Damage detection under ambient vibration by harmony search algorithm

Expert Systems with Applications 39 (2012) 9704–9714

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Damage detection under ambient vibration by harmony search algorithm Leandro Fleck Fadel Miguel a,⇑, Letícia Fleck Fadel Miguel b, João Kaminski Jr. c, Jorge Daniel Riera d a

Department of Civil Engineering, Federal University of Santa Catarina, Florianópolis, Brazil Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil c Department of Civil Engineering, Federal University of Santa Maria, Santa Maria, Brazil d Department of Civil Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil b

a r t i c l e

i n f o

Keywords: Structural health monitoring Ambient vibration Harmony search algorithm

a b s t r a c t Damage in structural systems induced by vibrations, alternating load cycles, temperature changes, corrosion, etc., constitute a serious technical problem. Smart methods of control and structural health monitoring (SHM) for large structures are, therefore, highly needed. In certain structural applications, moreover, a lack of access to the damaged area imposes an additional constraint on damage identification procedures. One method that may fulfill those requirements is dynamic nondestructive testing, which consists of monitoring changes in the structure’s natural frequencies, vibration modes and damping. In this paper, a new approach for vibration-based (SHM) procedures is presented, in an ambient vibration context; this method combines a time domain modal identification technique (SSI) with the evolutionary harmony search algorithm. A series of numerical examples with different damage scenarios and noise levels have been carried out under impact and ambient vibration. Thereafter, an experimental study of three cantilever beams with several different damage scenarios is conducted and the proposed methodology has shown potential for use in the damage diagnosis assessment of the remaining structural life. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Both the direct experience accumulated during the last two centuries, as well as the complete reliability assessments of representative structures, such as bridges, cranes, pressure vessels and towers, clearly show that to guarantee the system reliability throughout its life span, periodic maintenance are mandatory. In the absence of adequate inspection and maintenance, the failure probability would continue increasing until collapse occurs. Clearly, the damage accumulation process advances at a faster pace in those structures subjected to stress cycles, vibrations, harsh environment, or combinations of these conditions. Thus, unless subjected to a strict program of inspection and maintenance, bridges or pressure vessels, for example, generally present higher proneness to failure than building structures of the same age. In this context, it is evident that efficient methods to detect and quantify damage would have immediate application in much needed programs to restore the reliability of engineering structures, to initial design levels. One method that may fulfill those requirements is dynamic nondestructive testing, which consists of monitoring changes in the structure’s natural frequencies, ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (L.F.F. Miguel), [email protected] (L.F.F. Miguel), [email protected] (J. Kaminski Jr.), [email protected] (J.D. Riera). 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2012.02.147

vibration modes and damping. Any change in its dynamic properties may be due to a change in the system mass, stiffness or both, and may therefore be used to diagnose structural damage. In recent developments of vibration-based structural health monitoring (SHM) techniques, the focus has been placed on taking advantage of ambient vibrations such as traffic, wind or pedestrian induced vibrations in order to determine the spectral properties at any time without operational interference or the use of special equipment for the excitation, which would be required for the typical forced vibration tests. In such cases, the stochastic system identification field is of paramount importance since its main goal is to determine the modal parameters for output only measurement conditions (Miguel, Fadel Miguel, & Thomas, 2009). Modern time domain techniques, such as the Eigensystem Realization Algorithm (ERA) method (Juang & Pappa, 1985) coupled with the Natural Excitation Technique (NExT) or directly the Stochastic Subspace Identification (SSI) technique (Van Overschee & De Moor, 1993) can be successfully applied to deal with in situ measurements of ambient vibrations in long term SHM purposes. In spite of recent numerous studies with varying degrees of success (Barai & Pandey, 2000; Chen & Zang, 2011; Curadelli, Riera, Ambrosini, & Amani, 2008; Doebling, Farrar, Prime, & Shevitz, 1996; Fadel Miguel, Miguel, Riera, & Ramos de Menezes, 2007; Jaishia & Ren, 2006; Jiang, Zhang, & Zhang, 2011; Mehrjoo, Khaji, Moharrami, & Bahreininejad, 2008; Riera, 2004; Sohn et al., 2003) and the consequent increasing development of vibration-based

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structural health monitoring (SHM) techniques, this problem cannot be considered as completely addressed. Therefore, additional efforts must be carried out in order to assess robust and accurate techniques for monitoring and detecting damage through modal properties. Since SHM systems typically deal with the differences between the theoretically predicted behavior and the experimentally determined structural responses, its basic idea is to modify the properties of the numerical model in order to achieve an optimum fit with the experimental data. Hence, the process of locating and quantifying damage may be thought as an optimization procedure, in which the objective functions must be formulated in terms of modal properties. These solution methods can be mainly divided in gradient based numerical optimization methods and metaheuristic algorithms. Since gradient based methods can have difficulties in finding global minimum or maximum values when the initial solution is far different from the true solution, metaheuristic algorithms have recently received attention. Metaheuristic algorithms have been broadly used in these optimization problems, especially because they are not a gradient-based search, so they avoid most of the pitfalls of any gradient-based search algorithms. Thus, these algorithms have fewer mathematical requirements and they can be used to deal with complex objective functions. Many researchers claim that the metaheuristic algorithms are very efficient in the optimization area, and they attribute this efficiency to the fact that they imitate the best features in nature, especially the selection of the fittest in biological systems that have evolved by natural selection over millions of years. Among this class of algorithms, special attention was given to the use of Genetic Algorithms (GA) in structural health monitoring (SHM) procedures, resulting in a considerable number of studies (Au, Cheng, Tham, & Bai, 2003; Chou & Ghaboussi, 2001; Friswell, Penny, & Garvey, 1998; Gomes & Silva, 2008; Hao & Xia, 2002; Mares & Surace, 1996; Meruane & Heylen, 2010; Rao, Srinivas, & Murthy, 2004). This particular emphasis on GA algorithms can be explained since it was considered one of the first metaheuristic algorithms developed, being successfully applied in different engineering fields. However, they presented some drawbacks for SHM approaches, which are mainly linked to the great computational time that is required when dealing with large computational models. In this context, the harmony search algorithm (HS), developed by Geem, Kim, and Loganathan (2001), can handle some of these GA drawbacks. According to Yang (2008), HS could be more efficient than GA, for instance, because HS does not use binary encoding and decoding, but it does have multiple solution vectors. Therefore, HS is faster during each iteration. Furthermore, the implementation of HS algorithm is also easier. In addition, there is evidence to suggest that HS is less sensitive to the chosen parameters, which means that it is not necessary to fine-tune these parameters to get quality solutions. Recently, Degertekin (2008) and Degertekin, Hayalioglu, and Gorgun (2009) realized a comparative study of HS with other optimization methods for optimum design of steel frame structures and concluded that the HS algorithm yielded lighter designs for the examples presented and in many cases it also required less computational effort for the presented examples. Thus, the implementation of the HS algorithm in SHM context due to ambient vibration was not studied yet and seems to be promising. In this paper, a review of the previous developments is given, followed by numerical and experimental studies on damage assessment in the cases in which the (local) stiffness is affected by damage. In summary, a new methodology for detecting and quantifying structural damage under ambient vibration based on the harmony search algorithm (HS) and Stochastic System Identification is

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proposed. The study can be carried out in three major steps: (1) measurement of ambient vibration response; (2) modal identification using the time-domain output-only technique, the so-called SSI (Stochastic Subspace Identification) method and (3) damage localization and quantification using the HS algorithm through a minimization procedure. Firstly, a numerical assessment of a cantilever beam subjected to simulated impact and ambient vibration testing was conducted considering the noise effects. A series of numerical examples with different damage scenarios and noise levels have been carried out under impact and ambient vibration. Next, an experimental study of three cantilever beams with several different damage scenarios is conducted and the proposed methodology has shown potential for use in damage diagnosis and for use in finite model updating for model based condition assessment of the remaining structural life.

2. Damage detection as an optimization problem The theoretical basis in vibration-based structural health monitoring (SHM) procedures assumes that structural damage is typically related to changes in the structural physical parameters and, consequently, with the spectral properties of the structure. One classical approach to represent damage is considering a reduction in the stiffness properties of the structure, i.e., it can be assumed that structural damage can be modeled as a reduction of Young’s modulus in each finite element. Within this context, it is useful to introduce the damage in the structure through the consideration of an elemental stiffness reduction factor (ai), which enables the preservation of the original structural connectivity. Thus, the global stiffness matrix can be expressed as the summation of damaged and undamaged element stiffness matrices, where the local element stiffness is multiplied by a reduction factor as:

½k ¼

N X

/i ½Ki

ð1Þ

i¼1

In this equation, N is the total number of elements of the structure, [K] is the global stiffness matrix and it is assembled from the elemental stiffness matrices [K]i. The factor (ai) may be defined as the ratio of the elemental stiffness reduction to the initial stiffness. It ranges from 0 to 1, where 1 signifies no damage in the element and 0 means that the element loses its stiffness completely. In damage detection techniques through metaheuristic algorithms, structural damage is estimated from a model update process using damage-induced changes in the modal features. A numerical model is continuously updated until its difference with the experimental model is minimized. This process is defined as the minimization problem and it can be formulated as follows:

Finda Minimize PðaÞ ¼ kA  BðaÞk2

ð2Þ

Subjectto gðaÞ < 0 and hðaÞ ¼ 0

P(a) is the cost function, A is the experimental modal feature extracted from the structure, B(a) is the theoretical modal feature calculated from the numerical model of structure and g(a) and h(a) are inequality and equality constraints, respectively. In order to solve this problem, an objective function must be formulated in terms of the differences between the numerical and experimental values. A good choice of this cost function is of paramount importance in FE model updating, because it can influence the performance of the utilized optimization algorithm. A series of objective functions have been applied to FE model updating and damage assessment of structures, with different degrees of

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success. They usually adopt frequency and mode shape residuals as the starting point. In the sequence it is that shown some of the objective functions can be efficiently applied in a structural healthy monitoring context. 2.1. Frequency and mode shape changes An objective function may be constructed in terms of fractional changes in natural frequencies for the structures before and after damage, as shown in Eq. (3):

PðaÞ ¼

 A NM X dxi ðaÞ

xi

i¼1



 dxi

E !2 ð3Þ

xi

in which the superscripts A and E represent analytical and experimental, respectively, xi are the natural frequencies for the ith mode of undamaged or healthy condition for both experimental and analytical conditions and, finally, dxi is a fractional change of experimental and analytical natural frequencies for the ith mode of the structure. A FEM model should be used to represent the reference or healthy state of the target structure. Then, the stiffness reduction factor (a) of the FEM model should be updated until the differences of the numerical frequencies in the healthy and damaged state converged to those observed in experimental frequencies in the preand post-damaged state. Since the natural frequencies can be accurately measured, this objective function is practical in real-time structural healthy monitoring of structures under ambient vibration. However, it is difficult to differentiate the damage in symmetric locations for a symmetric structure. In this situation, mode shapes must be introduced in the cost functions, as shown in Eq. (4): PðaÞ ¼

 A NM X dxi ðaÞ i¼1

xi

 

dxi

xi

E ! 2 þ

NM X NP X ððd£ij ðaÞÞA  ðd£ij ÞE Þ2 i¼1

ð4Þ

j¼1

In practice, it is only possible to measure a few lower mode shapes and frequencies during vibration testing, even for free vibration tests. Thus, only those nodal displacements that are really measured can be picked out of the numerical FE mode shapes. Then, any mode expansion procedure can be avoided (Fadel Miguel, Ramos de Menezes, & Miguel, 2006). Another important feature is the possibility to relatively weigh the frequencies and mode shapes. Since the measured mode shapes are less accurate than the natural frequencies, the weights of the mode shapes are usually smaller than those adopted for the frequencies. Normally the weights are selected according to the variance of the measurements.

2.2. Dynamic residual force vectors An objective function may be formulated in terms of a dynamic residual force vector (Mares & Surace, 1996). This vector must be minimized when different dynamic properties from the healthy state are applied to the following equation:

fF r ðaÞgj ¼

N X

/i ½Ki f£j gexp  x2exp ½Mi f£j gexp

ð5Þ

i¼1

The dynamic residual force vector {Fr}j will be null if the experimental data are error free or if they represent the healthy state of the structure. However, if they represent the damaged state of the structure, the dynamic residual force vector {Fr}j will be different from zero and a minimization procedure must be carried out in order to locate and quantify the damage:

PðaÞ ¼

p X fF r ðaÞgTi fF r ðaÞgi

ð6Þ

i¼1

Since, a limited number of DOF measurements are available, a mode expansion procedure must be carried out and modeling errors are normally introduced. 2.3. Flexibility matrix The modal flexibility error residual may also be employed as a cost function. The modal flexibility matrix may be expressed as:

½F ¼ ½£½K1 ½£T

ð7Þ

where £ is a mode shape matrix and K represents a diagonal matrix containing the square of the modal frequencies. The difference between the experimental model and the numerical model can then be utilized as an objective function for damage quantification

PðaÞ ¼ k½FE  ½FðaÞE k2Fro

ð8Þ

in which a is the vector depicting the damage parameters, || ||Fro represents the Frobenius norm for the residual matrix, FE indicates the modal flexibility matrix from the identified results, and FA is the modal flexibility matrix calculated from the analytical model with vector damage parameters. It is not possible in practice to construct the flexibility matrix for the full DOFs, because only a limited number of measurements are available. Thus, the flexibility matrix may be obtained from only a few low-frequency modes in accordance with the measured DOFs. One important issue concerning Eq. (7) is the mass normalization of mode shapes obtained from ambient vibration tests, since the

Fig. 1. Pseudo code of Harmony search [adapted from Yang (2008)].

L.F.F. Miguel et al. / Expert Systems with Applications 39 (2012) 9704–9714

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Fig. 2. Beam modeled with 25 finite elements.

Table 1 Theoretical frequencies. Modes

Healthy condition (Hz)

Damage scenario (Hz)

1st 2nd 3rd 4th 5th 6th

26.5643 164.4624 451.7756 861.8124 1378.1590 1981.0781

26.5279 160.2891 418.6589 804.0409 1331.8619 1933.8760

input is not known. In general, there are three methods: FE model approach (Doebling et al., 1996), sensitivity-based method (Parloo, Verboven, Guillaume, & Van Overmeire, 2002) and the methods that use mass orthogonality condition (Bernal & Gunes, 2002).

3. Harmony search algorithm

Stiffness Reduction Factor (α)

Many classical or conventional algorithms for structural optimization are deterministics and most of them used the gradient information, i.e., they use the function values and their derivatives. They normally work extremely well for smooth unimodal problems; however, if there is some discontinuity in the objective function, they may not converge. Thus, in this kind of problem, a non-gradient algorithm is preferred. Non-gradient based or gradient-free algorithms do not use any derivative, but only the function values. Many different metaheuristic algorithms are in existence and new variants are continually being proposed. Among them, Harmony Search (HS) is a relatively new metaheuristic optimization algorithm, which was developed by Geem et al. (2001). HS is a music-based metaheuristic optimization algorithm, which is inspired by the observation that the aim of music is to search for a perfect state of harmony. This harmony in music is analogous to find the optimality in an optimization process. The search process in optimization can be compared to a musician’s improvisation process. Geem et al. (2001) observed that when musicians are improvising they used to (a) play famous piece of music (a series of pitches

in harmony) from their memory; or (b) play something similar to a known piece (thus adjusting the pitch slightly); or (c) compose new or random notes. Thus, in harmony search algorithm, these three options become (a) use of harmony memory; (b) pitch adjusting; and (c) randomization. The use of harmony memory will ensure that the best harmonies will be carried over to the new harmony memory. It is reached through the use of a parameter called Harmony Memory Considering Rate (HMCR), 0 6 HMCR 6 1. Therefore, if this parameter is too low, only a few of the best harmonies are selected and it may converge too slowly. On the other hand, if this parameter is extremely high (near 1), almost all the harmonies are used in the harmony memory, then other harmonies are not explored well, leading to potentially wrong solutions. Therefore, this parameter usually assumes values from 0.7 to 0.95. The second component is the pitch adjustment that is determined by a pitch bandwidth range (bw) and a pitch adjusting rate (PAR). In HS pitch adjustment corresponds to generating a slightly different solution. The PAR is used to control the degree of the adjustment. Thus, a low PAR with a narrow bandwidth can slow down the convergence of HS because the limitation in the exploration of only a small subspace of the whole search space. On the other hand, a very high PAR with a wide bandwidth may cause the solution to scatter around some potential optima as in a random search. Therefore, this parameter usually assumes values from 0.1 to 0.5 in most simulations (Yang, 2008). The third component is the randomization, which is to increase the diversity of the solutions. Although adjusting pitch has a similar role, it is limited to certain local pitch adjustment and thus corresponds to a local search. The use of randomization can drive the system further to explore various diverse solutions so as to find the global optimality (Yang, 2008). The three components in HS can be summarized as the pseudo code shown in Fig. 1 (Yang, 2008). Finally, according to Yang (2008), HS could be more efficient than genetic algorithms, for instance, because HS does not use binary encoding and decoding, but it does have multiple solution vectors. Therefore, HS is faster during each iteration. Moreover, the implementation of HS algorithm is also easier. In addition, there is evidence to suggest that HS is less sensitive to the chosen parameters,

1

Cost Function 1 Cost Function 2 Cost Function 3

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2

3 4

5

6 7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Element Fig. 3. Element 20 with 70% of damage for the three cost functions.

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Table 2 Identified frequencies for free vibration. Modes

Noise 3% (Hz)

1st 2nd 3rd 4th 5th

Noise 5% (Hz)

Healthy

Scenario 1

Scenario 2

Scenario 3

Healthy

Scenario 1

Scenario 2

26.563 164.458 451.723 861.458 1376.73

26.552 164.007 447.801 853.360 1370.11

26.521 160.285 418.614 803.738 1330.49

26.250 163.892 445.937 856.940 1370.87

26.552 164.459 451.723 861.453 1376.71

26.542 164.007 447.800 853.362 1370.11

26.518 160.286 418.611 803.733 1330.492

Table 3 Identified frequencies for ambient vibration. Modes

Noise 3% (Hz)

1st 2nd 3rd 4th 5th

Healthy

Scenario 1

Scenario 2

Scenario 3

26.563 164.505 451.969 862.400 1378.17

26.558 164.050 447.976 853.593 1369.87

26.527 160.303 418.836 803.749 1332.30

26.250 163.920 445.999 857.463 1371.33

which means that it is not necessary to fine-tune these parameters to obtain quality solutions. More details about HS may be found in Geem et al. (2001), Yang (2008), Geem (2009), among others. 4. Numerical examples A numerically simulated cantilever beam without damage and with several assumed damage elements is considered. The structure

Stiffness Reduction Factor (α)

(a)

is 750 mm long and it has square box cross section with external dimensions 25.4 mm and wall thickness equal to 1 mm. The structures were modeled with 25 Timoshenko beam elements, as shown in Fig. 2. The specific weight, elastic modulus of the material, Poisson’s coefficient and Timoshenko’s shear factor of the beam are 28 kN/m3, 68.6 GPa, 0.3 and 0.5, respectively. Since this beam is also used to experimentally validate the proposed damage detection procedure, as is detailed in Section 5, a concentrated mass of 18.2 g is also included in all DOFs of the numerical model, in order to correctly represent the presence of the accelerometers. Modal analysis is carried out through a FE Matlab code developed by the authors in order to obtain the frequencies and mode shapes. Then, a damage scenario is introduced in element 20 by means of a reduction in 70% of its elastic modulus and the modal analysis is again carried out aiming to obtain the simulated damaged experimental modal parameters. The frequencies are shown in Table 1. The HS model updating procedure explained in theoretical background is also implemented in a Matlab FE code. It is assumed that only the first 6 modes are available and measurements are

1

(b) 1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0

Element

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Element

Fig. 4. Free vibration and 3% noise: (a) damage Scenario 1 and (b) damage Scenario 2.

Stiffness Reduction Factor (α)

(a)

1

(b)1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Element

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Element

Fig. 5. Free vibration and 5% noise: (a) damage Scenario 1 and (b) damage Scenario 2.

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1

(b) 1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

Stiffness Reduction Factor (α)

(a)

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Element

Element

Fig. 6. Ambient vibration and 3% noise: (a) damage Scenario 1 and (b) damage Scenario 2.

Stiffness Reduction Factor (α)

(a)

(b)1

1 0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Element

Element

Fig. 7. Damage Scenario 3 and 3% noise: (a) free vibration and (b) ambient vibration.

Table 5 Experimental frequencies of Beam 2. Modes

1st 2nd 3rd 4th 5th

Healthy

26.26 161.22 434.49 783.97 1250.92

Cut depth 3.5 mm

5.0 mm

8.5 mm

120 mm

160 mm

26.25 158.67 414.14 739.75 1184.69

26.25 157.80 407.23 727.33 1182.89

26.25 155.19 387.80 698.84 1174.93

26.25 148.86 350.16 660.27 1170.82

26.19 133.95 294.39 632.29 1169.28

3.5 mm

5.0 mm

8.5 mm

120 mm

160 mm

25.25 157.51 415.58 759.20 1160.00

24.75 156.45 408.60 751.45 1149.49

23.72 154.67 394.21 732.58 1136.99

21.82 151.21 374.01 704.97 1119.61

18.62 145.95 350.61 665.28 1111.63

Table 6 Experimental frequencies of Beam 3. Modes Fig. 8. Experimental beam.

1st 2nd 3rd 4th 5th

Table 4 Experimental frequencies of Beam 1. Modes

1st 2nd 3rd 4th 5th

Healthy

26.01 159.95 430.62 772.15 1173.86

Healthy

25.97 159.05 428.16 767.48 1164.21

Cut depth

Cut depth 2.5 mm

5.0 mm

8.5 mm

120 mm

160 mm

26.00 154.01 424.92 767.41 1152.87

26.00 149.85 420.82 759.35 1142.93

25.75 142.04 413.95 753.73 1140.08

25.27 128.93 402.93 738.31 1129.46

23.69 102.15 385.89 701.72 1124.35

only obtained in the translational DOFs of the model, which is quite similar to a real condition. The three objective functions (Eqs. (3), (6), and (8)) presented in Section 2 are constructed from the dynamic properties determined in the theoretical modal analysis. Thus, there are 25 updating parameters, which represent each element of the beam.

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Fig. 10. Right side of a beam element.

Fig. 9. First 5 mode shapes of experimental healthy condition for Beam 1.

The updating parameters were estimated through the HS algorithm. After some iteration, the procedure is converged with excellent detection of damaged location and severity, as shown in Fig. 3. It is clearly seen that the detection of damage on element 20 is exact for the three cost functions with no errors on other elements. An important aspect in the development of any damage detection procedure is its sensitivity to uncertainty in the measurements. Experimental modal testing is always associated with some kinds of measurement noise or imprecision. Therefore, it seems to be not safe to test a SHM proposed approach by adopting the numerical eigenvectors and eigenvalues. Furthermore, to take the advantage of ambient vibrations the stochastic system identification field becomes fundamental, since its main goal is to determine modal parameters for output only measurement conditions. Thus, numerically simulated dynamic tests were conducted through the consideration of two standard excitations: an impulsive and an ambient excitation. The latter was modeled by 75 uncorrelated Gaussian white noise signals (generated with Matlab), with zero mean and standard deviation equal to one, applied at all the generalized coordinates of the structure. This representation seems adequate to simulate a broad band, ambient excitation of the structure, as suggested in several experimental studies (Brownjohn, Lee, & Cheong, 1999; Peeters and de Roeck, 2001). The impulsive loading is represented by the application of an impact at node 8 in the vertical direction. Three different damage scenarios are considered for both excitations, in order to verify the capability of the localization and quantification of the proposed SHM procedure. The elastic modulus is reduced 20% and 70% for element 20% and 30% for element 8, which are defined as damage Scenarios 1–3, respectively. Then, the structure is numerically modeled using the FE Matlab code, considering the different excitations and damage scenarios. The dynamic problem is solved by numerical integration of the equations of motion using the Newmark method, with an integration time step equal to 105 s for the free vibration case and 106 s for ambient vibrations. Damping of the structure is assumed as

proportional to the mass and stiffness matrices. The proportionality constants were determined to yield damping ratios in the 1st mode equal to n = 1%. Seeking to improve the identification, the output data are filtered with an eight-order Chebyshev type I lowpass filter with a cutoff frequency of 1600 Hz and the data is re-sampled at a rate of 4000 Hz, in order to reduce the number of data points and to make the procedure more accurate in the range of frequency of interest. In order to determine the variation of structural modal parameters due to noise effects and to evaluate the robustness of the damage detection procedure in situations closer to field conditions, noise levels were simulated through the addition of white noise signals with RMS amplitude of 3% and 5% of the mean measured response for free vibration and 3% for ambient vibration. In real dynamic testing, this is consistent with the assumption of uncorrelation between the primarily electronic noises with the actual measurement signal. In many papers in the technical literature, noise proportional to the signal is assumed, which may grossly misrepresent the effect, by eliminating noise in channels in which the measurement signal is weak, for example, those corresponding to transducers close to modal nodes or to fixed supports. The damage Scenario 3 is not considered for free vibration and 5% noise level.

Table 7 Bending stiffness reduction. Cut depth (mm)

2.5 3.5 5 8.5 12 16

Bending stiffness reduction (%) Uniform reduction

FEM analysis

60.205 64.764 70.970 82.678 90.896 96.657

42.924 52.61 64.368 79.937 88.572 93.955

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(b) 1

0.9

0.9

0.8

0.8

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0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

Stiffness Reduction Factor (α)

(a) 1

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Element

Element Fig. 11. Beam 1: (a) 25 mm and (b) 50 mm.

(b)1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

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0.3

0.3

0.2

0.2

0.1

0.1

Stiffness Reduction Factor (α)

(a) 1

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Element

Element Fig. 12. Beam 1: (a) 85 mm and (b) 120 mm.

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Element Fig. 13. Beam 1: 160 mm.

After getting the output for each excitation, damage scenario and noise level, the output-only system identification is carried out using the Stochastic Subspace Identification method (SSI) and it is assumed that only the first 5 modes are available and measurements are only obtained in the translational DOFs of the model. The SSI algorithm presents the main advantage of avoiding any preprocessing to obtain spectra or covariances, identifying models directly from time signals. As earlier described by Fadel Miguel et al. (2007) the performance of the two different algorithms and three different variants is quite similar, thus a combination of the second algorithm together with the variant PC was chosen to carry

out the identification approach for the two damage scenarios. The identified frequencies for both damage scenarios, both excitations and two noise levels are shown in Tables 2 and 3. After the identification, the structure is updated again in order to assess the changes caused by damage. Since the three cost functions presented quite similar behavior and considering that Eq. (3) presents clear experimental advantages, avoiding mode shape expansion or its mass normalization, the HS optimization procedure is carried out only through this objective function. The identified damage after HS updating for both excitations and noise levels are shown in Figs. 4–7. As can be seen from the figures, the damage locations and severities may be accurately identified for both excitations and noise levels regardless of the damage position in the beam. Even for small damage of 20% in Scenario 1, the damage location and severity can also be detected out. Some negligible values of damage have appeared on the undamaged elements, which did not disturb the diagnosis since they are in the neighborhood of the true damage element.

5. Experimental program A series of three cantilever aluminum beams (Fig. 8) with the same physical and geometrical characteristics of the structure assessed in the numerical examples were constructed in order to experimentally evaluate the HS based damage procedure. Damage was introduced in element 15 of Beam 1, element 20 of Beam 2 and element 8 of Beam 3, by means of a cut at the midsection of the

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Element Fig. 14. Beam 2: (a) 35 mm and (b) 50 mm.

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Element Fig. 15. Beam 2: (a) 85 mm and (b) 120 mm.

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Element Fig. 16. Beam 2: 160 mm.

elements, with varying depth (2.5 mm, 3.5 mm, 5mm, 8.5 mm and 16 mm). Acceleration time-histories were vertically measured at every 0.3m of the length of the beams with accelerometers, which are in accordance with all the vertical DOFs adopted in the numerical model of the structure. Since no rotational and longitudinal DOFs were measured, a total of 25 responses were recorded. The first five vertical frequencies and vibration modes were determined as the average of three measurements in all beams before damage, and, after the damage level was reached, in impact induced free vibration tests. Because of the low amplitude of the

response, it was assumed that there was no damage built up during tests at lower damage levels. The experimental frequencies for three damage scenarios are shown in Tables 4–6. For the clearest visualization, only the 5 mode shapes for the healthy condition of Beam 1 is shown in Fig. 9. It is important to point out that, since the damage is introduced by means of a cut at the midsection of an element, the simple numerical representation of the damage level through a uniform reduction in its moment of inertia would lead to a lower evaluation of its stiffness. Thus, the moment of inertia of the midsection of the element, i.e., the section with the cut, does not represent the bending stiffness of the damaged element of the structure. Within this context, the reduction of the bending stiffness of an element due to the increase in the intensity of the damage is evaluated through a FE plane stress model. The analysis consists in the application of a bending moment in the extremity of a beam element, without damage and with several damage levels. Thus, the curvature of the longitudinal axis of the structure may be determined and its variation indicates the decrease of the bending stiffness. Aiming to take advantage of the beam element symmetry, only its right side is represented in the model, which comprises 375 8node quadrilateral plane stress elements, as shown in Fig. 10. Since the beams have a hollow cross section, the thickness of the top and bottom horizontal lines of the mesh is taken as 25.4 mm, while all the other elements have this value equals to 2 mm. The bending moment around the z axis (Mz) can be represented by the application of axial forces in the nodes of free end of the element. These forces lead to a straight line deformation of the element extremity, i.e., the cross section remains plane after the deformation.

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Element Fig. 17. Beam 3: (a) 25 mm and (b) 50 mm.

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Element Fig. 18. Beam 3: (a) 85 mm and (b) 120 mm.

important values of damage have also appeared in the neighborhood elements, which does not interfere the diagnosis. Concerning the severity, in the same way as for Beam 1, it can be said that the method was able to quantify the damage. Again, for small levels some differences were found with the true values that disappeared with the increasing of the cut depth. The damage scenarios in Beam 3 can be analyzed in Figs. 17–19. The damage location and quantification can be accurately identified for small cut levels. For the deeper cuts, the HS algorithm has localized element 7 as the potential candidate. This cannot be considered a problem, since it is adjacent to the true damaged

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The curvature determined through the FEM approach differs in 0.0027% of the theoretical value, indicating that its model accurately represents the element without damage of the beam. Then, several damage levels are introduced by releasing the restrictions of the nodal displacements in the x direction, which were imposed by the supports that represent the symmetry of the model. The number of eliminated supports, from top to bottom, is proportional to the depth of the cut in the damaged element. Since the applied moment is known, the nodal displacements for the damage condition can be determined, which leads to a different curvature and, then, the bending stiffness may be assessed. It is important to point out that, for the damaged condition, the cross section does not remain plane after the deformation. In this case, the curvature can be appraised through the closer straight line of the deformed configuration of the element. The bending stiffness reduction of the element with several damage intensities, or cut depths, in relation to the element without damage, is presented in Table 7. As can be seen in Figs. 11–13 the damage locations may be accurately identified for the Beam 1 even for small cut levels. Some important values of damage also appeared in the neighborhood elements, which do not disturb the diagnosis. Concerning the severity, it can be also said that the method was able to correctly quantify the damage. For small levels, some differences were found when compared with the real values that disappeared with increasing cut depth. Figs. 14–16 show the damage identification of Beam 2. For the smallest level of damage, the damage was not correctly located; however, clearly, the free end of the beam could be identified as a potential damage region. For the other damage levels, some

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Element Fig. 19. Beam 3: 160 mm.

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element. Some wrong values of damage also appeared in two or three other elements along the beam. 6. Conclusions In this paper, a new approach for vibration-based (SHM) procedures was presented in an ambient vibration context. This method combines Stochastic System Identification and the evolutionary harmony search algorithm, for FE model updating. The proposed procedure was tested with a series of numerical examples with different damage scenarios and noise levels under impact and ambient vibration. Then, an experimental study of three cantilever beams with several different damage scenarios was conducted in order to assess the proposed methodology. The identified damage pattern (location and severity) in all situations was excellent and it has been shown that the behavior of the proposed algorithm on noise condition is very promising. Recent damage detection studies on beams (Gomes & Silva, 2008) based on Genetic Algorithms (GA) pointed out that it cannot correctly converge for the actual damage extent, even using the theoretical eigenvalues, indicating an advantage of the proposed approach. Thus, this approach may be very attractive for on-line or realtime damage diagnosis of structures in the framework of structural health monitoring. Acknowledgments The authors gratefully acknowledge the financial support from CNPq and CAPES. References Au, F. T. K., Cheng, Y. S., Tham, L. G., & Bai, Z. (2003). Structural damage detection based on a micro genetic algorithm using incomplete and noisy modal test data. Journal of Sound and Vibration, 259(5), 1081–1094. Barai, S. V., & Pandey, P. C. (2000). Integration of damage assessment paradigms of steel bridges on a blackboard architecture. Expert Systems with Applications, 19, 193–207. Bernal, D., & Gunes, B. (2002). Flexibility based approach for damage characterization: Benchmark application. Journal of Engineering Mechanics, 130(1), 61–70. Brownjohn, J. M. W., Lee., J., & Cheong, B. (1999). Dynamic performance of a curved cable-stayed bridge. Engineering Structures, 21, 1015–1027. Chen, B., & Zang, C. (2011). A hybrid immune model for unsupervised structural damage pattern recognition. Expert Systems with Applications, 38, 1650–1658. Chou, J. H., & Ghaboussi, J. (2001). Genetic algorithm in structural damage detection. International Journal of Computers and Structures, 79, 1335–1353. Curadelli, R., Riera, J., Ambrosini, D., & Amani, M. (2008). Damage detection by means of structural damping identification. Engineering Structures, 30(12), 3497–3504. Degertekin, S. O. (2008). Harmony search algorithm for optimum design of steel frame structures: A comparative study with other optimization methods. Structural Engineering and Mechanics, 29(4), 391–410.

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