a novel artificial bee colony algorithm for structural

Research ArticleA Novel Artificial Bee Colony Algorithm for StructuralDamage Detection

Yinghao Zhao Quansheng Yan Zheng Yang Xiaolin Yu and Buyu Jia

School of Civil Engineering and Transportation South China University of Technology Guangzhou China

Correspondence should be addressed to Buyu Jia ctjbyscuteducn

Received 3 July 2019 Revised 13 December 2019 Accepted 27 December 2019 Published 17 January 2020

Academic Editor Hossein Moayedi

Copyright copy 2020 Yinghao Zhao et al+is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel artificial bee colony (ABC) algorithm to detect structural damage via modal and frequency analyses is proposed (named asTCABC algorithm) Compared to the standard ABC algorithm tabu search method and chaotic search method are adopted in theproposed algorithm to enhance the exploration and exploitation ability +e tabu search method uses a memory function to avoidthe solution being trapped in a local minimum which increases the exploitation ability Chaotic search method generates moresearching points for finding the global minimum which increases the exploration ability Additionally the first roulette wheelselection is replaced by the tournament selection to enhance the global searching ability of the TCABC algorithm Several explicittest functions and an implicit damage detection function are employed to check the numerical results obtained from ABC andTCABC algorithms Afterward the damage detection accuracy of the TCABC algorithm is verified under different circumstancesand several recommendations are given for using the TCABC algorithm to detect structural damages under actual conditionsFinally an experimental study is applied to examine the performance of TCABC algorithm for damage detection+e results showthe following (1) compared to traditional ABC algorithm TCABC algorithm performs better (2) fewer groups lead to fasterconvergence as demonstrated by both algorithms used in the same damage situation (3) TCABC algorithm can infer the locationsand extents of the damage when the groupings are inaccurate (4) the accuracy of the field test data profoundly affects the precisionof the damage detection results In other words stronger noises result in worse identification results (5) whether or not the noisesexist the more data are measured the more accurate the results can be achieved (6) the TCABC algorithm can efficiently detectstructural damage in the experimental study

1 Introduction

Early-stage damage detection is one of the main tasks instructural health monitoring [1] and this task is accom-plished by fundamental damage detection method andparameter identification Once the structural parameters areidentified or the structural uncertainties are reduced thecorresponding structural numerical model can be updatedand used to predict relevant structural responses underpossible catastrophic events such as earthquake and hur-ricane [2 3] Besides with the detected damages structuralrepair cost and maintenance parameters can be dynamicallyevaluated [4] However traditional local damage detectionmethods such as the visual detection method and x-raymethod require specific prior knowledge which is difficultto obtain Hence global damage detection methods based on

physical quantities such as natural frequencies [5 6] modalshapes [7] accelerations [8 9] static displacements andstrain responses [10 11] become popular

For these global damage detection methods an idea toobtain structural damages is to transfer the damage detec-tion problem to the finite element (FE) model updatingproblem the theoretical basis of the updating is that innumerical calculation a real structure with infinite degreesof freedom can be discretized into the FE model +estructural vibration properties (ie frequencies and modalshapes) vary with the changes in physical parameters (iestiffness and mass) of the FEmodel+us obtaining physicalparameters via vibration properties is an inverse problem ofmodal and frequency analyses or a pattern-recognitionproblem [12] Onemain difficulty of this pattern-recognitionproblem lies in a large number of unknowns To simplify the

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 3743089 21 pageshttpsdoiorg10115520203743089

problem Moslem and Nafaspour [13] assumed that anychange in the properties of each element due to the damageis equivalent to a change in its elastic modulus By thissimplification the FE model updating-based damage de-tection problem can be formulated as an optimizationproblem with a simple-structured implicit objective func-tion For example the norm of the residuals between thenumerical results of the FE model and the test data of theactual structure is defined as the objective function Once theobjective function reaches the global minimum the inputparameters of the FE model will be treated as the physicalparameters of the actual structure

+e structural damage detection problem is transferredto find the extreme values of the corresponding objectivefunction However these extrema cannot be seen by thetraditional method because the objective function isexpressed by implicit functions To overcome this problemseveral metaheuristic algorithms and artificial intelligence(AI) are used Perera and Torres [14] identified the damageof a simply supported beam by genetic algorithm (GA) andthese detection results are better than those calculated withthe method that Hu et al [15] proposed Park et al [16]updated bridge boundary conditions using artificial neuralnetworks (ANN) and verified the results by both laboratoryand field tests it was concluded that the ANN reduces theuncertainties of the boundary conditions in FE modelupdating Pathirage et al [12] introduced dimensionalityreduction and relationship learning in the ANN to optimizethe weights in neural networks the laboratory test indicatedthat the detection results of this method are better than theresults of the traditional ANN Qiao and Yang [17] intro-duced a quantum search dolphin swarm algorithm (QDSA)in finding the optimal solution of large-scale functions theyconfirmed that the QSDA has outstanding advantages thandolphin swarm algorithm (DSA) Kang et al [18] proposedan improved particle swarm optimization (PSO) algorithmto identify structural damages using vibration data and theresults showed that compared to differential evolution (DE)algorithm the improved PSO algorithm is more efficient indetermining the sites and the extents of structural damagesQiao et al [19] combined the Volterra adaptive filter and animproved whale optimization algorithm (WOA) to predictthe short-term natural gas consumption the study showedthat the proposed prediction model is better than someadvanced prediction models

Besides the abovementioned algorithms the artificial beecolony (ABC) algorithm also attracted the attention of manyscholars in the optimization area since it was proposed ABCalgorithm was suggested by Karaboga [20] and this algo-rithm is motivated by the intelligent behavior of honey beeswhen seeking a high-quality food source In 2009 Karabogaand Akay [21] first conducted a detailed and comprehensiveanalysis of ABC algorithm and tested it with 50 numericalbenchmark functions and other well-known evolutionaryalgorithms such as GA PSO DE and ant colony algorithm(ACO) the results indicate ABC algorithm has several ad-vantages in getting the extrema of the function than othermethods In the same year the influences of changing pa-rameters in the ABC algorithm were analyzed by Akay and

Karaboga [22] With the benefits such as simple structureeasy implementation and outstanding performance theABC algorithm is widely used in many fields Awan et al[23] applied the ABC algorithm to optimize the connectionweights of neurons in the ANN for electric load forecastingUzlu et al [24] combined ABC and ANN to predict theannual output of hydroelectric power in Turkey Menon andRamakrishnan [25] used the ABC algorithm to process braintumors segmentation in magnetic resonance images Wenet al [26] applied the ABC algorithm to achieve globaloptimization of controlled-source audio-frequency mag-netotelluric data However there are limited researchersutilizing the ABC algorithm in structural damage detectionCombining the ABC algorithm with the NelderndashMeadsimplex method Sun and Betti [27] identified the structuralparameters and the input forces of buildings even when thestructural responses are noise contaminated By applying thehybrid search strategy in the ABC algorithm for structuraldamage detection Ding et al [28] got better identificationresults compared with those from GA and traditional ABCalgorithms Nevertheless there are some flaws in Sun andBetti [27] and Ding et al [28] +e calculations of Sun andBetti [27] are purely numerical-based and lack of experi-mental verification +e data from the experiment are dif-ferent from those artificially assumed and used in thenumerical models So an experimental study should be usedas a verification after the numerical study Ding et al [28] didnot consider several factors that influence the target functionand corresponding identification results However theimplicit target function highly relies on the observed fre-quencies and modal shapes and these observed data areeasily affected by the test conditions such as grouping noiseeffect mode shape order and sensor location which shouldbe taken into account

Based on the fact that there are limited research studiesin ABC-based structural damage detection and the assumedidentification condition is relatively simple In this studyABC algorithm combined with the tabu search method [29]and chaotic search method [30] (named as TCABC algo-rithm) is used to detect structural damages under differentidentify conditions such as grouping noise effect modeshape order sensor location and is also evaluated by severalexplicit test functions Recommendations for realisticallyusing the TCABC algorithm in damage detection are alsoput forward Furthermore an experimental study is used toexamine the damage detection results of TCABC algorithm

+e rest of the paper is organized as follows Section 2describes how the structural damage detection problem istransferred to an optimization problem traditional ACBalgorithm and proposed TCABC algorithm are presented inSection 3 and their numerical results are compared inSection 4 also in Section 4 the accuracy of TCABC algo-rithm for damage detection is verified under different cir-cumstances (ie influences of grouping size noise effectmode shape order and sensor location) and several rec-ommendations are given for using TCABC algorithm todetect structural damages under actual conditions in Sec-tion 5 an experimental study is used to examine the per-formance of TCABC algorithm Section 6 is the conclusions

2 Advances in Civil Engineering

2 Theoretical Background

21 Parameterization of Damages When a real structure isdiscretized into an N-DOF system the modal characteristicscan be obtained by

K minus ω2j middot M1113960 1113961 middot Φj1113966 1113967 0 j 1 2 N (1)

where M and K are the NtimesN global mass and stiffnessmatrices of the structure respectively ωj is the jth eigen-value and Φj is the corresponding modal vector

For actual structures the M (or K) for the damaged andthe undamaged ones are different However in damagedetection problems the mass matrix is assumed constantbecause the internal structural damage usually does notresult in losses of materials and the stiffness of the ith el-ement is expressed as follows [13]

Ki f Ai Ii Ji ti Li Ei Gi( 1113857 (2)

in which Ai Ii and Ji are cross-sectional area bending andpolar moments of the inertia of the ith element ti and Liare the element dimensions and Ei andGi are respectivelythe elastic and shear modulus To reduce the number ofinput parameters in the FE model Moslem and Nafaspour[13] mentioned that any change in the properties of eachelement due to the damage is equivalent to the change in itselastic modulus (Ei) According to the abovementionedstatements the stiffness of the ith element can be expressedas follows

Ki 1 minus ai( 1113857 middot Ei (3)

where ai is defined as the damage index ai 0 represents nodamage and ai 1 represents absolute damage of the ithelement

22 Implicit Objective Function +eoretically when theinput parameters of the FE model are exactly the same as thephysical parameters of the damaged structure for both theFEmodel and real structure their natural frequencies shouldbe the same and the value of modal assurance criterion(MAC) [31] should be equal to one So the objectivefunction can be set as a combination of the natural fre-quencies and the values ofMAC Perera and Torres [14] usedGA with two objective functions to detect the damage of abeam and the objective function with better performance isused in this paper +e used MAC is defined as follows

MAC ΦTj1113960 1113961 ΦF

j1113960 11139611113872 1113873 ΦT

j1113960 1113961t[W] ΦF

j1113960 11139611113874 11138752

ΦTj1113960 1113961

t[W] ΦT

j1113960 11139611113874 1113875 ΦFj1113960 1113961

t[W] ΦF

j1113960 11139611113874 1113875

(4)

whereΦTj andΦF

j are respectively the jth test and FE modalvectors t represents transpose and [W] is a diagonal squareweight matrix to ensure all test data are effectively used +ekth diagonal element of [W] is defined as follows

Wkk ΦTjk1113872 1113873

minus 1111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 (5)

where ΦTjk is the kth coordinate of the jth test modal vector

+e objective function is defined as follows

F 1 minus 1113945m

j1

MAC ΦTj1113960 1113961 ΦF

j1113960 11139611113872 1113873

1 + aj1113872 1113873 (6)

in which F and m respectively represent the objectivefunction and the number of the total measured orders and ajis a penalty function to account for differences between thetest and the FE model frequencies

aj fF

j1113872 11138732

minus fTj1113872 1113873

2

fFj1113872 1113873

2+ fT

j1113872 11138732

11138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868 (7)

fFj and fT

j are the jth frequencies of the FE model andthe test the factor aj accelerates the convergence of theobjective function as well

+eoretically when the global minimum of this implicitobjective function equals to zero the input parameters of theFE model equally represent the physical parameters of theactual structure

3 Algorithms for Damage Detection

31 StandardABCAlgorithm ABC algorithm is proposed byKaraboga [20] and the colony of artificial bees is categorizedinto three main groups employed bees onlooker bees andscout bees +e number of the employed bees and the on-looker bees is the same as the number of the food sources Insome cases an employed bee should be transformed into ascout bee Each bee can only exploit one food source eachtime +e employed bees first explore the food sourcesaccording to their memories and explore possible foodsources in corresponding neighborhoods and then they flyback to the hive and share the information of the foodsources by dancing After digesting the information theonlooker bees select the food sources one by one and becomeemployed bees (the food sources with more nectar have ahigher probability of being chosen) When a food source isexploited several times and no other better food sources arefound in the neighborhood the employed bee of this foodsource transforms into a scout bee +e scout bee randomlyselects a new food source and continues its work as anemployed bee +e main parts of ABC algorithm are asfollows [32]

Initialization the location of an initial food source or acandidate solution is a multidimensional parametervector +e parameters of this vector are generated bythe following equation

xij xminj + rand(0 1) times x

maxj minus x

minj1113872 1113873 (8)

where i and j are individually selected from [1 SN] and[1 D] SN is the number of the food sources D is thenumber of the dimensions xij is the jth dimension ofthe ith food source xmin

j and xmaxj are respectively the

lower and upper bound of the jth dimension and rand(0 1) is a random real number in the range [0 1]

Advances in Civil Engineering 3

Employed bee phase first each employed bee is as-sociated with a food source (xi) and then a new foodsource (vi) is produced+e parameters of the new foodsource are as follows

vij xij + rand(minus 1 1) times xij minus xkj1113872 1113873 (9)

where xkj represents the jth dimension of another foodsource (xk) selected randomly from the remainingpopulation +e employed bee compares the results ofthe two food sources (ie xi and vi) and brings back theinformation of the better one for the optimizationproblem of finding the global minimum the bettermeans a smaller objective value (the greedy selectionstrategy)Onlooker bee phase before the onlooker bees startworking the fitness values of the food sources broughtback by the employed bees are calculated as follows

fiti

11 + fi

fi ge 0

1 + abs fi( 1113857 fi lt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)

where fiti and fi are respectively the fitness value andthe objective value of the ith food source +e selectionprobability of the ith food source equals to pi (roulettewheel selection)

pi fiti

1113936SNm1fitm

(11)

After the selection the onlooker bees fly to the selectedfood sources one by one and generate correspondingnew food sources in a similar way as in equation (9)Again each generated food source is compared with thecorresponding previous food source via the greedyselection strategy+e updating process of the employed bees and theonlooker bees is the same except that the employedbees update all food sources while the onlooker beesonly update the selected food sourcesScout bee phase a food source may be visited severaltimes without a better food source founded nearby+is kind of food source is abandoned after a pre-defined number of visits +en the correspondingemployed bee transforms into a scout bee to randomlygenerate a food source via equation (8) and continuesits work as an employed bee +e schematic of the ABCalgorithm is shown in Figure 1

32 TCABC Algorithm In this paper two searchingmethods the tabu search method [29] and chaotic searchmethod [30] are adopted to enhance the exploration andexploitation ability of the traditional ABC algorithm (namedas TCABC algorithm) Additionally the original roulettewheel selection is replaced by the tournament selection toenhance the global searching ability of TCABC algorithm

321 Tabu SearchMethod +e tabu search a simulation ofthe human intelligence process is first proposed by Glover[29] as an extension of local neighborhood search As aglobal stepwise optimization algorithm it effectivelyavoids the solution falling into the local optima andprevents the algorithm from entering an infinite loop +eso-called tabu is to prevent exploiting the previous visitedand bad food sources By using a tabu list to record thesevisited food sources the repetition of the previous workcan be prohibited and the total exploitation ability isimproved +e tabu search method avoids roundaboutsearch by introducing a flexible storage structure andcorresponding tabu criteria Simultaneously some pro-hibited good food sources are ldquounlockedrdquo by the aspirationcriterion

In this paper the tabu search is applied throughout thewhole process of the TCABC algorithm and two tabu lists(ie T1 and T2) are introduced into the algorithm T1 is atabu list with a limited length which stores the recentlyvisited food sources By applying the T1 list other beescannot search recently visited food sources In detail the

Parameter initialization

Generate initial population

Employed bee phase

Roulette wheel selection

Onlooker bee phase

Abandon the food source

Generate new iterative population

Scout bee phase

Record the best solution

Complete iteration

Yes

No

Finish

Yes

No

Start

Figure 1 Schematic of the ABC algorithm


length of T1 is assumed as n When the n+ 1 food source isrecorded in T1 the first food source in T1 is taken out T2saves the deserted food sources (ie the food sources wherethe scout bees are generated) those food sources areexploited several times and no improvement is found whichmeans their qualities are not good Unlike T1 which onlyban finite food sources within a limited time those aban-doned food sources in T2 are banned eternally and thelength of the T2 list is infinite

322 2e Tournament Selection Strategy In the standardABC algorithm the roulette wheel selection is used to decidethe probability of each food source selected by the onlookerbees However some food sources may have immense fitnessvalues According to the roulette wheel selection in equation(11) many onlooker bees may fly to these food sourceswhich reduces the global searching ability of the algorithmTo avoid this flaw the tournament selection is employed toreplace the original roulette wheel selection in TCABCalgorithm

In the tournament selection strategy the fitness valuesof every two food sources are compared +e one with abigger fitness value gains 1 point and the other gains 0point and each food source is also compared to itself (toguarantee the worst food source gains 1 point) After thecomparison each food source (xi) has a correspondingscore (si) and the possibility of food source xi being se-lected equals

pi si

1113936SNm1sm

(12)

where SN means the number of food sources +is strategyguarantees that those food sources with significant fitnessvalues do not overly affect the selection tendency +ereforethe onlooker bees will not fly to the same food source indroves

323 Chaotic Search Method Mathematically logisticmapping is a straightforward chaotic mapping with severalcharacteristics such as uncertainty ergodicity and regu-larity +e logistic mapping is expressed as follows [30]

βn+1 μβn 1 minus βn( 1113857 (13)

Equation (13) is chaotic when μ 4 and 0le β0le1 Figure 2displays the calculation results when μ 4 and β0 027As can be seen the initial information is lost after severaltimes of iterations and the output is like a stochasticoutput

In this paper a chaotic search method is applied duringthe onlooker beesrsquo phase and the scout beesrsquo phase +eoperation steps are as follows (1) capturing the new foodsource generated by the onlooker bee (or capture theabandoned food source) (2) mapping this food source into auniform space (3) applying the chaotic search method to theuniform space-based food source several times record allresults (4) transferring all these results back to the normal

space (5) calculating the fitness values of all these foodsources and bringing back the information of the best one

Compared to the standard ABC algorithm the chaoticsearch method allows each bee fly to several food sourcessimultaneously by doing so more food sources are visitedat the same time and the population diversity of the al-gorithm is improved OpenSees platform [33] a widelyused software in civil engineering is also used to establishthe numerical model of the studied structure in this paper+e schematic of the TCABC algorithm is shown inFigure 3

4 Numerical Study

41Descriptionof theBenchmarkBeam WithMATLAB [34]and Opensees [33] the algorithmrsquos main procedure andcorresponding FEmodel are established and the accuracy ofthe TCABC algorithm in structural damage detection isvalidated and verified

In this paper a twenty-four-meter steel beam with twofixed ends shown in Figure 4 is modeled as the benchmarkcase +e density and the elastic modulus of steel are set as7850 kgm3 and 206GPa respectively +is beam is dis-cretized into a twenty-four-element structure every twoconnected elements are assigned as a group and share thesame elastic modulus

42 Comparison of Standard ABC and TCABC Algorithms

421 Comparison by Explicit Test Functions Before directlyapplying the TCABC algorithm to the damage detectionproblem several explicit test functions are used to com-pare the performance between ABC and TCABC algo-rithms By comparing the calculation results differencesbetween the two algorithms can be obviously observedand the generality and accuracy of the TCABC algorithmcan also be verified +e used explicit test functions arelisted in Table 1

To purely compare the differences between ABC andTCABC algorithms and to simplify the calculation

1510 200 5Iteration times

0

02

04

06

08

1

Itera

tive r

esul

ts

Figure 2 +e output of equation (13) when μ 4 and β0 027


Parameterinitialization

Generate initial population Start

Update tabu list T1 Is the bee in the list T1 or T2

Update tabu list T1

Employed bee phase

Yes

No

TournamentselectionOnlooker bee phase

Is the bee in the list T1 or T2

Yes

Chaotic searchmechanism

No Opensees parallelanalysis

Update tabu list T1Abandon the food sourceUpdate tabu list T2

Scout bee phase Chaotic search mechanism

Opensees parallel analysis


Update tabu list T1Generate new iterative population

No

Record the best solution Complete iteration Finish

Yes

Yes

No

Yes

No

Figure 3 Schematic of the TCABC algorithm

141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

02m 04m

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12

Figure 4 +e 24m benchmark beam

Table 1 Used explicit test functions

No Test functions Searching range Global minimum

F1 f1(x) 1113936Di1x

2i (minus 10 10)D 0

F2 f2(x) 1113936Di1(10

6)(iminus 1)(Dminus 1)x2i (minus 10 10)D 0

F3 f3(x) 1113936Di1ix

2i (minus 10 10)D 0

F4 f4(x) 1113936Di1|xi| + 1113937

Di1|xi| (minus 10 10)D 0

F5 f5(x) (14000)1113936Di1x

2i minus 1113937

Di1cos(xi

i

radic) + 1 (minus 10 10)D 0

F6 f6(x) 1113936Di1ix

4i (minus 10 10)D 0


procedure corresponding input parameters of the two al-gorithms in the six test functions are set as the same +ecalculation generations are set as 500 and each generationcontains 100 food sources the abandoned limitation of thefood sources is set as 1800 which is the same as 06times SNtimesDin [21] Besides the dimensions (D) of all six test functionsare set to 30 After trial calculations (which is introducedlater) two more parameters (ie the length of T1 and thetimes for chaotic search) of the TCABC algorithm areseparately set to 100 and 30 Each test function is calculated20 times and the corresponding results are displayed inTable 2 It can be observed that except for the worst solutionand the standard deviation of F5 the TCABC algorithmperforms better than the ABC algorithm

To study how the generation number the length of T1and the times of chaotic search influence the performanceof TCABC algorithm a single variable method is appliedand corresponding results are compared +e benchmarkparameters and the calculation times are set the sameas previously defined Different generation numbersare analyzed (ie conduct the analysis every 50 from 200to 700 generations) and the results are shown in Table 3and Figure 5 +e growth rate in Figure 5 is defined asfollows

Growth rate Vn minus Vp

Vp

(14)

where Vn and Vp respectively represent the value of presentand previous studied generations

In Figure 5 the growth rates of both mean and standarddeviation are always negative whichmeans by increasing thegenerations the results are closer to the global minimum

Similar to the generation number different chaoticsearch times and different lengths of T1 are studied Figures 6and 7 display the relevant mean results of the targetfunctions All results are normalized by the identified valuewhen the times of chaotic search (or the length of T1)equals zero (ie the dashed line) As can be observedexcept for 2 results in Figure 7 all other results are belowthe dashed line which means by introducing the tabusearch method and chaotic search method the algorithmperforms better However the two parameters show dif-ferently to the result of each target function trial analysesare recommended to find appropriate T1 and chaoticsearch times when using this method

422 Comparison by the Implicit Objective Function+e comparison between ABC and TCABC algorithms forthe implicit damage detection function (ie equation (6))is studied in this section Similar as in Section 421 theinput parameters of the two algorithms are set as the sameand after trial analyses the used parameters are listed inTable 4 It is noteworthy that to damage detection prob-lems the recognition accuracy is not required as high as inpreviously studied test functions +e reason is that due tothe limitation of the existed experimental devices thoseerrors less than one percent are unrecognizable So theldquoextremely preciserdquo to the identified parameters are

meaningless (eg for the benchmark beam the elasticmodulus of group one that equals from 10003 E0 to 10007E0 may result in very close vibration properties and thesevalues cannot be distinguished under test conditions dueto the instrument accuracy) Nevertheless evolution-based algorithms intend to achieve the global minimumand find input parameters as accurately as possible Toachieve a balance between them the identified parametersare rounded to five decimal places in the damage detectionproblem

For the benchmark case in Section 41 the first tenorders of modal vectors and frequencies are used for theobjective function (ie equation (6)) the modal vectorsand corresponding MACs are captured by all nodal dis-placements (ie 23 nodes here) +e assumed damagelevel of this beam is listed in Table 5 and the meaning ofdamage index is described in equation (3) Negative valueshere for the assumed damages are meaningful because foractual tests before detecting the damages the structureparameters should be modified first and the range of themodification is beyond the boundary (ie 0 and 1) of justconsidering the damage To verify the applicability of theproposed algorithm under the implicit damage detectionfunction the searching space is changed and the negativedamage indexes are used By combining the damage in-dexes (Table 5) the lower bound and the upper boundof the indexesrsquo searching space are set as minus 05 and 05respectively

+e convergence performance of the implicit objectivefunction is shown in Figure 8(a) In general the results ofthe TCABC algorithm are much closer to zero comparedto the results of the standard ABC algorithm which meansthe TCABC algorithm has a better performanceFigure 8(b) displays the identified parameters the y-axisrepresents the corresponding damage index ai +e resultsindicate that to damage detection problems here bothalgorithms work well but the TCABC algorithm is moreaccurate

43 Consideration of the Influences of Different GroupingTechniques For actual damage detection problems how toset the benchmark groups may influence the accuracy of theidentification results +is section investigates the precisionof the TCABC algorithm through two cases with differentgrouping techniques

431 2e Influences of Different Grouping SizesGrouping sizes may alter even in the same damage situationFor example as shown in Figure 9 the elastic modulus ofelement 9 to 12 are reduced to 065 of the undamaged ones(the benchmark beam in Section 41 is used here and E0 isthe elastic modulus of the intact beam) and three types ofgrouping are used to detect the damage in this section thebeam is respectively divided into 6 12 and 24 groups inFigures 9(a)ndash9(c) Same as in Section 41 the elements ineach group share the same elastic modulus the input pa-rameters and the searching space are set the same as inSection 422


+e numerical results are shown in Figure 10Figure 10(a) indicates that (1) in the same damage situationboth ABC and TCABC algorithms converge faster withfewer groupings (2) it is demonstrated again that theTCABC algorithm performs better than the ABC algorithmfor the implicit damage detection function Because thefocus here is the influences of the grouping sizesFigure 10(b) only displays the results of the TCABC algo-rithm under different scales of grouping it can be observedthat even though there are some noises the damages aredetected by all three dimensions of grouping

4322e Influences of Inappropriate Groupings In previousstudies the assumed damage groups are set exactly the same

as the actual damage groups However it is complicated tolocate the damage position in actual tests precisely In thissection three scenarios are studied to demonstrate whetherthe TCABC algorithm is able to detect structural damages ininappropriate groupings As shown in Figure 11 thebenchmark beam is divided into 12 groups Figures 11(a)ndash11(c) represent damage scenario 1 to 3 individually

In Figure 11(a) the stiffness damage index of element 9 is035 and the position of this damaged element is in group 5(G5) +is sort of damage is named as ldquoin-grouprdquo damageScenario 1 is used to check the detection results of theTCABC algorithm in this ldquoin-grouprdquo damage situation InFigure 11(b) both elements 8 and 9 are damaged and theyare exactly across the boundary of G4 and G5 Damage likethis is named as ldquocross-grouprdquo damage +e performance of

Table 2 Performance comparison between ABC and TCABC algorithms

NoBest solution Worst solution Mean Standard deviation

ABC TCABC ABC TCABC ABC TCABC ABC TCABCF1 0017068 0005334 0037956 0024644 0024267 0013655 0007486 0002975F2 3554567 1468162 7301145 3746693 535974 2534457 1228749 0774593F3 0283409 0124682 0382502 0201409 0325852 0147296 0030473 0022702F4 0 0 0 0 0 0 0 0F5 0006508 0005878 0067564 0136364 0042717 0035809 0019248 0044153F6 8134684 2758961 4050607 1224841 1882364 6891574 1049996 2536104

Table 3 Performance of TCABC with different generation numbers

Calculation generation Results based on 20 calculations F1 F2 F3 F4 F5 F6

200 Mean 3659166 8445289 3646164 0087539 0511423 8752596Standard deviation 0520355 1523502 7336005 0001679 0088312 3677924


ndash100

ndash80

ndash60

ndash40

ndash20

0

Gro

wth

rate

()

700350 400 450 500 550 600 650300250

Mean

Generation number

F1F2F3

F4F5F6

(a)

ndash100

ndash80

ndash60

ndash40

ndash20

0G

row

th ra

te (

)

300 350 400 450 500 550 600 650 700250Generation number

Standard deviation

F1F2F3

F4F5F6

(b)

Figure 5 +e growth rate of the TCABC algorithm with different generations for (a) mean and (b) standard deviation


the TCABC algorithm in this ldquocross-grouprdquo damage situa-tion is studied in scenario 2 Multidamage condition (ieshown in Figure 11(c)) which combines ldquoin-grouprdquo damageand ldquocross-grouprdquo damage is reviewed in scenario 3

After trial calculations and setting reasonable inputparameters the detection results of scenario 1 to 3 areseparately shown in Figures 12(a)ndash12(c) Due to the inap-propriate groupings the accuracy of the proposed algorithmcannot be guaranteed However by checking the detectionresults the detected damaged positions (enclosed in thoseblack dash boxed) contain or near the real damage positionswhich means even though the groupings are inaccurateTCABC algorithm still contributes to locating the damageunder all three circumstances as well as under actual testconditions Compared to Figures 8(b) and 10(b) the damagedetection results in Figure 12 have a lower damage index+e reason is that due to the inappropriate groupings thetheoretical global minimum (ie zero) of the objectivefunction cannot be achieved So a local minimum is foundinstead +e local minimum is obtained by adjusting thedamage indexes of all elements to let the value of equation(6) be as small as possible which causes the damage indexesof the undamaged and damaged elements respectivelyincreased and reduced

44 Consideration of the Influences of the Field Test DataBesides grouping size the accuracy of the damage detectionresults is affected by the field test data as well It is easy toobtain all the vibration properties by numerical analysesHowever there are several obstacles to gain modal andfrequencies data in the field test Firstly the values of nodaldisplacements and MACs are highly influenced by thesensors and usually only limited numbers of sensors areused secondly unlike numerical analyses only first feworders of frequencies and modal vectors can be obtained inthe field test which controls the m value in equation (6)thirdly environmental electromagnetic and instrumentalnoises are everywhere during the measurement and thenoise level is highly uncertain

Subjectively the more data are measured the moreaccurate the results will be however this is not confidentwhen these noises are introduced by increasing the amountof measured data the amount of obtained noises also in-creases So a quantitative analysis is needed +is sectionfocuses on how the measured data influence the detectionresults

441 Measuring Noises In an experimental study mea-sured data are all contaminated by noises +erefore whendifferent kinds of measured data are used to account for theinfluences on the damage detection results the artificialnoises should simultaneously be included +e randomnoises are applied to the theoretical frequencies and modalvectors by equations (15) and (16) separately

fNi fi 1 + η1φ

Ni1113872 1113873 (15)

where fNi and fi separately represent the noise-contami-

nated and noise-free ith frequency η1 is a random real valuein the interval [minus 1 1] and φN

i represents the noise level of ithfrequency

F1F2F3

F4F5F6

10 20 30 40 50 600Times of chaotic search

03

04

05

06

07

08

09

1

11Re

lativ

e ide

ntifi

ed v

alue

Figure 6 Influences of the times of chaotic search for TCABCalgorithm

F1F2F3

F4F5F6

03

04

05

06

07

08

09

1

11

Rela

tive i

dent

ified

val

ue

1755025 100 125 150 2000 75Length of T1

Figure 7 Influences of the length of T1 for TCABC algorithm

Table 4 Input parameters of the damage detection

Input parameters ABC TCABCCalculation generations 200 200Number of food sources 60 60Abandoned limitations 360 360Length of T1 mdash 100Times of chaotic search mdash 15


ΦNij Φij 1 + η2ξ

Nij1113872 1113873 (16)

where ΦNij and Φij separately represent the modal dis-

placement of the noise-contaminated and noise-free jthposition of ith modal vector η2 is a different random realvalue in the same interval as η1 and ξN

ij represents thecorresponding noise level

During the field test the noise level may be influencedby a lot of factors (eg test condition the precision ofthe instrument and testersrsquo skill) To verify the robustness ofthe proposed algorithm three levels of noise listed in Table 6are studied because the frequency measurements are morestable its noise levels are always smaller than those obtainedby modal displacements

Table 5 Assumed damage level of the elastic modulus of the benchmark beamGroup number G1 () G2 () G3 () G4 () G5 () G6 () G7 () G8 () G9 () G10 () G11 () G12 ()Damage index minus 30 0 10 minus 14 minus 42 0 25 minus 26 30 0 0 34

ABCTCABC

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue

50 100 150 2000Generation number

(a)

Assumed damageABCTCABC

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number

(b)

Figure 8 (a) +e convergence performances of the objective function (b) +e identified parameters of the damage detection

141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

E = 065E0

G1 G2 G3 G4 G5 G6

24m

(a)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12

(b)

G8

(c)

G1 G7 G24

Figure 9 +e same damage situation with different sizes of grouping


442 2e Total Orders of the Measured Modal Shapes andFrequencies +e total orders of the measured frequenciesand modal shapes influence the format of the objectivefunction and the accuracy of the proposed algorithm underdifferent objective functions may be different In order tocompare the effects of different numbers of measured ordersthe first three six and ten orders of modal shapes andfrequencies are used separately to check the damage de-tection results of the benchmark case in Section 422 (iedamage described in Table 5) the noises mentioned inSection 441 are simultaneously added It is noteworthy thatthe modal displacements here are captured by all existednodes (ie 25 nodes in this case) and the influences of thenumber of the measured nodes are introduced laterFigures 13(a)ndash13(c) show the damage detection results of

only using the first three six and ten orders of modal shapesand frequencies

It can be found that (1) when there is no noise thedetection results are not influenced by the number of themeasured orders +is can be seen from the similar height ofthe assumed damage bars and the non-noise bars in all threecases as shown in Figure 13 (2) when noises are includedthe damage detection results become better as the orders ofmodal vectors and frequencies increase+is can be obtainedby comparing the LV3-noise bars of Figures 13(a) and 13(c)In Figure 13(a) the LV3-noise bars are far different from theassumed-damage bars to some groups such as group 2 3 10and 11 but in Figure 13(c) their differences are not thatmuch (3) when noises are included the stronger the noisesare the worse the identification results become +is

ABC-6 groupsABC-12 groupsABC-24 groups

TCABC-6 groupsTCABC-12 groupsTCABC-24 groups

10ndash4

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)

Assumed damageTCABC-6 groups

TCABC-12 groupsTCABC-24 groups

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

179 19 236 168 1810 15133 201211742 14 21 22 241 5Element number

(b)

Figure 10 (a) +e convergence performance under different sizes of grouping (b) +e identified parameters under different sizes ofgrouping


phenomenon is easily observed by checking the LV1-noisebars (which represent the lowest noise level) and the LV3-noise bars (which serve the highest noise level) In generalthe damage indexes of the LV1-noise bars are closer to theassumed-damage bars than the LV3-noise bars

443 2e Number of the Measured Nodes Unlike in nu-merical analyses whose information can be abstracted outquickly the number of sensors for field tests is usuallylimited In this section twenty five thirteen seven and fivesensors are assumed to be used in the field test for thebenchmark case in Section 422 (ie damage described inTable 5) +ese sensors are applied at the nodes of thebenchmark beam as displayed in Figure 14 Figures 11(a)ndash11(d) correspond to the cases with twenty five thirteenseven and five sensors individually Since this section fo-cuses on the influences of the number of the measuredmodal displacements other parameters are set the same as inSection 422 including that the first ten orders of modalshapes and frequencies are used Figures 15(a)ndash15(c) re-spectively represent the numerical results of Figures 14(c)ndash14(d) +e scenario given in Figure 14(a) is not displayedagain because it is the same as that in Figure 13(c)

+e numerical results show (1) the more the points aremeasured the more accurate the damage detection resultswill be combining this with conclusion (2) in Section 442 itis concluded that regardless of noises the results will bemore accurate when more data are measured (2) it alsoverifies conclusion (3) in Section 442 that stronger noisesresult in worse identification results +us in the field testthe measurement error should be reduced as much aspossible to guarantee detection accuracy

It is noteworthy that in Section 442 when noises arenot included the accuracy of the damage detection resultsis not influenced by the measured orders however in thissection no matter whether the noises are included or not

as more points are measured the results will be better +isindicates that the damage detection problem with theproposed objective function is more sensitive to thenumber of the measured points than the number of thedetermined orders +erefore ensuring sufficient sensorshas a higher priority to guarantee the accuracy of thedamage detection

5 Experimental Study

All the case studies in Section 4 are numerically based+ough artificial noises are applied in them these noisescannot correctly represent the modeling errors between thereal structure and FE model +us an experimental study isnecessary for checking the accuracy and validity of TCABCalgorithm As an example the experimental study conductedby Hu et al [15] is used to verify the accuracy of TCABCalgorithm +e experimental beam is shown in Figure 16+e used material constants are E 70Gpa v 03 andρ 2700 kgm3 As shown in Figure 16 saw-cut damage atthe middle of the ninth element is applied and the depth ofthe crack is one-quarter of the total height of the beamAccording to the section properties and the assumption ofMoslem and Nafaspour [13] the elastic modulus of element9 is between 0125E0 and 1E0 where E0 is the undamagedelastic modulus of the beam +is experimental studyconsidered the errors of inappropriate groupings noisesand lack of measured data (only the first three orders ofvibration data are obtained) which can influence the de-tection results

Hu et al [15] abstracted the first three orders of vibrationdata of both the damaged and the undamaged models andused two identification algorithms to detect the damage inthe first algorithm the global analytical stiffness and massmatrices are not needed and the algorithm is cited asDDNKM for the second one only the analytical massmatrix is employed which is named as DDNK Perera and

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8

Figure 11 +e inappropriate groupings


Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)

Figure 12 (a) +e identified parameters of scenario 1 (b) +e identified parameters of scenario 2 (c) +e identified parameters of scenario 3


Table 6 Different noise levels

Noise level φNi () ξN

ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number

Assumed damageNo noiseLV1 noise

LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued


Torres [14] used the same case but GA to identify thedamage GA shows a significant improvement comparedwith DDNK However both Hu et al [15] and Perera andTorres [14] did not establish a baseline model before thedamage detection the baseline model which reduces theinfluences of lacking the prior knowledge of the structure isalways developed first in actual tests So to compare thedetection results with Hu et al [15] and Perera and Torres[14] the unmodified numerical model is used first and thenthe baseline model is adopted as well to simulate a moregeneral situation +e input parameters are the same asshown in Table 4 +e indexesrsquo searching spaces are indi-vidually set as [minus 05 05] and [0 1] in different trial

calculations and their final identification results are similarTo maintain consistency with Hu et al [15] and Perera andTorres [14] the identification results based on indexesrsquosearching space of [0 1] are listed By applying the samedomain the accuracy of the proposed algorithm can be fairlycompared

Figures 17 and 18 display the first three orders of modalshapes of the undamaged and damaged beams Table 7shows the corresponding MAC values Tables 8 and 9contain the related frequencies information When com-paring the data obtained from the baseline model and theoriginal model with the test data as shown in Figures 17 and18 and Table 7 the modal data based on the baseline model

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)

Figure 13 (a)+e identified parameters with the first 3 orders of modal shapes and frequencies (b)+e identified parameters with the first 6orders of modal shapes and frequencies (c) +e identified parameters with the first 10 orders of modal shapes and frequencies

24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)

Figure 14 +e applied sensors and corresponding positions



LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise

ndash50ndash40ndash30ndash20ndash10

01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)

Figure 15 (a)+e identified parameters with 13 sensors (b)+e identified parameters with 7 sensors (c)+e identified parameters with 5 sensors


are closer to the test data which is consistent with theexpectation that the adoption of the baseline model im-proves the performance of the FE model However as asimilar reason is shown in Section 432 the global minimumis replaced by a local minimum due to test noises andmodeling errors which leads to the frequencies error of thebaseline model more significant than the error of the originalmodel in this specific case

+e final identification results obtained from Hu et al[15] (DDNK) Perera and Torres [14] (GA) original modeland baseline model are listed simultaneously in Figure 19 Asexpected due to the inappropriate grouping and noisesseveral elements besides the damaged one (ie element 9)are identified as damaged which is similar to the results inSection 432 In this case a smaller number of damagedelements means better performance (because there is only

141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm

Figure 16 +e structure for the experimental study

Measured (intact)FEM (original model)FEM (baseline model)

0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)

Figure 17 Modal shapes of the undamaged beam of (a) the first modal (b) the second modal and (c) the third modal


00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)

Damaged (measured)FEM (identified-original model)FEM (identified-baseline model)

(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)

Figure 18 Modal shapes of the damaged beam of (a) the first modal (b) the second modal and (c) the third modal

Table 7 MAC of the experimental beam

MAC of +e first model +e second model +e third model

Undamaged beam Original model 0999225688 0997261687 0996034642Baseline model 0999423431 0998165522 0997671728

Damaged beam Original model 0994524097 0996194531 0950126061Baseline model 0995630316 0997770042 0960814259

Table 8 Frequencies of the undamaged beam

First-order frequency (Hz) Second-order frequency (Hz) +ird-order frequency (Hz)Numerical Test Error () Numerical Test Error () Numerical Test Error ()

Original model 87232 85726 176 240454 238572 079 471359 467694 078Baseline model 84638 123 234937 152 461950 117lowastError |numerical minus test|test


one damaged element) and a more extensive damage indexof the element suggests a higher damage probability at thislocation In other words the two identified models using theTCABC algorithm work better in this case and elements 8and 9 have a more significant probability of being the realdamage part +us it can be conducted that the TCABCalgorithm as a better method can precisely locate damageposition in a smaller approximate range And the identifiedbaseline model has the best identification ability

6 Conclusions

A novel ABC algorithm (ie TCABC algorithm) forstructural damage detection via modal and frequency ana-lyses is presented two searching methods tabu searchmethod and chaotic search method are adopted to enhancethe exploration and exploitation ability +e tabu searchmethod uses a memory function to avoid the solution beingtrapped in a local minimum which increases the exploi-tation ability +e chaotic search method generates moresearching points for finding the global minimum whichincreases the exploration ability Additionally the originalroulette wheel selection is replaced by the tournament se-lection to enhance the global searching ability of TCABCalgorithm Several explicit test functions and an implicitdamage detection function are employed for comparing thenumerical results obtained from ABC and TCABC

algorithms+e numerical results show that compared to thestandard ABC algorithm the performance of TCABC al-gorithm is improved +e applicability of the TCABC al-gorithm for damage detection is verified under differentnumerical circumstances +e results show the following (1)fewer groups lead to faster convergence as demonstrated byboth ABC and TCABC algorithms in the same damagesituation (2) even though the precision of TCABC algo-rithm cannot be guaranteed when inappropriate groupingsare assigned to the structure the approximate damageposition can still be identified (3) the environmentalelectromagnetic and instrumental noises in the field testinfluence the damage detection results Stronger noisesresult in worse identification results (4) regardless ofwhether or not the noises exist the more data are measuredthe more accurate the results can be Based on the abovestatements to obtain better-damaged detection results inactual tests a large amount of measured data good test andequipment conditions and reasonable groupings are allrequired A detailed and complete analysis of setting rea-sonable groupings will be considered in subsequent researchAn experimental study is used for checking the accuracy andvalidity of the TCABC algorithm for structural damagedetection Compared with DDNK and GA the proposedTCABC algorithm performs much better and the detectionresults based on the baseline model are better than theoriginal model

Table 9 Frequencies of the damaged beam

Identified byFirst-order frequency (Hz) Second-order frequency (Hz) +ird-order frequency (Hz)

Numerical Test Error () Numerical Test Error () Numerical Test Error ()Original model 82978 83984 120 232007 235792 161 458107 462712 100Baseline model 81876 251 229305 275 453064 209lowastError |numerical minus test|test

0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA

Identified-original modelIdentified-baseline model

Figure 19 +e identified parameters of the experimental study


Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was financially supported by the National NaturalScience Foundation of China (nos 51608207 and 51478193)

References

[1] J Li and H Hao ldquoA review of recent research advances onstructural health monitoring in western Australiardquo StructuralMonitoring and Maintenance vol 3 no 1 pp 33ndash49 2016

[2] K Liu J Yan M S Alam and C Zou ldquoSeismic fragilityanalysis of deteriorating recycled aggregate concrete bridgecolumns subjected to freeze-thaw cyclesrdquo EngineeringStructures vol 187 pp 1ndash15 2019

[3] T Choudhury and H B Kaushik ldquoTreatment of uncertaintiesin seismic fragility assessment of RC frames with masonryinfill wallsrdquo Soil Dynamics and Earthquake Engineeringvol 126 Article ID 105771 2019

[4] B Jia X Yu Q Yan and Z Yang ldquoAnalysis of bridge time-dependent performance based on dynamic Bayesian net-worksrdquo International Journal of Earth Sciences and Engi-neering vol 9 no 6 pp 2427ndash2436 2016

[5] H-P Zhu B He and X-Q Chen ldquoDetection of structuraldamage through changes in frequencyrdquo Wuhan UniversityJournal of Natural Sciences vol 10 no 6 pp 1069ndash1073 2005

[6] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journalfor Numerical Methods in Engineering vol 40 no 23pp 4451ndash4468 1997

[7] C P Ratcliffe ldquoDamage detection using a modified Laplacianoperator on mode shape datardquo Journal of Sound and Vi-bration vol 204 no 3 pp 505ndash517 1997

[8] S Sehgal and H Kumar ldquoStructural dynamic model updatingtechniques a state of the art reviewrdquo Archives of Computa-tional Methods in Engineering vol 23 no 3 pp 515ndash5332016

[9] S-S Jin and H-J Jung ldquoSequential surrogate modeling forefficient finite element model updatingrdquo Computers ampStructures vol 168 pp 30ndash45 2016

[10] W-X Ren S-E Fang and M-Y Deng ldquoResponse surfa-cendashbased finite-element-model updating using structuralstatic responsesrdquo Journal of Engineering Mechanics vol 137no 4 pp 248ndash257 2011

[11] M Sanayei J E Phelps J D Sipple E S Bell andB R Brenner ldquoInstrumentation nondestructive testing andfinite-element model updating for bridge evaluation usingstrain measurementsrdquo Journal of Bridge Engineering vol 17no 1 pp 130ndash138 2012

[12] C S N Pathirage J Li L Li H Hao W Liu and P NildquoStructural damage identification based on autoencoderneural networks and deep learningrdquo Engineering Structuresvol 172 pp 13ndash28 2018

[13] KMoslem and R Nafaspour ldquoStructural damage detection bygenetic algorithmsrdquo AIAA Journal vol 40 no 7 pp 1395ndash1401 2002

[14] R Perera and R Torres ldquoStructural damage detection viamodal data with genetic algorithmsrdquo Journal of StructuralEngineering vol 132 no 9 pp 1491ndash1501 2006

[15] N Hu X Wang H Fukunaga Z H Yao H X Zhang andZ S Wu ldquoDamage assessment of structures using modal testdatardquo International Journal of Solids and Structures vol 38no 18 pp 3111ndash3126 2001

[16] Y-S Park S Kim N Kim and J-J Lee ldquoFinite elementmodel updating considering boundary conditions usingneural networksrdquo Engineering Structures vol 150 pp 511ndash519 2017

[17] W Qiao and Z Yang ldquoSolving large-scale function optimi-zation problem by using a new metaheuristic algorithm basedon quantum dolphin swarm algorithmrdquo IEEE Access vol 7pp 138972ndash138989 2019

[18] F Kang J-J Li and Q Xu ldquoDamage detection based onimproved particle swarm optimization using vibration datardquoApplied Soft Computing vol 12 no 8 pp 2329ndash2335 2012

[19] W Qiao Z Yang Z Kang and Z Pan ldquoShort-term naturalgas consumption prediction based on Volterra adaptive filterand improved whale optimization algorithmrdquo EngineeringApplications of Artificial Intelligence vol 87 Article ID103323 2020

[20] D Karaboga ldquoAn idea based on honey bee swarm for nu-merical optimizationrdquo Technical Report-TR06 Departmentof Computer Engineering Engineering Faculty ErciyesUniversity Kayseri Turkey 2005

[21] D Karaboga and B Akay ldquoA comparative study of artificialbee colony algorithmrdquo Applied Mathematics and Computa-tion vol 214 no 1 pp 108ndash132 2009

[22] B Akay and D Karaboga ldquoParameter tuning for the artificialbee colony algorithmrdquo Computational Collective IntelligenceSemantic Web Social Networks and Multiagent Systemspp 608ndash619 2009

[23] S M Awan M Aslam Z A Khan and H Saeed ldquoAn efficientmodel based on artificial bee colony optimization algorithmwith neural networks for electric load forecastingrdquo NeuralComputing and Applications vol 25 no 7-8 pp 1967ndash19782014

[24] E Uzlu A Akpınar H T Ozturk S Nacar and M KankalldquoEstimates of hydroelectric generation using neural networkswith the artificial bee colony algorithm for Turkeyrdquo Energyvol 69 pp 638ndash647 2014

[25] N Menon and R Ramakrishnan ldquoBrain tumor segmentationin MRI images using unsupervised artificial bee colony al-gorithm and FCM clusteringrdquo in Proceedings of the 2015International Conference on Communications and SignalProcessing (ICCSP) pp 6ndash9 Chengdu China October 2015

[26] L Wen J Cheng F Li J Zhao Z Shi and H Zhang ldquoGlobaloptimization of controlled source audio-frequency magne-totelluric data with an improved artificial bee colony algo-rithmrdquo Journal of Applied Geophysics vol 170 Article ID103845 2019

[27] H Sun and R Betti ldquoSimultaneous identification of structuralparameters and dynamic input with incomplete output-onlymeasurementsrdquo Structural Control and Health Monitoringvol 21 no 6 pp 868ndash889 2013

[28] Z H Ding M Huang and Z R Lu ldquoStructural damagedetection using artificial bee colony algorithm with hybridsearch strategyrdquo Swarm and Evolutionary Computationvol 28 pp 1ndash13 2016


[29] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquo Computers amp Operations Researchvol 13 no 5 pp 533ndash549 1986

[30] B Li and W Jiang ldquoOptimizing complex functions by chaossearchrdquo Cybernetics and Systems vol 29 no 4 pp 409ndash4191998

[31] D EwinsModal Testing Research Studies Press Baldock UK2000

[32] S Ghambari and A Rahati ldquoAn improved artificial bee colonyalgorithm and its application to reliability optimizationproblemsrdquo Applied Soft Computing vol 62 pp 736ndash7672018

[33] F McKenna M H Scott and G L Fenves ldquoNonlinear finite-element analysis software architecture using object compo-sitionrdquo Journal of Computing in Civil Engineering vol 24no 1 pp 95ndash107 2010

[34] A Gilat MATLAB Wiley New York NY USA 2016


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problem Moslem and Nafaspour [13] assumed that anychange in the properties of each element due to the damageis equivalent to a change in its elastic modulus By thissimplification the FE model updating-based damage de-tection problem can be formulated as an optimizationproblem with a simple-structured implicit objective func-tion For example the norm of the residuals between thenumerical results of the FE model and the test data of theactual structure is defined as the objective function Once theobjective function reaches the global minimum the inputparameters of the FE model will be treated as the physicalparameters of the actual structure

+e structural damage detection problem is transferredto find the extreme values of the corresponding objectivefunction However these extrema cannot be seen by thetraditional method because the objective function isexpressed by implicit functions To overcome this problemseveral metaheuristic algorithms and artificial intelligence(AI) are used Perera and Torres [14] identified the damageof a simply supported beam by genetic algorithm (GA) andthese detection results are better than those calculated withthe method that Hu et al [15] proposed Park et al [16]updated bridge boundary conditions using artificial neuralnetworks (ANN) and verified the results by both laboratoryand field tests it was concluded that the ANN reduces theuncertainties of the boundary conditions in FE modelupdating Pathirage et al [12] introduced dimensionalityreduction and relationship learning in the ANN to optimizethe weights in neural networks the laboratory test indicatedthat the detection results of this method are better than theresults of the traditional ANN Qiao and Yang [17] intro-duced a quantum search dolphin swarm algorithm (QDSA)in finding the optimal solution of large-scale functions theyconfirmed that the QSDA has outstanding advantages thandolphin swarm algorithm (DSA) Kang et al [18] proposedan improved particle swarm optimization (PSO) algorithmto identify structural damages using vibration data and theresults showed that compared to differential evolution (DE)algorithm the improved PSO algorithm is more efficient indetermining the sites and the extents of structural damagesQiao et al [19] combined the Volterra adaptive filter and animproved whale optimization algorithm (WOA) to predictthe short-term natural gas consumption the study showedthat the proposed prediction model is better than someadvanced prediction models

Besides the abovementioned algorithms the artificial beecolony (ABC) algorithm also attracted the attention of manyscholars in the optimization area since it was proposed ABCalgorithm was suggested by Karaboga [20] and this algo-rithm is motivated by the intelligent behavior of honey beeswhen seeking a high-quality food source In 2009 Karabogaand Akay [21] first conducted a detailed and comprehensiveanalysis of ABC algorithm and tested it with 50 numericalbenchmark functions and other well-known evolutionaryalgorithms such as GA PSO DE and ant colony algorithm(ACO) the results indicate ABC algorithm has several ad-vantages in getting the extrema of the function than othermethods In the same year the influences of changing pa-rameters in the ABC algorithm were analyzed by Akay and

Karaboga [22] With the benefits such as simple structureeasy implementation and outstanding performance theABC algorithm is widely used in many fields Awan et al[23] applied the ABC algorithm to optimize the connectionweights of neurons in the ANN for electric load forecastingUzlu et al [24] combined ABC and ANN to predict theannual output of hydroelectric power in Turkey Menon andRamakrishnan [25] used the ABC algorithm to process braintumors segmentation in magnetic resonance images Wenet al [26] applied the ABC algorithm to achieve globaloptimization of controlled-source audio-frequency mag-netotelluric data However there are limited researchersutilizing the ABC algorithm in structural damage detectionCombining the ABC algorithm with the NelderndashMeadsimplex method Sun and Betti [27] identified the structuralparameters and the input forces of buildings even when thestructural responses are noise contaminated By applying thehybrid search strategy in the ABC algorithm for structuraldamage detection Ding et al [28] got better identificationresults compared with those from GA and traditional ABCalgorithms Nevertheless there are some flaws in Sun andBetti [27] and Ding et al [28] +e calculations of Sun andBetti [27] are purely numerical-based and lack of experi-mental verification +e data from the experiment are dif-ferent from those artificially assumed and used in thenumerical models So an experimental study should be usedas a verification after the numerical study Ding et al [28] didnot consider several factors that influence the target functionand corresponding identification results However theimplicit target function highly relies on the observed fre-quencies and modal shapes and these observed data areeasily affected by the test conditions such as grouping noiseeffect mode shape order and sensor location which shouldbe taken into account

Based on the fact that there are limited research studiesin ABC-based structural damage detection and the assumedidentification condition is relatively simple In this studyABC algorithm combined with the tabu search method [29]and chaotic search method [30] (named as TCABC algo-rithm) is used to detect structural damages under differentidentify conditions such as grouping noise effect modeshape order sensor location and is also evaluated by severalexplicit test functions Recommendations for realisticallyusing the TCABC algorithm in damage detection are alsoput forward Furthermore an experimental study is used toexamine the damage detection results of TCABC algorithm

+e rest of the paper is organized as follows Section 2describes how the structural damage detection problem istransferred to an optimization problem traditional ACBalgorithm and proposed TCABC algorithm are presented inSection 3 and their numerical results are compared inSection 4 also in Section 4 the accuracy of TCABC algo-rithm for damage detection is verified under different cir-cumstances (ie influences of grouping size noise effectmode shape order and sensor location) and several rec-ommendations are given for using TCABC algorithm todetect structural damages under actual conditions in Sec-tion 5 an experimental study is used to examine the per-formance of TCABC algorithm Section 6 is the conclusions












MAC ΦTj1113960 1113961 ΦF

j1113960 11139611113872 1113873 ΦT

j1113960 1113961t[W] ΦF

j1113960 11139611113874 11138752

ΦTj1113960 1113961

t[W] ΦT

j1113960 11139611113874 1113875 ΦFj1113960 1113961

t[W] ΦF

j1113960 11139611113874 1113875

(4)

whereΦTj andΦF


Wkk ΦTjk1113872 1113873

minus 1111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 (5)



F 1 minus 1113945m

j1

MAC ΦTj1113960 1113961 ΦF

j1113960 11139611113872 1113873

1 + aj1113872 1113873 (6)


aj fF

j1113872 11138732

minus fTj1113872 1113873

2

fFj1113872 1113873

2+ fT

j1113872 11138732

11138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868 (7)

fFj and fT







maxj minus x

minj1113872 1113873 (8)








fiti

11 + fi

fi ge 0

1 + abs fi( 1113857 fi lt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)


pi fiti

1113936SNm1fitm

(11)







Employed bee phase


Onlooker bee phase



Scout bee phase


Complete iteration

Yes

No

Finish

Yes

No

Start






pi si

1113936SNm1sm

(12)



βn+1 μβn 1 minus βn( 1113857 (13)





4 Numerical Study







0

02

04

06

08

1

Itera

tive r

esul

ts






Update tabu list T1

Employed bee phase

Yes

No



Yes








No


Yes

Yes

No

Yes

No


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

02m 04m

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12




F1 f1(x) 1113936Di1x

2i (minus 10 10)D 0

F2 f2(x) 1113936Di1(10


F3 f3(x) 1113936Di1ix

2i (minus 10 10)D 0

F4 f4(x) 1113936Di1|xi| + 1113937


F5 f5(x) (14000)1113936Di1x

2i minus 1113937

Di1cos(xi

i


F6 f6(x) 1113936Di1ix

4i (minus 10 10)D 0





Vp

(14)






















ndash100

ndash80

ndash60

ndash40

ndash20

0

Gro

wth

rate

()

700350 400 450 500 550 600 650300250

Mean

Generation number

F1F2F3

F4F5F6

(a)

ndash100

ndash80

ndash60

ndash40

ndash20

0G

row

th ra

te (

)


Standard deviation

F1F2F3

F4F5F6

(b)








fNi fi 1 + η1φ

Ni1113872 1113873 (15)




F1F2F3

F4F5F6


03

04

05

06

07

08

09

1

11Re

lativ

e ide

ntifi

ed v

alue


F1F2F3

F4F5F6

03

04

05

06

07

08

09

1

11

Rela

tive i

dent

ified

val

ue

1755025 100 125 150 2000 75Length of T1






Nij1113872 1113873 (16)






ABCTCABC

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)


ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number

(b)


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

E = 065E0

G1 G2 G3 G4 G5 G6

24m

(a)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12

(b)

G8

(c)

G1 G7 G24








10ndash4

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)



ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

179 19 236 168 1810 15133 201211742 14 21 22 241 5Element number

(b)











141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8



Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












MAC ΦTj1113960 1113961 ΦF

j1113960 11139611113872 1113873 ΦT

j1113960 1113961t[W] ΦF

j1113960 11139611113874 11138752

ΦTj1113960 1113961

t[W] ΦT

j1113960 11139611113874 1113875 ΦFj1113960 1113961

t[W] ΦF

j1113960 11139611113874 1113875

(4)

whereΦTj andΦF


Wkk ΦTjk1113872 1113873

minus 1111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 (5)



F 1 minus 1113945m

j1

MAC ΦTj1113960 1113961 ΦF

j1113960 11139611113872 1113873

1 + aj1113872 1113873 (6)


aj fF

j1113872 11138732

minus fTj1113872 1113873

2

fFj1113872 1113873

2+ fT

j1113872 11138732

11138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868 (7)

fFj and fT







maxj minus x

minj1113872 1113873 (8)








fiti

11 + fi

fi ge 0

1 + abs fi( 1113857 fi lt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)


pi fiti

1113936SNm1fitm

(11)







Employed bee phase


Onlooker bee phase



Scout bee phase


Complete iteration

Yes

No

Finish

Yes

No

Start






pi si

1113936SNm1sm

(12)



βn+1 μβn 1 minus βn( 1113857 (13)





4 Numerical Study







0

02

04

06

08

1

Itera

tive r

esul

ts






Update tabu list T1

Employed bee phase

Yes

No



Yes








No


Yes

Yes

No

Yes

No


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

02m 04m

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12




F1 f1(x) 1113936Di1x

2i (minus 10 10)D 0

F2 f2(x) 1113936Di1(10


F3 f3(x) 1113936Di1ix

2i (minus 10 10)D 0

F4 f4(x) 1113936Di1|xi| + 1113937


F5 f5(x) (14000)1113936Di1x

2i minus 1113937

Di1cos(xi

i


F6 f6(x) 1113936Di1ix

4i (minus 10 10)D 0





Vp

(14)






















ndash100

ndash80

ndash60

ndash40

ndash20

0

Gro

wth

rate

()

700350 400 450 500 550 600 650300250

Mean

Generation number

F1F2F3

F4F5F6

(a)

ndash100

ndash80

ndash60

ndash40

ndash20

0G

row

th ra

te (

)


Standard deviation

F1F2F3

F4F5F6

(b)








fNi fi 1 + η1φ

Ni1113872 1113873 (15)




F1F2F3

F4F5F6


03

04

05

06

07

08

09

1

11Re

lativ

e ide

ntifi

ed v

alue


F1F2F3

F4F5F6

03

04

05

06

07

08

09

1

11

Rela

tive i

dent

ified

val

ue

1755025 100 125 150 2000 75Length of T1






Nij1113872 1113873 (16)






ABCTCABC

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)


ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number

(b)


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

E = 065E0

G1 G2 G3 G4 G5 G6

24m

(a)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12

(b)

G8

(c)

G1 G7 G24








10ndash4

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)



ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

179 19 236 168 1810 15133 201211742 14 21 22 241 5Element number

(b)











141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8



Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





fiti

11 + fi

fi ge 0

1 + abs fi( 1113857 fi lt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)


pi fiti

1113936SNm1fitm

(11)







Employed bee phase


Onlooker bee phase



Scout bee phase


Complete iteration

Yes

No

Finish

Yes

No

Start






pi si

1113936SNm1sm

(12)



βn+1 μβn 1 minus βn( 1113857 (13)





4 Numerical Study







0

02

04

06

08

1

Itera

tive r

esul

ts






Update tabu list T1

Employed bee phase

Yes

No



Yes








No


Yes

Yes

No

Yes

No


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

02m 04m

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12




F1 f1(x) 1113936Di1x

2i (minus 10 10)D 0

F2 f2(x) 1113936Di1(10


F3 f3(x) 1113936Di1ix

2i (minus 10 10)D 0

F4 f4(x) 1113936Di1|xi| + 1113937


F5 f5(x) (14000)1113936Di1x

2i minus 1113937

Di1cos(xi

i


F6 f6(x) 1113936Di1ix

4i (minus 10 10)D 0





Vp

(14)






















ndash100

ndash80

ndash60

ndash40

ndash20

0

Gro

wth

rate

()

700350 400 450 500 550 600 650300250

Mean

Generation number

F1F2F3

F4F5F6

(a)

ndash100

ndash80

ndash60

ndash40

ndash20

0G

row

th ra

te (

)


Standard deviation

F1F2F3

F4F5F6

(b)








fNi fi 1 + η1φ

Ni1113872 1113873 (15)




F1F2F3

F4F5F6


03

04

05

06

07

08

09

1

11Re

lativ

e ide

ntifi

ed v

alue


F1F2F3

F4F5F6

03

04

05

06

07

08

09

1

11

Rela

tive i

dent

ified

val

ue

1755025 100 125 150 2000 75Length of T1






Nij1113872 1113873 (16)






ABCTCABC

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)


ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number

(b)


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

E = 065E0

G1 G2 G3 G4 G5 G6

24m

(a)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12

(b)

G8

(c)

G1 G7 G24








10ndash4

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)



ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

179 19 236 168 1810 15133 201211742 14 21 22 241 5Element number

(b)











141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8



Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





pi si

1113936SNm1sm

(12)



βn+1 μβn 1 minus βn( 1113857 (13)





4 Numerical Study







0

02

04

06

08

1

Itera

tive r

esul

ts






Update tabu list T1

Employed bee phase

Yes

No



Yes








No


Yes

Yes

No

Yes

No


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

02m 04m

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12




F1 f1(x) 1113936Di1x

2i (minus 10 10)D 0

F2 f2(x) 1113936Di1(10


F3 f3(x) 1113936Di1ix

2i (minus 10 10)D 0

F4 f4(x) 1113936Di1|xi| + 1113937


F5 f5(x) (14000)1113936Di1x

2i minus 1113937

Di1cos(xi

i


F6 f6(x) 1113936Di1ix

4i (minus 10 10)D 0





Vp

(14)






















ndash100

ndash80

ndash60

ndash40

ndash20

0

Gro

wth

rate

()

700350 400 450 500 550 600 650300250

Mean

Generation number

F1F2F3

F4F5F6

(a)

ndash100

ndash80

ndash60

ndash40

ndash20

0G

row

th ra

te (

)


Standard deviation

F1F2F3

F4F5F6

(b)








fNi fi 1 + η1φ

Ni1113872 1113873 (15)




F1F2F3

F4F5F6


03

04

05

06

07

08

09

1

11Re

lativ

e ide

ntifi

ed v

alue


F1F2F3

F4F5F6

03

04

05

06

07

08

09

1

11

Rela

tive i

dent

ified

val

ue

1755025 100 125 150 2000 75Length of T1






Nij1113872 1113873 (16)






ABCTCABC

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)


ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number

(b)


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

E = 065E0

G1 G2 G3 G4 G5 G6

24m

(a)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12

(b)

G8

(c)

G1 G7 G24








10ndash4

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)



ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

179 19 236 168 1810 15133 201211742 14 21 22 241 5Element number

(b)











141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8



Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Update tabu list T1

Employed bee phase

Yes

No



Yes








No


Yes

Yes

No

Yes

No


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

02m 04m

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12




F1 f1(x) 1113936Di1x

2i (minus 10 10)D 0

F2 f2(x) 1113936Di1(10


F3 f3(x) 1113936Di1ix

2i (minus 10 10)D 0

F4 f4(x) 1113936Di1|xi| + 1113937


F5 f5(x) (14000)1113936Di1x

2i minus 1113937

Di1cos(xi

i


F6 f6(x) 1113936Di1ix

4i (minus 10 10)D 0





Vp

(14)






















ndash100

ndash80

ndash60

ndash40

ndash20

0

Gro

wth

rate

()

700350 400 450 500 550 600 650300250

Mean

Generation number

F1F2F3

F4F5F6

(a)

ndash100

ndash80

ndash60

ndash40

ndash20

0G

row

th ra

te (

)


Standard deviation

F1F2F3

F4F5F6

(b)








fNi fi 1 + η1φ

Ni1113872 1113873 (15)




F1F2F3

F4F5F6


03

04

05

06

07

08

09

1

11Re

lativ

e ide

ntifi

ed v

alue


F1F2F3

F4F5F6

03

04

05

06

07

08

09

1

11

Rela

tive i

dent

ified

val

ue

1755025 100 125 150 2000 75Length of T1






Nij1113872 1113873 (16)






ABCTCABC

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)


ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number

(b)


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

E = 065E0

G1 G2 G3 G4 G5 G6

24m

(a)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12

(b)

G8

(c)

G1 G7 G24








10ndash4

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)



ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

179 19 236 168 1810 15133 201211742 14 21 22 241 5Element number

(b)











141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8



Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Vp

(14)






















ndash100

ndash80

ndash60

ndash40

ndash20

0

Gro

wth

rate

()

700350 400 450 500 550 600 650300250

Mean

Generation number

F1F2F3

F4F5F6

(a)

ndash100

ndash80

ndash60

ndash40

ndash20

0G

row

th ra

te (

)


Standard deviation

F1F2F3

F4F5F6

(b)








fNi fi 1 + η1φ

Ni1113872 1113873 (15)




F1F2F3

F4F5F6


03

04

05

06

07

08

09

1

11Re

lativ

e ide

ntifi

ed v

alue


F1F2F3

F4F5F6

03

04

05

06

07

08

09

1

11

Rela

tive i

dent

ified

val

ue

1755025 100 125 150 2000 75Length of T1






Nij1113872 1113873 (16)






ABCTCABC

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)


ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number

(b)


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

E = 065E0

G1 G2 G3 G4 G5 G6

24m

(a)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12

(b)

G8

(c)

G1 G7 G24








10ndash4

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)



ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

179 19 236 168 1810 15133 201211742 14 21 22 241 5Element number

(b)











141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8



Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia













ndash100

ndash80

ndash60

ndash40

ndash20

0

Gro

wth

rate

()

700350 400 450 500 550 600 650300250

Mean

Generation number

F1F2F3

F4F5F6

(a)

ndash100

ndash80

ndash60

ndash40

ndash20

0G

row

th ra

te (

)


Standard deviation

F1F2F3

F4F5F6

(b)








fNi fi 1 + η1φ

Ni1113872 1113873 (15)




F1F2F3

F4F5F6


03

04

05

06

07

08

09

1

11Re

lativ

e ide

ntifi

ed v

alue


F1F2F3

F4F5F6

03

04

05

06

07

08

09

1

11

Rela

tive i

dent

ified

val

ue

1755025 100 125 150 2000 75Length of T1






Nij1113872 1113873 (16)






ABCTCABC

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)


ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number

(b)


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

E = 065E0

G1 G2 G3 G4 G5 G6

24m

(a)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12

(b)

G8

(c)

G1 G7 G24








10ndash4

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)



ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

179 19 236 168 1810 15133 201211742 14 21 22 241 5Element number

(b)











141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8



Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







fNi fi 1 + η1φ

Ni1113872 1113873 (15)




F1F2F3

F4F5F6


03

04

05

06

07

08

09

1

11Re

lativ

e ide

ntifi

ed v

alue


F1F2F3

F4F5F6

03

04

05

06

07

08

09

1

11

Rela

tive i

dent

ified

val

ue

1755025 100 125 150 2000 75Length of T1






Nij1113872 1113873 (16)






ABCTCABC

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)


ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number

(b)


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

E = 065E0

G1 G2 G3 G4 G5 G6

24m

(a)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12

(b)

G8

(c)

G1 G7 G24








10ndash4

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)



ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

179 19 236 168 1810 15133 201211742 14 21 22 241 5Element number

(b)











141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8



Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Nij1113872 1113873 (16)






ABCTCABC

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)


ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number

(b)


141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24

24m

E = 065E0

G1 G2 G3 G4 G5 G6

24m

(a)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12

(b)

G8

(c)

G1 G7 G24








10ndash4

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)



ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

179 19 236 168 1810 15133 201211742 14 21 22 241 5Element number

(b)











141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8



Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







10ndash4

10ndash3

10ndash2

10ndash1

100

Obj

ectiv

e fun

ctio

n va

lue


(a)



ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

179 19 236 168 1810 15133 201211742 14 21 22 241 5Element number

(b)











141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8



Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

24 m

E = 065E0

E = 065E0

E = 065E0

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

141 2 3 4 5 6 7 8 9 10 111213 1516 17 1819 20 2122 23 24

E = 085E0

(a)

(b)

(c)

G1 G2 G3 G4 G5 G6 G7 G9 G10 G11 G12G8



Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

(a)

Assumed damageTCABC

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241Element number

0

10

20

30

40

Dam

age i

ndex

()

(b)

Assumed damageTCABC

ndash10

0

10

20

30

40

Dam

age i

ndex

()

179 19 236 168 1810 15133 207 11 1242 14 21 22 241 5Element number

(c)





ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




ij ()

LV1 3 8LV2 5 15LV3 8 20

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(a)

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

93 6 8 10742 11 121 5Group number


LV2 noiseLV3 noise

(b)

Figure 13 Continued





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 7 8 9 10 11 121Group number


LV2 noiseLV3 noise

(c)


24m

1 25

242 4 6 8 10 12 14 16 18 20 22

3 5 7 9 11 13 15 17 19 21 23

1 253 5 7 9 11 13 15 17 19 21 23

1 255 9 13 17 21

1 257 13 19

(a)

(b)

(d)

(c)




LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(a)


LV2 noiseLV3 noise


01020304050

Dam

age i

ndex

()

2 3 4 5 6 118 9 10 121 7Group number

(b)


LV2 noiseLV3 noise

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

30

40

50

Dam

age i

ndex

()

83 65 9 1042 7 11 121Group number

(c)





141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




141 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

60cm06cm

255cmSaw cut 5cm



0

02

04

06

08

1

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(a)


ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)

(b)


10 20 30 40 50 600X axis (cm)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

(c)



00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


00

02

04

06

08

1D

efle

ctio

n

20 30 40 50 6010X axis (cm)


(a)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(b)

ndash15

ndash1

ndash05

0

05

1

15

Def

lect

ion

10 20 30 40 50 600X axis (cm)


(c)











6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



6 Conclusions






0

10

20

30

40

50

60

Dam

age i

ndex

()

17 1993 6 158 1610 13 18 207 11 1442 121 5Element number

DDNKGA




Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


a novel artificial bee colony algorithm for structural

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