected] http://dx.doi.org/10.1016/j.mechrescom.2016.05.009 0093-6413/© 2016 Elsevier Ltd. All rights reserved.

Some other authors used the change in mode shape to detect the location of cracks in structures since the cracks influence the mode shape locally. Pandey et al. [11] and Abdel Wahab [12] applied the change in curvature mode shapes to determine the location of cracks in beam like structures. Zhong and Oyadiji [13] presented a wavelet-based method for crack detection of simply-supported beams by continuous wavelet transform of reconstructed modal data. Zhang et al. [14] presented the crack identification method combining wavelet analysis with transform matrix using mode shapes of a cracked cantilever beam. The author of this paper [15] used 3D finite elements to calculate the mode shapes of a cracked beam. This study showed that the distortion in mode shapes using 3D element model can be applied for detection small cracks. In literature, only few explicit expressions for crack identification are available, however, most of them refer to the case of a single crack. While, exact identification procedures of multiple concentrated cracks are very rare. Caddemi and Calio [16] derived the exact closed-form solution for the vibration modes of the Euler-Bernoulli beam with multiple open cracks. In other work, Caddemi and Calio [17] presented an exact procedure using natural frequencies and mode shapes for the reconstruction of multiple concentrated damages on a straight beam applied for the identification of multiple concentrated cracks. However, in practice, the measurement of mode shapes is more complicated with lower precision in comparison with the measurement of natural frequencies. Recently, the elastically connected beam structures are widely used in various engineering fields, such as multiple-walled carbon nanotubes, tall building, continuous dynamic vibration absorber, ect [18,19] due to performances of these complex continuous systems: weight reduction, strength and stiffness increase, and

K.V. Nguyen / Mechanics Research Communications 75 (2016) 20–28

21

Fig. 1. A double-beam element carrying a mass.

vibration absorption. The dynamic characteristics of the elastically connected beam structures, especially the double beams have been investigated by many researchers and obtained interesting achievement. Oniszczuk [18] presented the complete exact theoretical solutions of the free vibrations of two parallel simply supported beams continuously joined by a Winkler elastic layer. In this study, the effect of physical parameters characterizing the vibrating system on the natural frequencies was investigated. Mao [19] used the Adomian modified decomposition method to analyze the free vibration of elastically multiple-beams. Chen and Sheu [20] modelled a composite material by elastically connected beams to study the vibration of an axially loaded double Timoshenko beam. Vu et al. [21] proposed an exact method for analyzing the vibration of a double-beam system subjected to a harmonic excitation. Oniszczuk [22] studied undamped forced transverse vibrations of an elastically connected simply supported double-beam system subjected to arbitrarily distributed continuous loads using the modal expansion method. Seelig and Hoppmann [23] developed a method to solve the differential equations of motion of an elastically connected double-beam system subjected to an impulsive load. Rao [24] investigated the free response of Timoshenko beam systems in which the effects of rotary inertia and shear deformation were taken into account. Shamalta and Metrikine [25] studied the steady-state dynamic response of an embedded railway track subjected to a moving train. The track consists of two beams connected to a plate by continuous viscoelastic elements and an elastic foundation that supports the plate. In some other works, doublestring and double-rod systems were investigated [26–28]. In these studies, only dynamic properties of the intact double-beams were analyzed, while the effect of cracks on the dynamic characteristics and the application of this effect for crack detection have not been addressed yet. Although the change in the natural frequency caused by cracks is small and difficult to be measured when the crack size is small, this small change might be amplified by combining with other factors that also affect the natural frequency to establish an efficient method for crack detection of structures. For this purpose, the concentrated mass can be applied since it can influence the natural frequency of structures. The change in the natural frequency of structure caused by concentrated masses has been investigated in some works [29–37]. However, these works focused mainly on intact single beams. This work aims to present a method for damage detection of double-beams by using the change in natural frequency caused by cracks and a concentrated mass. When the concentrated mass is attached on the double-beams there is a change in the natural frequency and this change depends on the mass location. The relationship between the natural frequency and the mass location is established and called “Frequency–Mass Location” (FML). It is interesting that when the concentrated mass is located at

the crack positions there are irregular changes in the FML at the crack positions. This is not only property of double beams but it is also encountered in simple beams since the crack and concentrated mass affect the natural frequency of both single and double beams. Therefore, inspecting this irregular change in the FML, the crack can be localized. In this study, the finite element model of the cracked double-beam carrying a concentrated mass is presented and numerical simulations are provided. 2. Free vibration equation of a beam with concentrated mass 2.1. Intact double-beam In this study, the finite element model of a double-beam system carrying a concentrated mass m at section xm is presented in Fig. 1. The double-beam system consists of two different Euler–Bernoulli beams with rectangular sections connected by a Winkler elastic layer with stiffness modulus km per unit length. The length of the double-beam is L and the length of an element is l. In this study, the undamped vibrations of the system are considered. The free motion equation of an element of the double-beam system can be derived by using Hamilton’s principle as follows [38]. Using a local coordinate system having its origin at the center of the element, and the element is defined from −l/2 to +l/2, the kinetic energy of a double element carrying the concentrated mass can be written as:

⎛

T=

1 ˙T ⎜ d ⎝ 2 e1

⎛ 1 T ⎜ + d˙ e2 ⎝ 2

⎞

l/2



−l/2

l/2



⎟ mı(x − xm ) + 1 NT Ndx⎠ d˙ e1 ⎞

⎟ 2 NT Ndx⎠ d˙ e2

(1)

−l/2

where 1 and 2 are the material densities of the main and auxiliary elements per unit length, respectively; ı is delta Dirac function; d˙ e1 and d˙ e2 are the velocity vectors of the main and auxiliary elements; N is shape function. Denote that:

l/2 m∗

mı(x − xm )NT Ndx = mNT N;

=

−l/2 l/2



me1 =

(2)

l/2 1 NT Ndx; me2 =

−l/2

2 NT Ndx −l/2

22

K.V. Nguyen / Mechanics Research Communications 75 (2016) 20–28

ai

Substitute Eq. (2) into Eq. (1) we have: 1 T 1 T 1 T T = d˙ e1 m∗ d˙ e1 + d˙ e1 me1 d˙ e1 + d˙ e2 me2 d˙ e2 2 2 2

Mi

(3)

Mi+1 Pi

Here: me1 and me2 are the element mass matrices of the main and auxiliary beams; m∗ is the additional matrix of the concentrated mass. Denote: m∗e1 = me1 + m∗

1 T 1 T T = d˙ e1 m∗e1 d˙ e1 + d˙ e2 me2 d˙ e2 2 2

Fig. 2. Model of cracked element.

(4) M=

Substitute Eq. (4) into Eq. (3), we have:

Pi+1



M∗1

; K=

−K∗m

M2



−K∗m

K1 + K∗m

K2 +

;

K∗m

D1

D=



¨  ¨ D=

D2

 

D1 ¨2 D

;

˜ O=

O O

(14) (5)

The potential energy of the system can be obtained: =

1 T 1 T 1 de1 − dTe2 k∗m (de1 − de2 ) (6) d ke1 de1 + dTe2 ke2 de2 + 2 2 e1 2

where ke1 and ke2 are element stiffness matrices of the main and auxiliary beam; de1 and de2 are the displacement vectors of the main and auxiliary elements; and: k∗m

l/2 =

km (x) NT Ndx

(7)

−l/2

(8)

or

1 1 1 T − dTe1 ke1 de1 − dTe2 ke2 de2 − de1 − dTe2 k∗m (de1 − de2 ) (9) 2 2 2

Applying Hamilton’s principle:

(15)

where

T=

1 T 1 T 1 T L = d˙ e1 me1 d˙ e1 + d˙ e1 m∗ d˙ e1 + d˙ e2 me2 d˙ e2 2 2 2

−1

−l

1

0

0

−1

0

1

T (16)

˜ is flexibility matrix of the cracked element which is the sum C of the flexibility matrix of the intact element C0 and the additional flexibility matrix C1 caused by the crack. The generic components of the flexibility matrices C0 and C1 can be calculated from the fracture mechanics as follows: 2

t2

(0)

Ldt = 0

ı

Fig. 2 shows the cracked element model. It is assumed that the crack only affects the stiffness, while the mass of the beam remains constant. A brief description for deriving the element stiffness matrix of a cracked element is presented here, more details can be found from the previous papers [39–41]. The stiffness matrix of the open cracked element is derived as follows: The stiffness matrix of the cracked element is derived by applying the principle of virtual work: ˜ kc = TT CT

The Lagrangian can be established: L =T −

2.2. Cracked double-beam

(10)

∂ W (0) ; i, j = 1, 2; P1 = P; P1 = M ∂Pi ∂Pj

(17)

2

t1

(1)

with the initial conditions ıde1 = 0 and ıde2 = 0 at moments t = t1 and t = t2 , the governing equations for free vibration of an element can be obtained as follows: m∗e1 d¨ e1 + ke1 de1 + k∗m (de1 − de2 ) = o me2 d¨ e2 + ke2 de2 − k∗m (de1 − de2 ) = o

(11)

Here, d¨ e1 and d¨ e2 are the acceleration vectors of the main and auxiliary elements; o is the zero column vector consisting of four elements. Finally, the governing equations of free vibration of the doublebeam in the global coordinate system can be written as: ¨ 1 + K1 D1 + K∗m (D1 − D2 ) = O M∗1 D ¨ 2 + K2 D2 − K∗m (D1 − D2 ) = O M2 D

(12)

Here M2 , K1 , K2 are global structural mass and stiffness matrices of the main and auxiliary beams, respectively; K∗m is the global stiffness matrix of the elastic medium; D1 and D2 are column vectors which denote the nodal displacements of the main and auxiliary beams, respectively; O is the zero column vector assembled from column vectors o. Eq. (12) can be rewritten as follows: ¨ + KD = O ˜ MD

cij =

∂ W (1) i, j = 1, 2 P1 = P ; P1 = M ∂Pi ∂Pj

(18)

where W (0) is the strain energy of the intact element; W (1) is the additional energy due to the crack; P and M are the shear and bending internal forces at the right node of the element (Fig. 2). Considering the bending only, W (0) and W (1) are obtained as: W

(0)

W

(1)

1 = 2EI



a  =b

P 2 L3 M l + MPl + 3 2



2

2 (KIM + KIP )2 + KIIP  E

(19)

 da

(20)

0

M∗1 ,

Here:

cij =

(13)

where: KIM =

√ 6M aFI (s) ; bh2



FI (s) =



2 tg s

KIP =

√ 3Pl aFI (s) ; bh2

KIIP =

√ P aFII (s) (21) bh

 s  0.923 + 0.1991 − sin s/2 4 

FII (s) = 3s − 2s2

2

cos s/2

1.122 − 0.561s + 0.085s2 + 0.18s3 √ 1−s

(22)

(23)

Here, a is the crack depth; h is the thickness; b is the width of the beam; s = a/h; FI (s) and FII (s) are the correction functions for stress intensity factor.

K.V. Nguyen / Mechanics Research Communications 75 (2016) 20–28 Table 1 Natural frequencies of a double-beam. Natural frequencies (rad/s)

Ref. [18]

Ref. [19]

Present paper

ω ¯1 ω ¯2 ω ¯3 ω ¯4 ω ¯5 ω ¯6

19.7 43.5 79.0 87.9 177.7 181.8

19.7392 43.4699 78.9568 87.9442 177.6529 181.8256

19.7392 43.4699 78.9568 87.9442 177.6529 181.8256

Element stiffness matrices are assembled to form the global stiffness matrix K of the cracked double-beam. Substituting this global matrix K into Eq. (13) and solving this eigenvalue equation, the frequencies and mode shapes of the cracked double-beam carrying a concentrated mass will be obtained.

23

from 5% to 30%. As can be seen from this figure, there are two sharp changes at the crack positions in the graph of df as expected. This result is useful for crack detection: the crack position can be detected by the positions of the irregular changes in the graph of df. However, the FMLs of an intact double-beam which can be considered as a baseline data may not always available in practice. To overcome this, the wavelet transform can be applied for analyzing the FMLs since the wavelet transform uses small wavelike functions which have local properties that are useful to analyse the hidden details or irregular changes contained in the FMLs. The wavelet transform is defined as follows [42]:







+∞



1 W ˛, ˇ = √ ˛

f (x)

∗

x−ˇ ˛

 dx

(24)

−∞

3.1. Reliability of the theory In order to check the reliability of the theory, a double simply supported beam with parameters adopted from Refs. [18,19] is considered as follows. E 2 I 2 = 4 × 106 Nm2 , E 1 I 1 = 2 × E 2 I 2 , 2 A2 = 100 kg/m, 1 A1 = 2 × 2 A2 , km = 1 × 105 N/m2 , L = 10 m Table 1 lists the six lowest natural frequencies of the doublebeam without a concentrated mass obtained by three methods. As can be seen from Table 1, the first six natural frequencies of the present work are in close agreement with Refs. [18,19]. Especially, the natural frequencies obtained by the present method are in excellent agreement with Ref. [19].

where ˛ is a real number called scale or dilation, ˇ is a real number coefficients at scale ˛ and posicalled position, W(˛,ˇ) are wavelet  tion ˇ, f(x) is input signal, (x − ˇ)/˛ is wavelet function and ∗







20

3.2. Influence of the concentrated mass on the natural frequencies of the cracked double-beams and its application for crack detection using wavelet transform

19.5

19

18.5

0

0.2

0.4 0.6 0.8 Mass position (x/L)

1

a) FML of the 1st natural frequency

Frequency (Hz)

44

43

42

41

0

0.2

0.4 0.6 Mass position (x/L)

0.8

1

b) FML of the 2nd natural frequency 79 Frequency (Hz)

3.2.1. Double simply supported beam In order to investigate the influence of the position of concentrated mass on natural frequencies of the double-beam, the relationship between the natural frequency and the location of concentrated mass is established. In this simulation, the concentrated mass of 100 kg is applied. Fig. 3 presents the FMLs of the first three natural frequencies of the intact double simply supported beam. As can be seen from this figure, the natural frequencies change when the concentrated mass is moved from the left to the right of the double beam. As can be found from the literature, when there is a crack, the natural frequencies of a beam also change and the change in natural frequencies depends on the position of the crack. Therefore, it is expected that when the concentrated mass is moved to the crack position there might be an irregular change in the FMLs. However, numerical simulations have shown that this irregular change in FMLs is small and difficult to be inspected visually. Fig. 4 presents the FMLs of the first three natural frequencies of the simply supported double-beam having two cracks located at positions 3 m and 6.5 m with depths of 30% of the main beam height. Obviously, the irregular changes at the crack positions cannot be observed visually from this figure. However, using the difference df between FMLs of the intact and cracked double beams the irregular changes in FMLs can be observed. Fig. 5 presents the graph of df versus the mass location of FMLs of the first frequency of the intact and cracked double simply supported beams with four levels of the crack depth ranging



(x − ˇ)/˛ is complex conjugate of (x − ˇ)/˛ . Let us apply the wavelet transform for the FML of the first natural frequency of the double-beam having two cracks located arbitrary in the main beam. In this simulation, the two cracks with identical depths are assumed to be located at 3 m and 6.5 m. Four levels of the crack depth from 5% to 30% of the main beam height are applied.

Frequency (Hz)

3. Numerical simulation

78 77 76 75

0

0.2

0.4 0.6 Mass position (x/L)

0.8

1

c) FML of the 3rd natural frequency Fig. 3. FMLs of the intact double simply supported beam.

24

K.V. Nguyen / Mechanics Research Communications 75 (2016) 20–28 -3

x 10 9.6

19 df (Hz)

Frequency (Hz)

19.5

18.5

9.4 9.2

18

0

0.2

0.4 0.6 0.8 Mass position (x/L)

9

1

0

0.2

a) FML of the 1st natural frequency

0.4 0.6 0.8 Mass position (x/L)

1

a) Crack depth 5%

43 df (Hz)

Frequency (Hz)

44

42

41

0

0.2

0.4 0.6 0.8 Mass position (x/L)

0.036

0.035

0.034

1

0

b) FML of the 2nd natural frequency

0.4 0.6 0.8 Mass position (x/L)

1

b) Crack depth 10%

78

0.14

76 df (Hz)

Frequency (Hz)

0.2

74

72

0

0.2

0.4 0.6 0.8 Mass position (x/L)

0.135

0.13

1

0

c) FML of the 3rd natural frequency

0.2

0.4 0.6 0.8 Mass position (x/L)

1

c) Crack depth 20%

Fig. 4. FMLs of the cracked double simply supported beam having two cracks located at positions 3 m and 6.5 m, crack depths of 30%.

0.315

Fig. 6 shows the wavelet transforms of the FMLs of the first natural frequency with different crack depth levels. As can be seen from Graph (a) in this figure when there are two small cracks with depths of 5%, there are two peaks in the wavelet transform of the FMLs at positions of 3 m and 6.5 m which are the locations of the cracks. When the crack depths increase from 10% to 30%, the values of these peaks increase significantly as illustrated in Graphs (b)–(d). These results imply that when there are cracks, the first frequency changes suddenly when the concentrated mass is located at the positions of the cracks. This means that the crack locations can be detected by the locations of the significant peaks in the wavelet transform of the FML of the first natural frequency. The intensity of the mass also affects the frequency changes. In order to investigate the effect of the intensity of the mass on the frequency changes, three concentrated masses of 1 kg, 10 kg and 100 kg are applied. Fig. 7 presents the wavelet transform of the FML of the first frequency with the crack depth of 30%. It is observed that when the intensity of the mass increases, the two peaks in the wavelet transform at the crack positions become clearer and the values of the peaks increase. This result implies that when the intensity of the mass is high the proposed method for crack detection is more efficient. Therefore, the intensity of the mass should

df (Hz)

0.31 0.305 0.3 0.295 0

0.2

0.4 0.6 0.8 Mass position (x/L)

1

d) Crack depth 30% Fig. 5. FMLs of the cracked double simply supported beam with different crack depths.

be chosen suitably for specific structures for the crack detection purpose. The numerical simulations have shown that the accuracy of the detection of crack locations is affected by the finite discretization. Fig. 8 presents the close-up of the first peak in the wavelet transform of the FML of the first natural frequency with different finite discretization. As can be seen from this figure, when the number of finite elements applied for the double-beam model is large, the width of the peak in the wavelet transform at the crack position is small and vise versa. Similar result is obtained for the second

K.V. Nguyen / Mechanics Research Communications 75 (2016) 20–28 -5

-6

x 10

4 Wavelet coefficient

Wavelet coefficient

1 0.5 0 -0.5 -1

0

25

0.2

0.4

0.6

0.8

2 0 -2 -4

1

x 10

x/L

0

0.2

0.4

0.6

0.8

1

x/L

a) Crack depth 5%

a) Concentrated mass is 1kg

-5

-5

x 10

2

Wavelet coefficient

Wavelet coefficient

x 10

0

-2 0

0.2

0.4

0.6

0.8

2

0

-2

1

x/L

0

0.2

0.4

b) Crack depth 10%

2

0 -0.5 -1 0

0.2

0.4

0.6

0.8

1

x 10

1 0 -1 -2 0

x/L

c) Crack depth 20%

0.2

-4

0.6

0.8

1

c) Concentrated mass is 100kg Fig. 7. Wavelet transforms of the FMLs of the first natural frequency with different intensity of the mass.

1 0

-4

x 10 2

-1

0.2

0.4

0.6

0.8

Wavelet coefficient

Wavelet coefficient

0.4 x/L

x 10

-2 0

1

-4

Wavelet coefficient

Wavelet coefficient

x 10

0.5

2

0.8

b) Concentrated mass is 10kg

-4

1

0.6 x/L

1

x/L

d) Crack depth 30% Fig. 6. Wavelet transforms of the FMLs of the first natural frequency.

peak but it is omitted here for the sake of brevity. These results mean that the fine finite discretization provides a better accuracy for detecting the location of cracks in comparison with the coarse finite discretization. To demonstrate the applicability of the proposed method for multi-crack detection, three cracks at 3.3 m, 5.5 m and 7 m with crack depths of 20%, 24% and 30% are applied respectively. As can be seen from Fig. 9, there are three significant peaks in the wavelet transform of the FML at the crack positions. The heights of peaks depend on their corresponding crack depths: the low peaks correspond to the small crack depths and the high peaks correspond to the large crack depths. This result shows that the proposed method can be applied effectively for multi-crack detection.

0 60 elements 80 elements 100 elements

-2 0.2

0.25

0.3

0.35 x/L

0.4

0.45

Fig. 8. The close-up of the first peak with different finite discretization, crack depth is 30%.

In order to simulate the corrupted measurements, white noise is added to the FML of the first frequency of the double beam having two cracks at 3 m and 6.5 m with depths of 30%. For this purpose, a white noise vector is calculated as following formula [43]:



noise =



2

×R

exp (SNR × ln(10))/10

(25)

26

K.V. Nguyen / Mechanics Research Communications 75 (2016) 20–28 -4

-4

1 0 -1

1 0 -1 -2

0

x 10

2 Wavelet coefficient

Wavelet coefficient

x 10

0.2

0.4

0.6

0.8

1

0

0.2

0.4

3.2.2. Double fixed-end beam Similar simulation results are obtained for the case of the cracked double fixed-end beam as presented in Fig. 11. As can be seen from this figure, there are two peaks in the wavelet transforms of the FMLs at the crack locations when the crack depths range from 5% to 30%. When the crack depths increase the values of these peaks also increase as presented in Graphs (a)–(d). As can be observed from Figs. 6 and 11, the peak values in the wavelet transform of the FMLs in the case of double fixed-end beam are larger than the case of double simply supported beam. Therefore, the cracks in the double fixed-end beam can be detected more efficiently than in the double simply supported beam. 3.2.3. Double cantilever beam In the case of cantilever double-beam, the cracks are more difficult to be detected in comparison with the cases of simply supported and double fixed-end beams. In this case, two significant peaks in the wavelet transform corresponding to the crack locations can only be observed when the crack depths are equal to or larger than 10% as shown in Graph (b) of Fig. 12. These peaks become clearer and more significant when the crack depths are equal to or larger than 20% as presented in Graphs (c)–(d) of this figure. The dependence of results on the double-beam boundary conditions can be explained by the fact that the natural frequencies of the double-beam with different boundary conditions are different, thus the change in natural frequencies when the mass is moved from the left end to the right end are different. In this study, the maximum changes in the first natural frequency when the mass is moved from the left end to the right end of the double-beam in the case of fixed-end, simply supported and cantilever boundary

0.8

1

0.8

1

0.8

1

a) SNR=70

Fig. 9. Wavelet transforms of the FML of the first natural frequency with three cracks.

-4

Wavelet coefficient

x 10 2 1 0 -1 -2 0

0.2

0.4

0.6 x/L

b) SNR=50 -3

x 10

2 Wavelet coefficient

where  2 is the variance of the FML, SNR is the desired signal to noise ratio and R is a standard normal distribution vector with zero mean value and unit standard deviation. The corrupted FML is simply the sum of the simulated FML and the noise vector calculated in Eq. (25). Fig. 10 shows the wavelet transform of the noisy FML of the first frequency with the SNR of 70, 50 and 30. For a NSR of 70, the plot is very similar to noise free case (Graph (d) of Fig. 6). When the noise level increases to a SNR of 50, the two peaks at the crack positions can still be detected, but less recognizable as can be observed from Graph (b) of Fig. 10. However, when the SNR equal to 30, the peaks at crack positions cannot be detected as presented in Graph (c) of Fig. 10. These results mean that the proposed method can be applied efficiently with low levels of noise, but it cannot be applied for very high levels of noise. Our numerical simulations have shown that similar results can be obtained from higher natural frequencies. Therefore, in this paper only results obtained from the first natural frequency are presented.

0.6 x/L

x/L

1 0 -1 -2

0

0.2

0.4

0.6 x/L

c) SNR=30 Fig. 10. Wavelet transform of the FML of the first natural frequency.

conditions are 2.7 Hz, 0.9 Hz, and 0.6 Hz, respectively. This means that the change in the FMLs is largest in the case of fixed-end condition, it is smaller in the case of simply supported condition, and it is smallest in the case of cantilever condition. As a result, the change in the FMLs in the case of fixed-end condition can be detected more efficient than the case of simply supported and cantilever conditions.

4. Conclusion In this paper, the influence of a concentrated mass on the natural frequencies of a double-beam consisting of two different beams with different boundary conditions is presented. When there is a concentrated mass the natural frequencies of the double-beam change. The concentrated mass causes irregular changes in natural frequencies of the cracked double-beam when it is located at the crack positions. The irregular changes in natural frequencies can be inspected by significant peaks in the wavelet transform of the FMLs. This can be useful for crack detection: the crack locations can be determined by the locations of the significant peaks in the wavelet transform of the FMLs.

K.V. Nguyen / Mechanics Research Communications 75 (2016) 20–28 -5

-6

x 10

5 Wavelet coefficient

Wavelet coefficient

5

27

0

-5 0

0.2

0.4

0.6

0.8

x 10

0

-5 0

1

0.2

0.4

-4

Wavelet coefficient

Wavelet coefficient

1

1

0

-1 0.2

0.4

0.6

0.8

x 10

0.5 0 -0.5 -1

0

0.2

x 10 Wavelet coefficient

Wavelet coefficient

1

b) Crack depth 10%

-4

2 0 -2

2

0

-2 0

0.6

0.8

0.2

0.4

0.6

0.8

1

0.8

1

x/L

1

c) Crack depth 20%

x/L -6

c) Crack depth 20%

x 10

-4

Wavelet coefficient

x 10 Wavelet coefficient

0.8

-6

x 10

0.4

0.6 x/L

x/L

0.2

0.4

1

b) Crack depth 10%

-4 0

1

-6

x 10

4

0.8

a) Crack depth 5%

a) Crack depth 5%

0

0.6 x/L

x/L

5 0 -5

5

0

-5 0

0.2

0.4

0.6 x/L

0

0.2

0.4

0.6

0.8

1

d) Crack depth 30%

x/L

d) Crack depth 30% Fig. 11. Wavelet transforms of the FMLs of the first natural frequency.

The proposed method can be applied for detecting cracks with depths as small as 5% in the cases of simply supported and double fixed-end beams, while it can only be applied for detecting cracks with depth of up to 10% in the case of double cantilever beam. The proposed method can be applied for the case of noncorrupted as well as corrupted measurements. However, for very high levels of noise, e.g. for a SNR of 30, the proposed method cannot be applied.

Fig. 12. Wavelet transform of the FML of the first natural frequency.

Acknowledgement This paper was sponsored by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) 107.022014.01. References [1] T.G. Chondros, A.D. Dimarogonas, Identification of cracks in welded joints of complex structures, J. Sound Vib. 69 (1980) 531–538. [2] R.Y. Liang, J. Hu, F. Choi, Theoretical study of crack-induced eigenfrequency changes on beam structures, J. Eng. Mech.: ASCE 118 (1992) 384–396. [3] Y. Lee, M. Chung, A study on crack detection using eigenfrequency test data, Comput. Struct. 77 (2000) 327–342.

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