Research Article Back Analysis of Geomechanical Parameters ... · Introduction Numerical analysis...

Research ArticleBack Analysis of Geomechanical Parameters in UndergroundEngineering Using Artificial Bee Colony

Changxing Zhu Hongbo Zhao and Ming Zhao

School of Civil Engineering Henan Polytechnic University Jiaozuo 454003 China

Correspondence should be addressed to Hongbo Zhao bxhbzhaohotmailcom

Received 13 April 2014 Accepted 26 June 2014 Published 17 July 2014

Academic Editor Minghuwi Horng

Copyright copy 2014 Changxing Zhu et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Accurate geomechanical parameters are critical in tunneling excavation design and supporting In this paper a displacementsback analysis based on artificial bee colony (ABC) algorithm is proposed to identify geomechanical parameters from monitoreddisplacements ABC was used as global optimal algorithm to search the unknown geomechanical parameters for the problem withanalytical solution To the problem without analytical solution optimal back analysis is time-consuming and least square supportvector machine (LSSVM) was used to build the relationship between unknown geomechanical parameters and displacement andimprove the efficiency of back analysisThe proposed method was applied to a tunnel with analytical solution and a tunnel withoutanalytical solution The results show the proposed method is feasible

1 Introduction

Numerical analysis plays an important role in constructionand design of geotechnical engineering [1] Geomechani-cal parameters such as Youngrsquos modulus and cohesion arecritical to numerical analysis and are difficult to determinebecause of the complexity and uncertainty of geotechnicalengineering Back analysis is a reliable approach to esti-mate the geomechanical parameters and is used widely ingeotechnical engineering [2] Because the deformation ofrock masses induced by excavation can be measured easilyand reliably displacement-based back analysis techniquesas a practical engineering tool are nowadays frequentlyused in geotechnical engineering problems to determine theunknown geomechanical parameters [3ndash9]

There are mainly three types of displacement back anal-ysis methods inverse solving method atlas method anddirect (ie optimal) method [7] Because of the specialadvantages the optimal methods are more and more exten-sively employed in solving engineering problems [10ndash12]Optimization method is important to optimal back anal-ysis Levenber-Marquardt method Gauss-Newton method

Bayesian method Powell method Rosenbork method softcomputing and particle swarm optimization have been pro-posed and applied to back analysis [12ndash14] To the practicalgeotechnical engineering optimal back analysis needs tocall numerical analysis many times This procedure is time-consuming Neural network and support vector machinewere applied to back analysis to replace the numerical analysis[14ndash17] This has been a new way for displacement backanalysis

In this paper artificial bee colony (ABC) algorithmwas chosen for its biological and evolutionary appeal infinding the set of unknown parameters that best matchesthe modeling prediction with the measured displacementdata Least square support vector machine (LSSVM) wasused to replace numerical analysis to present the relationshipbetween unknown geomechanical parameters and displace-ment of geotechnical structure Firstly the idea and algorithmof ABC were presented in Section 2 In Section 3 ABC wasadopted to search geomechanical parameters in displacementback analysisThe procedure of ABC-based back analysis waspresented to the tunnel with analytical solution and applied toa circular tunnel with hydrostatic stressThen to the complex

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 693812 13 pageshttpdxdoiorg1011552014693812

2 The Scientific World Journal

geotechnical engineeringwithout analytical solution LSSVMwas used to present the relationship between geomechanicalparameters and displacement LSSVM model replaced thenumerical analysis to improve the efficiency of back analysisBack analysis based on LSSVM and ABC combination wasproposed in Section 4 LSSVM and the procedure of theproposed method were presented in brief Lastly someconclusion was listed in Section 5

2 Artificial Bee Colony Algorithms

The artificial bee colony (ABC) algorithm was originallydeveloped in 2005 by Karaboga [18] In ABC algorithmthe colony of artificial bees contains three groups of beesemployed bees onlookers and scouts Employed bees searchfor specific food sources (solution) and calculate the amountof nectars (fitness value) Onlooker bees choose a food sourcebased on the nectars shared by employed bees and determinethe source to be abandoned and allocate its employed beeas scout bees Scout bees randomly search for a new foodsource The position of a food source represents a possiblesolution for the problem under consideration and the nectaramount of a food source represents the quality of the solutionrepresented by the fitness value [19 20] To the minimumproblem the fitness can be computed by the target function

In the algorithm the first half of the colony consists ofemployed artificial bees and the second half constitutes theonlookersThe number of the employed bees or the onlookerbees is equal to the number of solutions in the population Atthe first step theABCgenerates a randomly distributed initialpopulation of SN solutions and calculates the fitness of eachsolution Consider

119909 (119894 119895) = 119909119895

min + rand (0 1) (119909119895max minus 119909119895

min) (1)

where 119909(119894 119895) is the candidate solution of problem 119894 =

1 2 1198781198732 and 1198781198732 denotes the size of population 119895 =1 2 119863 and 119863 is the dimension number of each solutionrand(01) is a random number between [0 1] 119909119894min and 119909

119894max

are the upper and lower bound of each solutionOnce initialization is completed the artificial bees are

used to conduct the search for the best food resource(solution) Procedures can be described as follows [20]

(i) Employed bees determine a food source within theneighborhood of the food source through their mem-ory

(ii) Employed bees share their information with onlook-ers within the hive and then the onlookers select oneof the food sources

(iii) Onlookers select a food source within the neighbor-hood of the food sources chosen by them to produceand exploit the new food resources

(iv) An employed bee of the sources that have beenabandoned by onlookers becomes a scout and startsto search for a new food source randomly

In the ABC algorithm a candidate food position can beproduced from the memory of bees which is defined as

V (119894 119895) = 119909 (119894 119895) + 120593119894119895 (119909 (119894 119895) minus 119909 (119896 119895)) (2)

where k used to be different from 119894 is randomly chosenindexes from 1 2 1198781198732 j is also randomly chosenindexes from 1 2 119863 and 120593ij is a random numberin [minus1 1] and controls the generation of neighbor foodsources around 119909(119894 119895) and represents the comparison oftwo food positions seen by a bee As can be seen from(2) the perturbation on the position 119909(119894 119895) decreases whenthe difference between the parameters of 119909(119894 119895) and 119909(119896 119895)decreases so that the step length is adaptively reduced

An artificial onlooker bee chooses a food source basedon the probability of food source The probability of beingselected for fitness pi can be expressed as

119901119894 =fitness119894

sum119878119873119899=1 fitness119899

(3)

where fitness119894 is the fitness of the solutionIn ABC algorithm a food source whose position cannot

be improved further through a predetermined number ofcycles is assumed to be abandoned by onlookers 119909(119894 119895) usedto represent the abandoned source is replaced with 1199091015840(119894 119895)that is a new food source the scout bees find which isconducted by (1)

Each candidate source position V(119894 119895) produced by 119909(119894 119895)can be evaluated using the comparison between 119909(119894 119895) and itsold source positionThe old food source will be replaced withthe new food source when it is equal to or better than the oldfood source Otherwise the old food source is retained in thememory

There are three control parameters in the ABC thenumber of food sources which is equal to the number ofemployed or onlooker bees (SN2) the value of limit and themaximum cycle number (MCN) The following is the briefprocedure of artificial bee colony (ABC) algorithm

Step 1 Determine the value of control parametersSN2 MCN and ldquolimitrdquo of ABC algorithmStep 2 Generate the initial population 119909(119894 119895) by (1)and evaluate the fitness of each solutionStep 3 Produce new solution V(119894 119895) for each employedbee by using (2) In the meantime the fitness isevaluatedStep 4 Calculate the probability 119901119894 for the solution119909(119894 119895) by (3)Step 5 Select a solution 119909(119894 119895) for each onlookerbee according to 119901119894 Then a new solution V(119894 119895) isgenerated by (2)Step 6 Calculate the fitnessStep 7 If there is an abandoned solution for the scoutit will be replaced by using a new solution which israndomly produced by (2)Step 8 Trace the best solutionStep 9 Repeat Steps 3 to 8 until the cycle reaches themaximum cycle number (MCN)

The Scientific World Journal 3

p0

p0

pi

rP

r0

PlasticElastic

Figure 1 A circular tunnel subjected to hydrostatic far field stressand uniform support pressure

3 ABC-Based Back Analyses

Optimization algorithm is critical to back analysis In thissection ABC-based back analysis was presented to identifythe geomechanical parameters of a circular tunnel withanalytical solution

31 The Analytical Solution of Circular Tunnel A circulartunnel is excavated in a continuous homogeneous isotropicinitially elastic rock mass and subjected to a hydrostatic farfield stress p0 and uniform support pressure pi as shown inFigure 1

According to the Mohr-Coulomb criterion the normalstress pcr at the plastic-elastic zone interface is given [21] asfollows

119901119888119903 =2119901119900 minus 120590119888

119896 + 1

119896 =

1 + sin1205931 minus sin120593

120590119888 =119888 (119896 minus 1)

tan120593

(4)

where 120593 is the friction angle and c is the cohesion If theuniform support pressure pi is less than the critical pressurepcr the plastic zone exists The plastic zone radius R is given[22] as follows

119877 = 119903119900 lowast [2 (119901119900 + 119904)

(119896 + 1) (119901119894 + 119904)]

1(119896minus1)

(5)

in which

119904 =

120590119888

119896 minus 1

(6)

and 119903119900 is the radius of the tunnel

The deformation of surrounding rock of tunnel is asfollows

Elastic zone

119906119903 =

(119901119900 sin120593 + 119888 sdot cos120593) (1198772119903)

2119866

(7)

Plastic zone

119906119903 =119903

2119866

sdot 120594 (8)

where E is the deformation modulus and 120583 is Poissonrsquos ratio

120594 = (2120583 minus 1) (119901119900 + 119888 sdot ctg120593)

+ (1 minus 120583) [(1198702119901 minus 1) (119870119901 + 119870119901119904)]

times (119901119894 + 119888 sdot ctg120593) (119877

119903119900

)

(119870119901minus1)

(

119877

119903

)

(119870119901119904+1)

+ [

(1 minus 120583) (119870119901119870119901119904 + 1)

(119870119901 + 119870119901119904)

sdot 120583]

times (119901119894 + 119888 sdot ctg120593) (119903

119903119900

)

(119870119901minus1)

119870119901119904 =(1 + sin120595119904)(1 minus sin120595119904)

119866 =

119864

2 (1 + 120583)

(9)

32 Error Function An error function in this work isdefined as the minimum error between the displacementspredicted by the analyticalmodel based identified parametersand the actualmeasured displacements It can be expressed as

fitness = radicsum119899119894=1 (119910119901119894 minus 119910119894)

2

119899

(10)

where n is the number of key points 119910119894 is the monitoreddisplacement of the ith key points and 119910119901119894 is the predicteddisplacement of ith key point

33 The Procedure of ABC-Based Back Analysis ABC-basedback analysis is combined ABC with the analytical solution(see (7) and (8)) ABC produces population of artificial beesincluding employer bees onlooker bees and scout bees Thefitness values can be computed by (10) The displacementof (10) can be computed by (7) and (8) Based on the ABCalgorithm the new population was produced ABC-basedback analysis algorithm can be described as follows (seeFigure 2)

Step 1 Collect the information of engineering such asgeology conditions and engineering size


Start

Determine engineering condition andselect the computing model

Initiate the ABC algorithm

Generate the initial population by (1) andcompute the displacement by (7) and (8)and the fitness of each solution by (10)

Generate the new population by (2) and(3) and compute the displacement by (7)

and (8) the fitness of each solution by (10)

Memorize the best solution

Maximum cycle meets

Get the geomechanical parameters

End

No

Yes

Figure 2 Flowchart of ABC-based back analysis

Step 2 Select the appropriate model according to theabove informationStep 3 Determine the error functionStep 4 Activate the ABC algorithm (see Section 2) toproduce the initial population 119909(119894 119895) by (1) Displace-ments are computed using (7) and (8)Step 5 The fitness of each solution is calculated by(10)Step 6 Generate the new population based on ABCalgorithm (see (2) and (3)) and compute the displace-ment (see (7) and (8))Step 7 Trace the best solution according to the ABCalgorithm

Table 1 Parameters of tunnel model

1199010(MPa) 119864 (MPa) 119888 (MPa) Φ (∘) 119901119894 (Mpa) 120595 (∘)300000 70000000 34500 300000 0 0

Table 2 Identified parameters using ABC-based back analysis

119864 (Mpa) 119888 (Mpa) 120593 (∘)ABC-based back analysis 689304951 35065 2999284Actual value 70000000 34500 300000Relative error () 15279 minus16377 00239

Step 8Repeat Steps 5 to 7 until finding the solution orreaching the maximum cycle

34 Verification The displacement of monitored point oftunnel can be computed by the above formula In this studysix monitored points were used in circular tunnel to monitorthe displacements at the horizontal direction for ABC searchThe distance between central of tunnel and 6 monitoredpoints is 10m 11m 13m 15m 17m and 21m respectively(see Figure ) The radius of tunnel is 1m The parameterof rock is listed in Table 1 ABC-based back analysis is usedto identify geomechanical parameters (eg Youngrsquos modulusE cohesion c and friction angle 120593) from displacements ofsix monitored points The recognized parameters and theirerror are listed in Table 2 The maximum relative error is16 It shows the recognized parameters agree well withthe real parameters The comparison between recognizedand real parameters about the displacement and stress ofsurrounding rock of tunnel is shown in Figures and Theresults show stresses and displacements of surrounding rockidentified by ABC are in well agreement with real stresses anddisplacements of surrounding rock and ABC is an excellentoptimization method The relationship between fitness andcycle is shown in Figure 3The relationship between identifiedparameters and cycle is shown in Figure 4They show that theperformance and convergence of ABC are good and quick foridentification of geomechanical parameters using ABC

341 Effect of Searching Range Theperformances of ABC aredemonstrated with different searching ranges (Table 3) Theresults of different searching ranges are shown in Figure 5To the smaller range the convergence is quicker than thebigger range But to the bigger range the fitness is the sameas the smaller range It shows ABC has strong capabilityof global searching and makes it possible to find the rockmass parameters in a big global space which enables theback analysis to be applied to more complex engineeringproblems

342 Effect of Population Size Population size is key param-eters of ABC To study the effect of the colony size on the


1 2 3 4 5 6

1m 13m

11m

15m 17m 21m

Figure 3 Position of monitored point in circular tunnel

0

02

04

06

08

1

12

1 2 3 4 5 6 7 8 9 10Distance to the center of tunnel (m)

Computed displacement using recognized parametersComputed displacement using actual parameters

Disp

lace

men

t (10

minus2

m)

Figure 4 The comparison of displacement between actual andrecognized parameters

convergence rate of theABC algorithm five different coloniesthat consisted of 20 50 100 200 and 400 bees were usedThefitness versus cycle numbers is shown in Figure 6 It can beseen that the convergence rates increasewith greater numbersof bees and population size of 200 or 400 bees is enough inthis study

4 Back Analysis Based on LSSVM and ABC

In the above section ABC-based back analysis was used tothe circular tunnel with analytical solution To the practicalengineering it is difficult to get the analytical solutionThe procedure with numerical solution is time-consumingRegression analysis is a good approach to build the rela-tion between geomechanical parameters and field moni-tored information In this study least square support vectormachine (LSSVM) was adopted to present the relationship

0

5

10

15

20

25

30

35

40

45

50

1 3 5 7 9

Stre

ss (M

Pa)

Distance to the center of tunnel (m)

Computed radial stress using recognized parametersComputed radial stress using actual parametersComputed tangential stress using recognized parametersComputed tangential stress using actual parameters

Figure 5 The comparison of stress between actual and recognizedparameters

0 200 400 600 800 1000

Fitn

ess

Cycle

100E minus 04

900E minus 05

800E minus 05

700E minus 05

600E minus 05

500E minus 05

400E minus 05

300E minus 05

200E minus 05

100E minus 05

000E + 00

Figure 6 Relationship between fitness value and cycle

between geomechanical parameters and displacement basedon numerical analysis

41 Least Square Support Vector Machine The least squaresupport vector machine (LSSVM) was originally developedby Suykens andVandewalle [21] Consider a given training setofN data points 119909119896 119910119896 (119896 = 1 2 119873)with input data xk isinRN and output yk isin r where RN is the N-dimensional vector


600000

620000

640000

660000

680000

700000

720000

0 200 400 600 800 1000

Fitn

ess

Cycle

E (MPa)

(a) 119864

200

250

300

350

400

450

500

550

0 200 400 600 800 1000

Fitn

ess

Cycle

c (MPa)

(b) 119888

2000

2200

2400

2600

2800

3000

3200

3400

0 200 400 600 800 1000

Fitn

ess

Cycle

120593 (∘)

(c) 120593

Figure 7 The variation of identified parameter with the cycle

space and r is the one-dimensional vector space Accordingto the LSSVM algorithm LSSVMmodel becomes

119910 (119909) =

119873

sum

119896=1

120572119896119870(119909 119909119896) + 119887 (11)

where 119870(119909 119909119896) is kernel functions and 120572 and b meet thefollowing equation

[

0 1119879

1 Ω + 120574minus1119868

] [

119887

120572] = [

0

119910] (12)

where 119910 = [1199101 119910119873] 1 = [1 1] 120572 = [1205721 120572119873]and Mercerrsquos theorem is applied within the Ω matrix

Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the

analytical solution of 120572 and b is given by

[

119887

120572] = Φ

minus1[

0

119910] (13)

42 Representation of Nonlinear Relationship LSSVM is usedin this study to map the nonlinear relationship betweengeomechanical parameters such as Youngrsquos modulus cohe-sion geostress coefficients and monitored displacements


0

000005

00001

000015

00002

000025

0 200 400 600 800 1000

Fitn

ess

Cycle

Range 1Range 2Range 3

Figure 8The performance of ABCusing different searching ranges

0

000005

00001

000015

00002

0 200 400 600 800 1000

Fitn

ess

Cycle

SN2 = 20

SN2 = 50

SN2 = 100

SN2 = 200

SN2 = 400

Figure 9 The convergence of different population size

The mathematical model of least square support vectormachine is defined as

LSSVM (X) 119877119899 997888rarr 119877

Y = LSSVM (X) X = (1199091 1199092 119909119899)

Y = (1199101 1199102 119910119899)

(14)

0 15

minus5 10

minus5 0 5 0

5 10

10MPa 20MPa

30∘

Failure criterion Mohr-Coulomb

Youngrsquos modulus E 20000MPa

Cohesion c 105MPa

Friction angle 120593 35∘

Poissonrsquos ratio 120583 02

Figure 10 The cross section of tunnel and parameters

Table 3 The ranges of identified parameters

Range 1 Range 2 Range 3119864 (Mpa) [2000 12000] [4000 1000] [5000 8000]119888 (Mpa) [05 7] [1 6] [3 7]120593 (∘) [5 60] [10 50] [20 40]

Table 4 Identified in situ stress and angle in different stages

1198751 1198752 AngleActual 200000 100000 300000Stage 1 199583 100614 300104Stage 2 206493 108171 333676Stage 3 200252 100376 30623

where 119909119894(119894 = 1 2 119899) is geomechanical parameters forexample Youngrsquos modulus friction angle geostress coeffi-cients and so forth and 119910119894(119894 = 1 2 119899) is displacementsof the key points

In order to obtain LSSVM(X) a training process basedon the known data set is needed Necessary training samplesare created in this work by using numerical analysis (egFEM model) which is used to obtain displacements of rockmass of key points corresponding to the given set of tentativegeomechanical parameters The geomechanical parametersare defined as input of LSSVM The displacement is definedas output of LSSVM


Table5Training

samples

andmod

elparameterso

fLSSVM

Num

bero

fsamples

1198751(M

pa)

1198752(M

pa)

120593(∘ )

Disp

lacement

120572

MP1

MP2

MP3

MP1119909

MP1119910

MP2119909

MP2119910

MP3119909

MP3119910

119909119910

119909119910

119909119910

110000

0500

0020000

0minus08380

minus13

600

15500

minus00231

minus20200

minus15

100

14473

20149

minus08992

minus03815

15989

22484

210000

075

000

25000

0minus04990

minus23300

13900

minus006

87minus16

700

minus15

800

16424

08880

minus09801

minus03294

16348

19749

310000

010000

030000

0000

00minus31300

1400

0minus14

400

1400

0minus14

400

21479

02439

minus09786

minus16

870

49088

21843

412500

012500

035000

0000

00minus39100

17500

minus18

000

minus17

500

minus18

000

20307

minus03980

minus05684

minus18

560

14959

17655

515000

015000

040000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

200

40minus10

849

minus02202

minus21514

1240

414

127

615000

0500

0025000

0minus200

00minus14

700

20800

08610

minus31900

minus27200

02187

18194

minus03108

05215

03286

09391

715000

075

000

30000

0minus16

800

minus25600

18300

01890

minus27700

minus28200

05089

06915

minus05137

minus0119

105530

07972

815000

010000

035000

0minus12

300

minus34700

1740

0minus05740

minus24200

minus27500

06722

006

83minus05353

minus05142

07871

07058

915000

012500

040000

0minus064

20minus41900

18300

minus13

800

minus21800

minus25200

10483

minus03389

minus05100

minus10

033

10326

08752

1015000

015000

020000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

22964

minus12

063

minus04593

minus24334

16207

16580

1120000

0500

0030000

0minus34100

minus19

500

22700

18500

minus42500

minus43300

minus09584

14147

minus01741

13821

minus05279

minus04169

1220000

075

000

35000

0minus30700

minus32100

19200

1100

0minus360

00minus43700

minus04940

02093

minus046

0505409

01538

minus03109

1320000

010000

040000

0minus25800

minus42600

1740

002750

minus31500

minus43100

minus01430

minus06938

minus06499

minus01060

05071

minus03365

1420000

012500

020000

0minus12

600

minus36100

30300

minus07560

minus37300

minus29900

09442

minus00545

04200

minus09120

00125

07845

1520000

015000

025000

0minus09990

minus46500

27900

minus13

700

minus33400

minus31500

12917

minus11019

01791

minus16

037

044

3806994

1625000

0500

0035000

0minus50300

minus28100

22000

29600

minus53200

minus62900

minus23159

07126

minus02344

23232

minus14

578

minus20741

1725000

075

000

40000

0minus45700

minus43400

17200

20100

minus42700

minus62200

minus17

211

minus08042

minus07054

12559

minus03011

minus18

612

1825000

010000

020000

0minus25600

minus25500

40000

06760

minus53900

minus38400

minus006

8108422

12115

02427

minus13

147

01347

1925000

012500

025000

0minus25800

minus38300

35200

02050

minus48900

minus42900

minus01085

minus02739

07151

minus01537

minus08165

minus02612

2025000

015000

030000

0minus23100

minus50100

32000

minus03910

minus44100

minus45200

02387

minus13

780

05252

minus08035

minus04328

minus040

6121

30000

0500

0040000

0minus70

100

minus42700

19200

42500

minus63900

minus85200

minus44142

minus07485

minus05555

37206

minus25680

minus440

6722

30000

075

000

20000

0minus41800

minus15

000

51000

21700

minus74

200

minus48300

minus16

564

19159

23915

16943

minus34243

minus08180

2330000

010000

025000

0minus43200

minus306

0043800

18900

minus65200

minus55400

minus16

996

03741

15582

1364

6minus23453

minus14

311

2430000

012500

030000

0minus41800

minus45200

37600

13100

minus58600

minus59500

minus15

641

minus08404

10366

08632

minus18

027

minus17

517

2530000

015000

035000

0minus38900

minus58500

33200

05890

minus52300

minus61900

minus13

480

minus22716

07182

01455

minus13

269

minus21053

119887mdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

minus24124

minus34816

25241

03809

minus37541

minus39253


Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers

Monitored displacement (mm)

00

50

100

150

00 50 100 150minus150 minus100 minus50

minus150

minus100

minus50

(a) Stage 1

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(b) Stage 2

Stage 1Stage 2Stage 3

00

50

100

150

00 50 100 150

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers

Monitored displacement (mm)minus150 minus100 minus50

minus150

minus100

minus50

(c) Stage 3

Figure 11 Comparison between monitored displacement and predicted displacement using identified parameters

43 Procedure of Back Analysis Algorithm Based on LSSVMand ABC After the LSSVM model representing the non-linear relation between the displacement and a parameteris obtained it can be used to predict displacements atmonitored points instead of numerical analysis ABC is usedto search the optimal parameter to be identified based on theerror function (see (10)) The back analysis technique basedon LSSVM-ABC combination can be described as follows

Step 1 Determine ABC parameters and the range ofparameters to be recognized

Step 2 Generate randomly 119899 group of parameters attheir given range Each individual represents an initialsolution

Step 3 Input a set of rock mass parameters to themodel LSSVM(X) obtained above to calculate thedisplacement values at given monitoring points

Step 4 Use (10) to evaluate the fitness of the currentindividuals that is the reasonability of the parameterset


0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX

(a) 120590119909 using theory parameters

0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX

(b) 120590119909 using identified parameters

Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001

(c) 120590119910 using theory parameters

Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001

(d) 120590119910 using identified parameters

Figure 12 Calculated stress comparison between using theory value and identified value at stage 3

Step 5 If all individuals are evaluated then go to Step6 Otherwise go to Step 3Step 6 If the maximum cycle is reached or the bestindividuals (the parameter to be back recognized)are obtained then the cycle ends and outputs bestindividuals Otherwise go to Step 7Step 7Update the individuals according to (2) and (3)Step 8 Repeat Step 7 until all 119899 new individuals aregenerated They are used as offspringStep 9 Go to Step 3

44 Verification To verify the model we suppose there isa tunnel (see Figure 7) The size of tunnel geomechanicalparameters and in situ stress are listed in Figure 7 The valuein Figure 7 is theoretical values Displacement values for somekey points indicated by nodes are calculated by elastic finiteelement method The suggested algorithm above is used toidentify initial geostress components P1 and P2 and anglebetween P1 and P2 We used orthogonal experiment design

to create 25 sets of tentative geostresses P1 and P2 and anglebetween P1 and P2 The training samples will be obtainedthrough computing the displacement of each set of tentativegeostresses Then the LSSVMmodel was build based on (13)The training samples and model parameters of LSSVM arelisted in Table 5 In situ stresses P1 and P2 and angle atdifferent stages can be identified according to the procedureof Section 43 Identified in situ stress P1 and P2and angleat different stages are listed in Table 4 The comparisonbetween displacement of the key points using the theoreticalparameters and displacements identified by back analysisbased on ABC and LSSVM is shown in Figure 8 Stresses ofsurrounding rock are shown in Figure 9 after stage 3 Resultsshow the proposed method can effectively identify the in situstress

45 Discussions

451 Performance of LSSVM The performance of LSSVM isvery important to back analysis The predicted displacement


00000

10000

20000

30000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus10000

minus20000

minus30000

minus40000

minus50000

(a) Stage 1

00000

20000

40000

60000

80000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus20000

minus60000

minus80000

minus100000

minus40000

MP4

-x

MP4

-y

MP5

-x

MP5

-y

(b) Stage 2

Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM

00000

50000

100000

150000

minus50000

minus100000

minus150000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

MP4

-x

MP4

-y

MP5

-x

MP5

-y

MP6

-x

MP6

-y

MP7

-x

MP7

-y

(c) Stage 3

Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters

using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM

452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial

5 Conclusions

The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the


0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1

Figure 14 Fitness with different parameters of kernel function

00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000

minus150000 minus100000 minus50000

Figure 15The performance of LSSVMwith different parameters ofkernel function

efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was also supported by the National Fund ofScience in China (no 41072224 51104057)

References

[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003

[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005

[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981

[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983

[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986

[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987

[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987

[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003

[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011

[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998

[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981

[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983

[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012

[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009

[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004

[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007


[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001

[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005

[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011

[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999

[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



geotechnical engineeringwithout analytical solution LSSVMwas used to present the relationship between geomechanicalparameters and displacement LSSVM model replaced thenumerical analysis to improve the efficiency of back analysisBack analysis based on LSSVM and ABC combination wasproposed in Section 4 LSSVM and the procedure of theproposed method were presented in brief Lastly someconclusion was listed in Section 5

2 Artificial Bee Colony Algorithms

The artificial bee colony (ABC) algorithm was originallydeveloped in 2005 by Karaboga [18] In ABC algorithmthe colony of artificial bees contains three groups of beesemployed bees onlookers and scouts Employed bees searchfor specific food sources (solution) and calculate the amountof nectars (fitness value) Onlooker bees choose a food sourcebased on the nectars shared by employed bees and determinethe source to be abandoned and allocate its employed beeas scout bees Scout bees randomly search for a new foodsource The position of a food source represents a possiblesolution for the problem under consideration and the nectaramount of a food source represents the quality of the solutionrepresented by the fitness value [19 20] To the minimumproblem the fitness can be computed by the target function

In the algorithm the first half of the colony consists ofemployed artificial bees and the second half constitutes theonlookersThe number of the employed bees or the onlookerbees is equal to the number of solutions in the population Atthe first step theABCgenerates a randomly distributed initialpopulation of SN solutions and calculates the fitness of eachsolution Consider

119909 (119894 119895) = 119909119895

min + rand (0 1) (119909119895max minus 119909119895

min) (1)

where 119909(119894 119895) is the candidate solution of problem 119894 =

1 2 1198781198732 and 1198781198732 denotes the size of population 119895 =1 2 119863 and 119863 is the dimension number of each solutionrand(01) is a random number between [0 1] 119909119894min and 119909

119894max

are the upper and lower bound of each solutionOnce initialization is completed the artificial bees are

used to conduct the search for the best food resource(solution) Procedures can be described as follows [20]

(i) Employed bees determine a food source within theneighborhood of the food source through their mem-ory

(ii) Employed bees share their information with onlook-ers within the hive and then the onlookers select oneof the food sources

(iii) Onlookers select a food source within the neighbor-hood of the food sources chosen by them to produceand exploit the new food resources

(iv) An employed bee of the sources that have beenabandoned by onlookers becomes a scout and startsto search for a new food source randomly

In the ABC algorithm a candidate food position can beproduced from the memory of bees which is defined as

V (119894 119895) = 119909 (119894 119895) + 120593119894119895 (119909 (119894 119895) minus 119909 (119896 119895)) (2)

where k used to be different from 119894 is randomly chosenindexes from 1 2 1198781198732 j is also randomly chosenindexes from 1 2 119863 and 120593ij is a random numberin [minus1 1] and controls the generation of neighbor foodsources around 119909(119894 119895) and represents the comparison oftwo food positions seen by a bee As can be seen from(2) the perturbation on the position 119909(119894 119895) decreases whenthe difference between the parameters of 119909(119894 119895) and 119909(119896 119895)decreases so that the step length is adaptively reduced

An artificial onlooker bee chooses a food source basedon the probability of food source The probability of beingselected for fitness pi can be expressed as

119901119894 =fitness119894

sum119878119873119899=1 fitness119899

(3)

where fitness119894 is the fitness of the solutionIn ABC algorithm a food source whose position cannot

be improved further through a predetermined number ofcycles is assumed to be abandoned by onlookers 119909(119894 119895) usedto represent the abandoned source is replaced with 1199091015840(119894 119895)that is a new food source the scout bees find which isconducted by (1)

Each candidate source position V(119894 119895) produced by 119909(119894 119895)can be evaluated using the comparison between 119909(119894 119895) and itsold source positionThe old food source will be replaced withthe new food source when it is equal to or better than the oldfood source Otherwise the old food source is retained in thememory

There are three control parameters in the ABC thenumber of food sources which is equal to the number ofemployed or onlooker bees (SN2) the value of limit and themaximum cycle number (MCN) The following is the briefprocedure of artificial bee colony (ABC) algorithm

Step 1 Determine the value of control parametersSN2 MCN and ldquolimitrdquo of ABC algorithmStep 2 Generate the initial population 119909(119894 119895) by (1)and evaluate the fitness of each solutionStep 3 Produce new solution V(119894 119895) for each employedbee by using (2) In the meantime the fitness isevaluatedStep 4 Calculate the probability 119901119894 for the solution119909(119894 119895) by (3)Step 5 Select a solution 119909(119894 119895) for each onlookerbee according to 119901119894 Then a new solution V(119894 119895) isgenerated by (2)Step 6 Calculate the fitnessStep 7 If there is an abandoned solution for the scoutit will be replaced by using a new solution which israndomly produced by (2)Step 8 Trace the best solutionStep 9 Repeat Steps 3 to 8 until the cycle reaches themaximum cycle number (MCN)


p0

p0

pi

rP

r0

PlasticElastic






119901119888119903 =2119901119900 minus 120590119888

119896 + 1

119896 =

1 + sin1205931 minus sin120593

120590119888 =119888 (119896 minus 1)

tan120593

(4)


119877 = 119903119900 lowast [2 (119901119900 + 119904)

(119896 + 1) (119901119894 + 119904)]

1(119896minus1)

(5)

in which

119904 =

120590119888

119896 minus 1

(6)



Elastic zone

119906119903 =

(119901119900 sin120593 + 119888 sdot cos120593) (1198772119903)

2119866

(7)

Plastic zone

119906119903 =119903

2119866

sdot 120594 (8)


120594 = (2120583 minus 1) (119901119900 + 119888 sdot ctg120593)

+ (1 minus 120583) [(1198702119901 minus 1) (119870119901 + 119870119901119904)]

times (119901119894 + 119888 sdot ctg120593) (119877

119903119900

)

(119870119901minus1)

(

119877

119903

)

(119870119901119904+1)

+ [

(1 minus 120583) (119870119901119870119901119904 + 1)

(119870119901 + 119870119901119904)

sdot 120583]

times (119901119894 + 119888 sdot ctg120593) (119903

119903119900

)

(119870119901minus1)

119870119901119904 =(1 + sin120595119904)(1 minus sin120595119904)

119866 =

119864

2 (1 + 120583)

(9)



2

119899

(10)





Start







Maximum cycle meets


End

No

Yes




1199010(MPa) 119864 (MPa) 119888 (MPa) Φ (∘) 119901119894 (Mpa) 120595 (∘)300000 70000000 34500 300000 0 0








1 2 3 4 5 6

1m 13m

11m

15m 17m 21m


0

02

04

06

08

1

12



Disp

lace

men

t (10

minus2

m)





0

5

10

15

20

25

30

35

40

45

50

1 3 5 7 9

Stre

ss (M

Pa)




0 200 400 600 800 1000

Fitn

ess

Cycle

100E minus 04

900E minus 05

800E minus 05

700E minus 05

600E minus 05

500E minus 05

400E minus 05

300E minus 05

200E minus 05

100E minus 05

000E + 00





600000

620000

640000

660000

680000

700000

720000

0 200 400 600 800 1000

Fitn

ess

Cycle

E (MPa)

(a) 119864

200

250

300

350

400

450

500

550

0 200 400 600 800 1000

Fitn

ess

Cycle

c (MPa)

(b) 119888

2000

2200

2400

2600

2800

3000

3200

3400

0 200 400 600 800 1000

Fitn

ess

Cycle

120593 (∘)

(c) 120593



119910 (119909) =

119873

sum

119896=1

120572119896119870(119909 119909119896) + 119887 (11)


[

0 1119879

1 Ω + 120574minus1119868

] [

119887

120572] = [

0

119910] (12)


Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the


[

119887

120572] = Φ

minus1[

0

119910] (13)



0

000005

00001

000015

00002

000025

0 200 400 600 800 1000

Fitn

ess

Cycle



0

000005

00001

000015

00002

0 200 400 600 800 1000

Fitn

ess

Cycle

SN2 = 20

SN2 = 50

SN2 = 100

SN2 = 200

SN2 = 400



LSSVM (X) 119877119899 997888rarr 119877

Y = LSSVM (X) X = (1199091 1199092 119909119899)

Y = (1199101 1199102 119910119899)

(14)

0 15

minus5 10

minus5 0 5 0

5 10

10MPa 20MPa

30∘



Cohesion c 105MPa











Table5Training

samples

andmod

elparameterso

fLSSVM

Num

bero

fsamples

1198751(M

pa)

1198752(M

pa)

120593(∘ )

Disp

lacement

120572

MP1

MP2

MP3

MP1119909

MP1119910

MP2119909

MP2119910

MP3119909

MP3119910

119909119910

119909119910

119909119910

110000

0500

0020000

0minus08380

minus13

600

15500

minus00231

minus20200

minus15

100

14473

20149

minus08992

minus03815

15989

22484

210000

075

000

25000

0minus04990

minus23300

13900

minus006

87minus16

700

minus15

800

16424

08880

minus09801

minus03294

16348

19749

310000

010000

030000

0000

00minus31300

1400

0minus14

400

1400

0minus14

400

21479

02439

minus09786

minus16

870

49088

21843

412500

012500

035000

0000

00minus39100

17500

minus18

000

minus17

500

minus18

000

20307

minus03980

minus05684

minus18

560

14959

17655

515000

015000

040000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

200

40minus10

849

minus02202

minus21514

1240

414

127

615000

0500

0025000

0minus200

00minus14

700

20800

08610

minus31900

minus27200

02187

18194

minus03108

05215

03286

09391

715000

075

000

30000

0minus16

800

minus25600

18300

01890

minus27700

minus28200

05089

06915

minus05137

minus0119

105530

07972

815000

010000

035000

0minus12

300

minus34700

1740

0minus05740

minus24200

minus27500

06722

006

83minus05353

minus05142

07871

07058

915000

012500

040000

0minus064

20minus41900

18300

minus13

800

minus21800

minus25200

10483

minus03389

minus05100

minus10

033

10326

08752

1015000

015000

020000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

22964

minus12

063

minus04593

minus24334

16207

16580

1120000

0500

0030000

0minus34100

minus19

500

22700

18500

minus42500

minus43300

minus09584

14147

minus01741

13821

minus05279

minus04169

1220000

075

000

35000

0minus30700

minus32100

19200

1100

0minus360

00minus43700

minus04940

02093

minus046

0505409

01538

minus03109

1320000

010000

040000

0minus25800

minus42600

1740

002750

minus31500

minus43100

minus01430

minus06938

minus06499

minus01060

05071

minus03365

1420000

012500

020000

0minus12

600

minus36100

30300

minus07560

minus37300

minus29900

09442

minus00545

04200

minus09120

00125

07845

1520000

015000

025000

0minus09990

minus46500

27900

minus13

700

minus33400

minus31500

12917

minus11019

01791

minus16

037

044

3806994

1625000

0500

0035000

0minus50300

minus28100

22000

29600

minus53200

minus62900

minus23159

07126

minus02344

23232

minus14

578

minus20741

1725000

075

000

40000

0minus45700

minus43400

17200

20100

minus42700

minus62200

minus17

211

minus08042

minus07054

12559

minus03011

minus18

612

1825000

010000

020000

0minus25600

minus25500

40000

06760

minus53900

minus38400

minus006

8108422

12115

02427

minus13

147

01347

1925000

012500

025000

0minus25800

minus38300

35200

02050

minus48900

minus42900

minus01085

minus02739

07151

minus01537

minus08165

minus02612

2025000

015000

030000

0minus23100

minus50100

32000

minus03910

minus44100

minus45200

02387

minus13

780

05252

minus08035

minus04328

minus040

6121

30000

0500

0040000

0minus70

100

minus42700

19200

42500

minus63900

minus85200

minus44142

minus07485

minus05555

37206

minus25680

minus440

6722

30000

075

000

20000

0minus41800

minus15

000

51000

21700

minus74

200

minus48300

minus16

564

19159

23915

16943

minus34243

minus08180

2330000

010000

025000

0minus43200

minus306

0043800

18900

minus65200

minus55400

minus16

996

03741

15582

1364

6minus23453

minus14

311

2430000

012500

030000

0minus41800

minus45200

37600

13100

minus58600

minus59500

minus15

641

minus08404

10366

08632

minus18

027

minus17

517

2530000

015000

035000

0minus38900

minus58500

33200

05890

minus52300

minus61900

minus13

480

minus22716

07182

01455

minus13

269

minus21053

119887mdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

minus24124

minus34816

25241

03809

minus37541

minus39253


Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(a) Stage 1

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(b) Stage 2


00

50

100

150

00 50 100 150

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


minus150

minus100

minus50

(c) Stage 3








0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001






45 Discussions



00000

10000

20000

30000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus10000

minus20000

minus30000

minus40000

minus50000

(a) Stage 1

00000

20000

40000

60000

80000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus20000

minus60000

minus80000

minus100000

minus40000

MP4

-x

MP4

-y

MP5

-x

MP5

-y

(b) Stage 2


00000

50000

100000

150000

minus50000

minus100000

minus150000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

MP4

-x

MP4

-y

MP5

-x

MP5

-y

MP6

-x

MP6

-y

MP7

-x

MP7

-y

(c) Stage 3




5 Conclusions



0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1


00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000






Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










p0

p0

pi

rP

r0

PlasticElastic






119901119888119903 =2119901119900 minus 120590119888

119896 + 1

119896 =

1 + sin1205931 minus sin120593

120590119888 =119888 (119896 minus 1)

tan120593

(4)


119877 = 119903119900 lowast [2 (119901119900 + 119904)

(119896 + 1) (119901119894 + 119904)]

1(119896minus1)

(5)

in which

119904 =

120590119888

119896 minus 1

(6)



Elastic zone

119906119903 =

(119901119900 sin120593 + 119888 sdot cos120593) (1198772119903)

2119866

(7)

Plastic zone

119906119903 =119903

2119866

sdot 120594 (8)


120594 = (2120583 minus 1) (119901119900 + 119888 sdot ctg120593)

+ (1 minus 120583) [(1198702119901 minus 1) (119870119901 + 119870119901119904)]

times (119901119894 + 119888 sdot ctg120593) (119877

119903119900

)

(119870119901minus1)

(

119877

119903

)

(119870119901119904+1)

+ [

(1 minus 120583) (119870119901119870119901119904 + 1)

(119870119901 + 119870119901119904)

sdot 120583]

times (119901119894 + 119888 sdot ctg120593) (119903

119903119900

)

(119870119901minus1)

119870119901119904 =(1 + sin120595119904)(1 minus sin120595119904)

119866 =

119864

2 (1 + 120583)

(9)



2

119899

(10)





Start







Maximum cycle meets


End

No

Yes




1199010(MPa) 119864 (MPa) 119888 (MPa) Φ (∘) 119901119894 (Mpa) 120595 (∘)300000 70000000 34500 300000 0 0








1 2 3 4 5 6

1m 13m

11m

15m 17m 21m


0

02

04

06

08

1

12



Disp

lace

men

t (10

minus2

m)





0

5

10

15

20

25

30

35

40

45

50

1 3 5 7 9

Stre

ss (M

Pa)




0 200 400 600 800 1000

Fitn

ess

Cycle

100E minus 04

900E minus 05

800E minus 05

700E minus 05

600E minus 05

500E minus 05

400E minus 05

300E minus 05

200E minus 05

100E minus 05

000E + 00





600000

620000

640000

660000

680000

700000

720000

0 200 400 600 800 1000

Fitn

ess

Cycle

E (MPa)

(a) 119864

200

250

300

350

400

450

500

550

0 200 400 600 800 1000

Fitn

ess

Cycle

c (MPa)

(b) 119888

2000

2200

2400

2600

2800

3000

3200

3400

0 200 400 600 800 1000

Fitn

ess

Cycle

120593 (∘)

(c) 120593



119910 (119909) =

119873

sum

119896=1

120572119896119870(119909 119909119896) + 119887 (11)


[

0 1119879

1 Ω + 120574minus1119868

] [

119887

120572] = [

0

119910] (12)


Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the


[

119887

120572] = Φ

minus1[

0

119910] (13)



0

000005

00001

000015

00002

000025

0 200 400 600 800 1000

Fitn

ess

Cycle



0

000005

00001

000015

00002

0 200 400 600 800 1000

Fitn

ess

Cycle

SN2 = 20

SN2 = 50

SN2 = 100

SN2 = 200

SN2 = 400



LSSVM (X) 119877119899 997888rarr 119877

Y = LSSVM (X) X = (1199091 1199092 119909119899)

Y = (1199101 1199102 119910119899)

(14)

0 15

minus5 10

minus5 0 5 0

5 10

10MPa 20MPa

30∘



Cohesion c 105MPa











Table5Training

samples

andmod

elparameterso

fLSSVM

Num

bero

fsamples

1198751(M

pa)

1198752(M

pa)

120593(∘ )

Disp

lacement

120572

MP1

MP2

MP3

MP1119909

MP1119910

MP2119909

MP2119910

MP3119909

MP3119910

119909119910

119909119910

119909119910

110000

0500

0020000

0minus08380

minus13

600

15500

minus00231

minus20200

minus15

100

14473

20149

minus08992

minus03815

15989

22484

210000

075

000

25000

0minus04990

minus23300

13900

minus006

87minus16

700

minus15

800

16424

08880

minus09801

minus03294

16348

19749

310000

010000

030000

0000

00minus31300

1400

0minus14

400

1400

0minus14

400

21479

02439

minus09786

minus16

870

49088

21843

412500

012500

035000

0000

00minus39100

17500

minus18

000

minus17

500

minus18

000

20307

minus03980

minus05684

minus18

560

14959

17655

515000

015000

040000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

200

40minus10

849

minus02202

minus21514

1240

414

127

615000

0500

0025000

0minus200

00minus14

700

20800

08610

minus31900

minus27200

02187

18194

minus03108

05215

03286

09391

715000

075

000

30000

0minus16

800

minus25600

18300

01890

minus27700

minus28200

05089

06915

minus05137

minus0119

105530

07972

815000

010000

035000

0minus12

300

minus34700

1740

0minus05740

minus24200

minus27500

06722

006

83minus05353

minus05142

07871

07058

915000

012500

040000

0minus064

20minus41900

18300

minus13

800

minus21800

minus25200

10483

minus03389

minus05100

minus10

033

10326

08752

1015000

015000

020000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

22964

minus12

063

minus04593

minus24334

16207

16580

1120000

0500

0030000

0minus34100

minus19

500

22700

18500

minus42500

minus43300

minus09584

14147

minus01741

13821

minus05279

minus04169

1220000

075

000

35000

0minus30700

minus32100

19200

1100

0minus360

00minus43700

minus04940

02093

minus046

0505409

01538

minus03109

1320000

010000

040000

0minus25800

minus42600

1740

002750

minus31500

minus43100

minus01430

minus06938

minus06499

minus01060

05071

minus03365

1420000

012500

020000

0minus12

600

minus36100

30300

minus07560

minus37300

minus29900

09442

minus00545

04200

minus09120

00125

07845

1520000

015000

025000

0minus09990

minus46500

27900

minus13

700

minus33400

minus31500

12917

minus11019

01791

minus16

037

044

3806994

1625000

0500

0035000

0minus50300

minus28100

22000

29600

minus53200

minus62900

minus23159

07126

minus02344

23232

minus14

578

minus20741

1725000

075

000

40000

0minus45700

minus43400

17200

20100

minus42700

minus62200

minus17

211

minus08042

minus07054

12559

minus03011

minus18

612

1825000

010000

020000

0minus25600

minus25500

40000

06760

minus53900

minus38400

minus006

8108422

12115

02427

minus13

147

01347

1925000

012500

025000

0minus25800

minus38300

35200

02050

minus48900

minus42900

minus01085

minus02739

07151

minus01537

minus08165

minus02612

2025000

015000

030000

0minus23100

minus50100

32000

minus03910

minus44100

minus45200

02387

minus13

780

05252

minus08035

minus04328

minus040

6121

30000

0500

0040000

0minus70

100

minus42700

19200

42500

minus63900

minus85200

minus44142

minus07485

minus05555

37206

minus25680

minus440

6722

30000

075

000

20000

0minus41800

minus15

000

51000

21700

minus74

200

minus48300

minus16

564

19159

23915

16943

minus34243

minus08180

2330000

010000

025000

0minus43200

minus306

0043800

18900

minus65200

minus55400

minus16

996

03741

15582

1364

6minus23453

minus14

311

2430000

012500

030000

0minus41800

minus45200

37600

13100

minus58600

minus59500

minus15

641

minus08404

10366

08632

minus18

027

minus17

517

2530000

015000

035000

0minus38900

minus58500

33200

05890

minus52300

minus61900

minus13

480

minus22716

07182

01455

minus13

269

minus21053

119887mdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

minus24124

minus34816

25241

03809

minus37541

minus39253


Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(a) Stage 1

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(b) Stage 2


00

50

100

150

00 50 100 150

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


minus150

minus100

minus50

(c) Stage 3








0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001






45 Discussions



00000

10000

20000

30000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus10000

minus20000

minus30000

minus40000

minus50000

(a) Stage 1

00000

20000

40000

60000

80000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus20000

minus60000

minus80000

minus100000

minus40000

MP4

-x

MP4

-y

MP5

-x

MP5

-y

(b) Stage 2


00000

50000

100000

150000

minus50000

minus100000

minus150000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

MP4

-x

MP4

-y

MP5

-x

MP5

-y

MP6

-x

MP6

-y

MP7

-x

MP7

-y

(c) Stage 3




5 Conclusions



0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1


00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000






Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Start







Maximum cycle meets


End

No

Yes




1199010(MPa) 119864 (MPa) 119888 (MPa) Φ (∘) 119901119894 (Mpa) 120595 (∘)300000 70000000 34500 300000 0 0








1 2 3 4 5 6

1m 13m

11m

15m 17m 21m


0

02

04

06

08

1

12



Disp

lace

men

t (10

minus2

m)





0

5

10

15

20

25

30

35

40

45

50

1 3 5 7 9

Stre

ss (M

Pa)




0 200 400 600 800 1000

Fitn

ess

Cycle

100E minus 04

900E minus 05

800E minus 05

700E minus 05

600E minus 05

500E minus 05

400E minus 05

300E minus 05

200E minus 05

100E minus 05

000E + 00





600000

620000

640000

660000

680000

700000

720000

0 200 400 600 800 1000

Fitn

ess

Cycle

E (MPa)

(a) 119864

200

250

300

350

400

450

500

550

0 200 400 600 800 1000

Fitn

ess

Cycle

c (MPa)

(b) 119888

2000

2200

2400

2600

2800

3000

3200

3400

0 200 400 600 800 1000

Fitn

ess

Cycle

120593 (∘)

(c) 120593



119910 (119909) =

119873

sum

119896=1

120572119896119870(119909 119909119896) + 119887 (11)


[

0 1119879

1 Ω + 120574minus1119868

] [

119887

120572] = [

0

119910] (12)


Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the


[

119887

120572] = Φ

minus1[

0

119910] (13)



0

000005

00001

000015

00002

000025

0 200 400 600 800 1000

Fitn

ess

Cycle



0

000005

00001

000015

00002

0 200 400 600 800 1000

Fitn

ess

Cycle

SN2 = 20

SN2 = 50

SN2 = 100

SN2 = 200

SN2 = 400



LSSVM (X) 119877119899 997888rarr 119877

Y = LSSVM (X) X = (1199091 1199092 119909119899)

Y = (1199101 1199102 119910119899)

(14)

0 15

minus5 10

minus5 0 5 0

5 10

10MPa 20MPa

30∘



Cohesion c 105MPa











Table5Training

samples

andmod

elparameterso

fLSSVM

Num

bero

fsamples

1198751(M

pa)

1198752(M

pa)

120593(∘ )

Disp

lacement

120572

MP1

MP2

MP3

MP1119909

MP1119910

MP2119909

MP2119910

MP3119909

MP3119910

119909119910

119909119910

119909119910

110000

0500

0020000

0minus08380

minus13

600

15500

minus00231

minus20200

minus15

100

14473

20149

minus08992

minus03815

15989

22484

210000

075

000

25000

0minus04990

minus23300

13900

minus006

87minus16

700

minus15

800

16424

08880

minus09801

minus03294

16348

19749

310000

010000

030000

0000

00minus31300

1400

0minus14

400

1400

0minus14

400

21479

02439

minus09786

minus16

870

49088

21843

412500

012500

035000

0000

00minus39100

17500

minus18

000

minus17

500

minus18

000

20307

minus03980

minus05684

minus18

560

14959

17655

515000

015000

040000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

200

40minus10

849

minus02202

minus21514

1240

414

127

615000

0500

0025000

0minus200

00minus14

700

20800

08610

minus31900

minus27200

02187

18194

minus03108

05215

03286

09391

715000

075

000

30000

0minus16

800

minus25600

18300

01890

minus27700

minus28200

05089

06915

minus05137

minus0119

105530

07972

815000

010000

035000

0minus12

300

minus34700

1740

0minus05740

minus24200

minus27500

06722

006

83minus05353

minus05142

07871

07058

915000

012500

040000

0minus064

20minus41900

18300

minus13

800

minus21800

minus25200

10483

minus03389

minus05100

minus10

033

10326

08752

1015000

015000

020000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

22964

minus12

063

minus04593

minus24334

16207

16580

1120000

0500

0030000

0minus34100

minus19

500

22700

18500

minus42500

minus43300

minus09584

14147

minus01741

13821

minus05279

minus04169

1220000

075

000

35000

0minus30700

minus32100

19200

1100

0minus360

00minus43700

minus04940

02093

minus046

0505409

01538

minus03109

1320000

010000

040000

0minus25800

minus42600

1740

002750

minus31500

minus43100

minus01430

minus06938

minus06499

minus01060

05071

minus03365

1420000

012500

020000

0minus12

600

minus36100

30300

minus07560

minus37300

minus29900

09442

minus00545

04200

minus09120

00125

07845

1520000

015000

025000

0minus09990

minus46500

27900

minus13

700

minus33400

minus31500

12917

minus11019

01791

minus16

037

044

3806994

1625000

0500

0035000

0minus50300

minus28100

22000

29600

minus53200

minus62900

minus23159

07126

minus02344

23232

minus14

578

minus20741

1725000

075

000

40000

0minus45700

minus43400

17200

20100

minus42700

minus62200

minus17

211

minus08042

minus07054

12559

minus03011

minus18

612

1825000

010000

020000

0minus25600

minus25500

40000

06760

minus53900

minus38400

minus006

8108422

12115

02427

minus13

147

01347

1925000

012500

025000

0minus25800

minus38300

35200

02050

minus48900

minus42900

minus01085

minus02739

07151

minus01537

minus08165

minus02612

2025000

015000

030000

0minus23100

minus50100

32000

minus03910

minus44100

minus45200

02387

minus13

780

05252

minus08035

minus04328

minus040

6121

30000

0500

0040000

0minus70

100

minus42700

19200

42500

minus63900

minus85200

minus44142

minus07485

minus05555

37206

minus25680

minus440

6722

30000

075

000

20000

0minus41800

minus15

000

51000

21700

minus74

200

minus48300

minus16

564

19159

23915

16943

minus34243

minus08180

2330000

010000

025000

0minus43200

minus306

0043800

18900

minus65200

minus55400

minus16

996

03741

15582

1364

6minus23453

minus14

311

2430000

012500

030000

0minus41800

minus45200

37600

13100

minus58600

minus59500

minus15

641

minus08404

10366

08632

minus18

027

minus17

517

2530000

015000

035000

0minus38900

minus58500

33200

05890

minus52300

minus61900

minus13

480

minus22716

07182

01455

minus13

269

minus21053

119887mdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

minus24124

minus34816

25241

03809

minus37541

minus39253


Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(a) Stage 1

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(b) Stage 2


00

50

100

150

00 50 100 150

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


minus150

minus100

minus50

(c) Stage 3








0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001






45 Discussions



00000

10000

20000

30000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus10000

minus20000

minus30000

minus40000

minus50000

(a) Stage 1

00000

20000

40000

60000

80000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus20000

minus60000

minus80000

minus100000

minus40000

MP4

-x

MP4

-y

MP5

-x

MP5

-y

(b) Stage 2


00000

50000

100000

150000

minus50000

minus100000

minus150000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

MP4

-x

MP4

-y

MP5

-x

MP5

-y

MP6

-x

MP6

-y

MP7

-x

MP7

-y

(c) Stage 3




5 Conclusions



0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1


00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000






Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3 4 5 6

1m 13m

11m

15m 17m 21m


0

02

04

06

08

1

12



Disp

lace

men

t (10

minus2

m)





0

5

10

15

20

25

30

35

40

45

50

1 3 5 7 9

Stre

ss (M

Pa)




0 200 400 600 800 1000

Fitn

ess

Cycle

100E minus 04

900E minus 05

800E minus 05

700E minus 05

600E minus 05

500E minus 05

400E minus 05

300E minus 05

200E minus 05

100E minus 05

000E + 00





600000

620000

640000

660000

680000

700000

720000

0 200 400 600 800 1000

Fitn

ess

Cycle

E (MPa)

(a) 119864

200

250

300

350

400

450

500

550

0 200 400 600 800 1000

Fitn

ess

Cycle

c (MPa)

(b) 119888

2000

2200

2400

2600

2800

3000

3200

3400

0 200 400 600 800 1000

Fitn

ess

Cycle

120593 (∘)

(c) 120593



119910 (119909) =

119873

sum

119896=1

120572119896119870(119909 119909119896) + 119887 (11)


[

0 1119879

1 Ω + 120574minus1119868

] [

119887

120572] = [

0

119910] (12)


Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the


[

119887

120572] = Φ

minus1[

0

119910] (13)



0

000005

00001

000015

00002

000025

0 200 400 600 800 1000

Fitn

ess

Cycle



0

000005

00001

000015

00002

0 200 400 600 800 1000

Fitn

ess

Cycle

SN2 = 20

SN2 = 50

SN2 = 100

SN2 = 200

SN2 = 400



LSSVM (X) 119877119899 997888rarr 119877

Y = LSSVM (X) X = (1199091 1199092 119909119899)

Y = (1199101 1199102 119910119899)

(14)

0 15

minus5 10

minus5 0 5 0

5 10

10MPa 20MPa

30∘



Cohesion c 105MPa











Table5Training

samples

andmod

elparameterso

fLSSVM

Num

bero

fsamples

1198751(M

pa)

1198752(M

pa)

120593(∘ )

Disp

lacement

120572

MP1

MP2

MP3

MP1119909

MP1119910

MP2119909

MP2119910

MP3119909

MP3119910

119909119910

119909119910

119909119910

110000

0500

0020000

0minus08380

minus13

600

15500

minus00231

minus20200

minus15

100

14473

20149

minus08992

minus03815

15989

22484

210000

075

000

25000

0minus04990

minus23300

13900

minus006

87minus16

700

minus15

800

16424

08880

minus09801

minus03294

16348

19749

310000

010000

030000

0000

00minus31300

1400

0minus14

400

1400

0minus14

400

21479

02439

minus09786

minus16

870

49088

21843

412500

012500

035000

0000

00minus39100

17500

minus18

000

minus17

500

minus18

000

20307

minus03980

minus05684

minus18

560

14959

17655

515000

015000

040000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

200

40minus10

849

minus02202

minus21514

1240

414

127

615000

0500

0025000

0minus200

00minus14

700

20800

08610

minus31900

minus27200

02187

18194

minus03108

05215

03286

09391

715000

075

000

30000

0minus16

800

minus25600

18300

01890

minus27700

minus28200

05089

06915

minus05137

minus0119

105530

07972

815000

010000

035000

0minus12

300

minus34700

1740

0minus05740

minus24200

minus27500

06722

006

83minus05353

minus05142

07871

07058

915000

012500

040000

0minus064

20minus41900

18300

minus13

800

minus21800

minus25200

10483

minus03389

minus05100

minus10

033

10326

08752

1015000

015000

020000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

22964

minus12

063

minus04593

minus24334

16207

16580

1120000

0500

0030000

0minus34100

minus19

500

22700

18500

minus42500

minus43300

minus09584

14147

minus01741

13821

minus05279

minus04169

1220000

075

000

35000

0minus30700

minus32100

19200

1100

0minus360

00minus43700

minus04940

02093

minus046

0505409

01538

minus03109

1320000

010000

040000

0minus25800

minus42600

1740

002750

minus31500

minus43100

minus01430

minus06938

minus06499

minus01060

05071

minus03365

1420000

012500

020000

0minus12

600

minus36100

30300

minus07560

minus37300

minus29900

09442

minus00545

04200

minus09120

00125

07845

1520000

015000

025000

0minus09990

minus46500

27900

minus13

700

minus33400

minus31500

12917

minus11019

01791

minus16

037

044

3806994

1625000

0500

0035000

0minus50300

minus28100

22000

29600

minus53200

minus62900

minus23159

07126

minus02344

23232

minus14

578

minus20741

1725000

075

000

40000

0minus45700

minus43400

17200

20100

minus42700

minus62200

minus17

211

minus08042

minus07054

12559

minus03011

minus18

612

1825000

010000

020000

0minus25600

minus25500

40000

06760

minus53900

minus38400

minus006

8108422

12115

02427

minus13

147

01347

1925000

012500

025000

0minus25800

minus38300

35200

02050

minus48900

minus42900

minus01085

minus02739

07151

minus01537

minus08165

minus02612

2025000

015000

030000

0minus23100

minus50100

32000

minus03910

minus44100

minus45200

02387

minus13

780

05252

minus08035

minus04328

minus040

6121

30000

0500

0040000

0minus70

100

minus42700

19200

42500

minus63900

minus85200

minus44142

minus07485

minus05555

37206

minus25680

minus440

6722

30000

075

000

20000

0minus41800

minus15

000

51000

21700

minus74

200

minus48300

minus16

564

19159

23915

16943

minus34243

minus08180

2330000

010000

025000

0minus43200

minus306

0043800

18900

minus65200

minus55400

minus16

996

03741

15582

1364

6minus23453

minus14

311

2430000

012500

030000

0minus41800

minus45200

37600

13100

minus58600

minus59500

minus15

641

minus08404

10366

08632

minus18

027

minus17

517

2530000

015000

035000

0minus38900

minus58500

33200

05890

minus52300

minus61900

minus13

480

minus22716

07182

01455

minus13

269

minus21053

119887mdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

minus24124

minus34816

25241

03809

minus37541

minus39253


Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(a) Stage 1

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(b) Stage 2


00

50

100

150

00 50 100 150

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


minus150

minus100

minus50

(c) Stage 3








0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001






45 Discussions



00000

10000

20000

30000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus10000

minus20000

minus30000

minus40000

minus50000

(a) Stage 1

00000

20000

40000

60000

80000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus20000

minus60000

minus80000

minus100000

minus40000

MP4

-x

MP4

-y

MP5

-x

MP5

-y

(b) Stage 2


00000

50000

100000

150000

minus50000

minus100000

minus150000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

MP4

-x

MP4

-y

MP5

-x

MP5

-y

MP6

-x

MP6

-y

MP7

-x

MP7

-y

(c) Stage 3




5 Conclusions



0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1


00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000






Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










600000

620000

640000

660000

680000

700000

720000

0 200 400 600 800 1000

Fitn

ess

Cycle

E (MPa)

(a) 119864

200

250

300

350

400

450

500

550

0 200 400 600 800 1000

Fitn

ess

Cycle

c (MPa)

(b) 119888

2000

2200

2400

2600

2800

3000

3200

3400

0 200 400 600 800 1000

Fitn

ess

Cycle

120593 (∘)

(c) 120593



119910 (119909) =

119873

sum

119896=1

120572119896119870(119909 119909119896) + 119887 (11)


[

0 1119879

1 Ω + 120574minus1119868

] [

119887

120572] = [

0

119910] (12)


Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the


[

119887

120572] = Φ

minus1[

0

119910] (13)



0

000005

00001

000015

00002

000025

0 200 400 600 800 1000

Fitn

ess

Cycle



0

000005

00001

000015

00002

0 200 400 600 800 1000

Fitn

ess

Cycle

SN2 = 20

SN2 = 50

SN2 = 100

SN2 = 200

SN2 = 400



LSSVM (X) 119877119899 997888rarr 119877

Y = LSSVM (X) X = (1199091 1199092 119909119899)

Y = (1199101 1199102 119910119899)

(14)

0 15

minus5 10

minus5 0 5 0

5 10

10MPa 20MPa

30∘



Cohesion c 105MPa











Table5Training

samples

andmod

elparameterso

fLSSVM

Num

bero

fsamples

1198751(M

pa)

1198752(M

pa)

120593(∘ )

Disp

lacement

120572

MP1

MP2

MP3

MP1119909

MP1119910

MP2119909

MP2119910

MP3119909

MP3119910

119909119910

119909119910

119909119910

110000

0500

0020000

0minus08380

minus13

600

15500

minus00231

minus20200

minus15

100

14473

20149

minus08992

minus03815

15989

22484

210000

075

000

25000

0minus04990

minus23300

13900

minus006

87minus16

700

minus15

800

16424

08880

minus09801

minus03294

16348

19749

310000

010000

030000

0000

00minus31300

1400

0minus14

400

1400

0minus14

400

21479

02439

minus09786

minus16

870

49088

21843

412500

012500

035000

0000

00minus39100

17500

minus18

000

minus17

500

minus18

000

20307

minus03980

minus05684

minus18

560

14959

17655

515000

015000

040000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

200

40minus10

849

minus02202

minus21514

1240

414

127

615000

0500

0025000

0minus200

00minus14

700

20800

08610

minus31900

minus27200

02187

18194

minus03108

05215

03286

09391

715000

075

000

30000

0minus16

800

minus25600

18300

01890

minus27700

minus28200

05089

06915

minus05137

minus0119

105530

07972

815000

010000

035000

0minus12

300

minus34700

1740

0minus05740

minus24200

minus27500

06722

006

83minus05353

minus05142

07871

07058

915000

012500

040000

0minus064

20minus41900

18300

minus13

800

minus21800

minus25200

10483

minus03389

minus05100

minus10

033

10326

08752

1015000

015000

020000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

22964

minus12

063

minus04593

minus24334

16207

16580

1120000

0500

0030000

0minus34100

minus19

500

22700

18500

minus42500

minus43300

minus09584

14147

minus01741

13821

minus05279

minus04169

1220000

075

000

35000

0minus30700

minus32100

19200

1100

0minus360

00minus43700

minus04940

02093

minus046

0505409

01538

minus03109

1320000

010000

040000

0minus25800

minus42600

1740

002750

minus31500

minus43100

minus01430

minus06938

minus06499

minus01060

05071

minus03365

1420000

012500

020000

0minus12

600

minus36100

30300

minus07560

minus37300

minus29900

09442

minus00545

04200

minus09120

00125

07845

1520000

015000

025000

0minus09990

minus46500

27900

minus13

700

minus33400

minus31500

12917

minus11019

01791

minus16

037

044

3806994

1625000

0500

0035000

0minus50300

minus28100

22000

29600

minus53200

minus62900

minus23159

07126

minus02344

23232

minus14

578

minus20741

1725000

075

000

40000

0minus45700

minus43400

17200

20100

minus42700

minus62200

minus17

211

minus08042

minus07054

12559

minus03011

minus18

612

1825000

010000

020000

0minus25600

minus25500

40000

06760

minus53900

minus38400

minus006

8108422

12115

02427

minus13

147

01347

1925000

012500

025000

0minus25800

minus38300

35200

02050

minus48900

minus42900

minus01085

minus02739

07151

minus01537

minus08165

minus02612

2025000

015000

030000

0minus23100

minus50100

32000

minus03910

minus44100

minus45200

02387

minus13

780

05252

minus08035

minus04328

minus040

6121

30000

0500

0040000

0minus70

100

minus42700

19200

42500

minus63900

minus85200

minus44142

minus07485

minus05555

37206

minus25680

minus440

6722

30000

075

000

20000

0minus41800

minus15

000

51000

21700

minus74

200

minus48300

minus16

564

19159

23915

16943

minus34243

minus08180

2330000

010000

025000

0minus43200

minus306

0043800

18900

minus65200

minus55400

minus16

996

03741

15582

1364

6minus23453

minus14

311

2430000

012500

030000

0minus41800

minus45200

37600

13100

minus58600

minus59500

minus15

641

minus08404

10366

08632

minus18

027

minus17

517

2530000

015000

035000

0minus38900

minus58500

33200

05890

minus52300

minus61900

minus13

480

minus22716

07182

01455

minus13

269

minus21053

119887mdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

minus24124

minus34816

25241

03809

minus37541

minus39253


Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(a) Stage 1

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(b) Stage 2


00

50

100

150

00 50 100 150

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


minus150

minus100

minus50

(c) Stage 3








0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001






45 Discussions



00000

10000

20000

30000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus10000

minus20000

minus30000

minus40000

minus50000

(a) Stage 1

00000

20000

40000

60000

80000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus20000

minus60000

minus80000

minus100000

minus40000

MP4

-x

MP4

-y

MP5

-x

MP5

-y

(b) Stage 2


00000

50000

100000

150000

minus50000

minus100000

minus150000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

MP4

-x

MP4

-y

MP5

-x

MP5

-y

MP6

-x

MP6

-y

MP7

-x

MP7

-y

(c) Stage 3




5 Conclusions



0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1


00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000






Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

000005

00001

000015

00002

000025

0 200 400 600 800 1000

Fitn

ess

Cycle



0

000005

00001

000015

00002

0 200 400 600 800 1000

Fitn

ess

Cycle

SN2 = 20

SN2 = 50

SN2 = 100

SN2 = 200

SN2 = 400



LSSVM (X) 119877119899 997888rarr 119877

Y = LSSVM (X) X = (1199091 1199092 119909119899)

Y = (1199101 1199102 119910119899)

(14)

0 15

minus5 10

minus5 0 5 0

5 10

10MPa 20MPa

30∘



Cohesion c 105MPa











Table5Training

samples

andmod

elparameterso

fLSSVM

Num

bero

fsamples

1198751(M

pa)

1198752(M

pa)

120593(∘ )

Disp

lacement

120572

MP1

MP2

MP3

MP1119909

MP1119910

MP2119909

MP2119910

MP3119909

MP3119910

119909119910

119909119910

119909119910

110000

0500

0020000

0minus08380

minus13

600

15500

minus00231

minus20200

minus15

100

14473

20149

minus08992

minus03815

15989

22484

210000

075

000

25000

0minus04990

minus23300

13900

minus006

87minus16

700

minus15

800

16424

08880

minus09801

minus03294

16348

19749

310000

010000

030000

0000

00minus31300

1400

0minus14

400

1400

0minus14

400

21479

02439

minus09786

minus16

870

49088

21843

412500

012500

035000

0000

00minus39100

17500

minus18

000

minus17

500

minus18

000

20307

minus03980

minus05684

minus18

560

14959

17655

515000

015000

040000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

200

40minus10

849

minus02202

minus21514

1240

414

127

615000

0500

0025000

0minus200

00minus14

700

20800

08610

minus31900

minus27200

02187

18194

minus03108

05215

03286

09391

715000

075

000

30000

0minus16

800

minus25600

18300

01890

minus27700

minus28200

05089

06915

minus05137

minus0119

105530

07972

815000

010000

035000

0minus12

300

minus34700

1740

0minus05740

minus24200

minus27500

06722

006

83minus05353

minus05142

07871

07058

915000

012500

040000

0minus064

20minus41900

18300

minus13

800

minus21800

minus25200

10483

minus03389

minus05100

minus10

033

10326

08752

1015000

015000

020000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

22964

minus12

063

minus04593

minus24334

16207

16580

1120000

0500

0030000

0minus34100

minus19

500

22700

18500

minus42500

minus43300

minus09584

14147

minus01741

13821

minus05279

minus04169

1220000

075

000

35000

0minus30700

minus32100

19200

1100

0minus360

00minus43700

minus04940

02093

minus046

0505409

01538

minus03109

1320000

010000

040000

0minus25800

minus42600

1740

002750

minus31500

minus43100

minus01430

minus06938

minus06499

minus01060

05071

minus03365

1420000

012500

020000

0minus12

600

minus36100

30300

minus07560

minus37300

minus29900

09442

minus00545

04200

minus09120

00125

07845

1520000

015000

025000

0minus09990

minus46500

27900

minus13

700

minus33400

minus31500

12917

minus11019

01791

minus16

037

044

3806994

1625000

0500

0035000

0minus50300

minus28100

22000

29600

minus53200

minus62900

minus23159

07126

minus02344

23232

minus14

578

minus20741

1725000

075

000

40000

0minus45700

minus43400

17200

20100

minus42700

minus62200

minus17

211

minus08042

minus07054

12559

minus03011

minus18

612

1825000

010000

020000

0minus25600

minus25500

40000

06760

minus53900

minus38400

minus006

8108422

12115

02427

minus13

147

01347

1925000

012500

025000

0minus25800

minus38300

35200

02050

minus48900

minus42900

minus01085

minus02739

07151

minus01537

minus08165

minus02612

2025000

015000

030000

0minus23100

minus50100

32000

minus03910

minus44100

minus45200

02387

minus13

780

05252

minus08035

minus04328

minus040

6121

30000

0500

0040000

0minus70

100

minus42700

19200

42500

minus63900

minus85200

minus44142

minus07485

minus05555

37206

minus25680

minus440

6722

30000

075

000

20000

0minus41800

minus15

000

51000

21700

minus74

200

minus48300

minus16

564

19159

23915

16943

minus34243

minus08180

2330000

010000

025000

0minus43200

minus306

0043800

18900

minus65200

minus55400

minus16

996

03741

15582

1364

6minus23453

minus14

311

2430000

012500

030000

0minus41800

minus45200

37600

13100

minus58600

minus59500

minus15

641

minus08404

10366

08632

minus18

027

minus17

517

2530000

015000

035000

0minus38900

minus58500

33200

05890

minus52300

minus61900

minus13

480

minus22716

07182

01455

minus13

269

minus21053

119887mdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

minus24124

minus34816

25241

03809

minus37541

minus39253


Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(a) Stage 1

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(b) Stage 2


00

50

100

150

00 50 100 150

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


minus150

minus100

minus50

(c) Stage 3








0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001






45 Discussions



00000

10000

20000

30000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus10000

minus20000

minus30000

minus40000

minus50000

(a) Stage 1

00000

20000

40000

60000

80000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus20000

minus60000

minus80000

minus100000

minus40000

MP4

-x

MP4

-y

MP5

-x

MP5

-y

(b) Stage 2


00000

50000

100000

150000

minus50000

minus100000

minus150000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

MP4

-x

MP4

-y

MP5

-x

MP5

-y

MP6

-x

MP6

-y

MP7

-x

MP7

-y

(c) Stage 3




5 Conclusions



0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1


00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000






Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table5Training

samples

andmod

elparameterso

fLSSVM

Num

bero

fsamples

1198751(M

pa)

1198752(M

pa)

120593(∘ )

Disp

lacement

120572

MP1

MP2

MP3

MP1119909

MP1119910

MP2119909

MP2119910

MP3119909

MP3119910

119909119910

119909119910

119909119910

110000

0500

0020000

0minus08380

minus13

600

15500

minus00231

minus20200

minus15

100

14473

20149

minus08992

minus03815

15989

22484

210000

075

000

25000

0minus04990

minus23300

13900

minus006

87minus16

700

minus15

800

16424

08880

minus09801

minus03294

16348

19749

310000

010000

030000

0000

00minus31300

1400

0minus14

400

1400

0minus14

400

21479

02439

minus09786

minus16

870

49088

21843

412500

012500

035000

0000

00minus39100

17500

minus18

000

minus17

500

minus18

000

20307

minus03980

minus05684

minus18

560

14959

17655

515000

015000

040000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

200

40minus10

849

minus02202

minus21514

1240

414

127

615000

0500

0025000

0minus200

00minus14

700

20800

08610

minus31900

minus27200

02187

18194

minus03108

05215

03286

09391

715000

075

000

30000

0minus16

800

minus25600

18300

01890

minus27700

minus28200

05089

06915

minus05137

minus0119

105530

07972

815000

010000

035000

0minus12

300

minus34700

1740

0minus05740

minus24200

minus27500

06722

006

83minus05353

minus05142

07871

07058

915000

012500

040000

0minus064

20minus41900

18300

minus13

800

minus21800

minus25200

10483

minus03389

minus05100

minus10

033

10326

08752

1015000

015000

020000

0minus000

01minus47000

20900

minus21600

minus21000

minus21700

22964

minus12

063

minus04593

minus24334

16207

16580

1120000

0500

0030000

0minus34100

minus19

500

22700

18500

minus42500

minus43300

minus09584

14147

minus01741

13821

minus05279

minus04169

1220000

075

000

35000

0minus30700

minus32100

19200

1100

0minus360

00minus43700

minus04940

02093

minus046

0505409

01538

minus03109

1320000

010000

040000

0minus25800

minus42600

1740

002750

minus31500

minus43100

minus01430

minus06938

minus06499

minus01060

05071

minus03365

1420000

012500

020000

0minus12

600

minus36100

30300

minus07560

minus37300

minus29900

09442

minus00545

04200

minus09120

00125

07845

1520000

015000

025000

0minus09990

minus46500

27900

minus13

700

minus33400

minus31500

12917

minus11019

01791

minus16

037

044

3806994

1625000

0500

0035000

0minus50300

minus28100

22000

29600

minus53200

minus62900

minus23159

07126

minus02344

23232

minus14

578

minus20741

1725000

075

000

40000

0minus45700

minus43400

17200

20100

minus42700

minus62200

minus17

211

minus08042

minus07054

12559

minus03011

minus18

612

1825000

010000

020000

0minus25600

minus25500

40000

06760

minus53900

minus38400

minus006

8108422

12115

02427

minus13

147

01347

1925000

012500

025000

0minus25800

minus38300

35200

02050

minus48900

minus42900

minus01085

minus02739

07151

minus01537

minus08165

minus02612

2025000

015000

030000

0minus23100

minus50100

32000

minus03910

minus44100

minus45200

02387

minus13

780

05252

minus08035

minus04328

minus040

6121

30000

0500

0040000

0minus70

100

minus42700

19200

42500

minus63900

minus85200

minus44142

minus07485

minus05555

37206

minus25680

minus440

6722

30000

075

000

20000

0minus41800

minus15

000

51000

21700

minus74

200

minus48300

minus16

564

19159

23915

16943

minus34243

minus08180

2330000

010000

025000

0minus43200

minus306

0043800

18900

minus65200

minus55400

minus16

996

03741

15582

1364

6minus23453

minus14

311

2430000

012500

030000

0minus41800

minus45200

37600

13100

minus58600

minus59500

minus15

641

minus08404

10366

08632

minus18

027

minus17

517

2530000

015000

035000

0minus38900

minus58500

33200

05890

minus52300

minus61900

minus13

480

minus22716

07182

01455

minus13

269

minus21053

119887mdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

minus24124

minus34816

25241

03809

minus37541

minus39253


Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(a) Stage 1

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(b) Stage 2


00

50

100

150

00 50 100 150

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


minus150

minus100

minus50

(c) Stage 3








0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001






45 Discussions



00000

10000

20000

30000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus10000

minus20000

minus30000

minus40000

minus50000

(a) Stage 1

00000

20000

40000

60000

80000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus20000

minus60000

minus80000

minus100000

minus40000

MP4

-x

MP4

-y

MP5

-x

MP5

-y

(b) Stage 2


00000

50000

100000

150000

minus50000

minus100000

minus150000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

MP4

-x

MP4

-y

MP5

-x

MP5

-y

MP6

-x

MP6

-y

MP7

-x

MP7

-y

(c) Stage 3




5 Conclusions



0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1


00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000






Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(a) Stage 1

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


00

50

100

150


minus150

minus100

minus50

(b) Stage 2


00

50

100

150

00 50 100 150

Com

pute

d di

spla

cem

ent u

sing

iden

tified

par

amet

ers


minus150

minus100

minus50

(c) Stage 3








0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001






45 Discussions



00000

10000

20000

30000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus10000

minus20000

minus30000

minus40000

minus50000

(a) Stage 1

00000

20000

40000

60000

80000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus20000

minus60000

minus80000

minus100000

minus40000

MP4

-x

MP4

-y

MP5

-x

MP5

-y

(b) Stage 2


00000

50000

100000

150000

minus50000

minus100000

minus150000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

MP4

-x

MP4

-y

MP5

-x

MP5

-y

MP6

-x

MP6

-y

MP7

-x

MP7

-y

(c) Stage 3




5 Conclusions



0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1


00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000






Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


0000e + 000

5000e + 000

1000e + 001

1500e + 001

2000e + 001

2500e + 001

3000e + 001

3500e + 001

4000e + 001

4500e + 001

5000e + 001

5500e + 001

6000e + 001

Use

r dat

a120590

XX


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001


Use

r dat

a120590

YY

0000e + 000

4000e + 000

8000e + 000

1200e + 001

1600e + 001

2000e + 001

2400e + 001

2800e + 001

3200e + 001

3600e + 001

4400e + 001

4000e + 001

4800e + 001






45 Discussions



00000

10000

20000

30000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus10000

minus20000

minus30000

minus40000

minus50000

(a) Stage 1

00000

20000

40000

60000

80000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus20000

minus60000

minus80000

minus100000

minus40000

MP4

-x

MP4

-y

MP5

-x

MP5

-y

(b) Stage 2


00000

50000

100000

150000

minus50000

minus100000

minus150000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

MP4

-x

MP4

-y

MP5

-x

MP5

-y

MP6

-x

MP6

-y

MP7

-x

MP7

-y

(c) Stage 3




5 Conclusions



0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1


00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000






Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00000

10000

20000

30000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus10000

minus20000

minus30000

minus40000

minus50000

(a) Stage 1

00000

20000

40000

60000

80000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

minus20000

minus60000

minus80000

minus100000

minus40000

MP4

-x

MP4

-y

MP5

-x

MP5

-y

(b) Stage 2


00000

50000

100000

150000

minus50000

minus100000

minus150000

MP1

-x

MP1

-y

MP2

-x

MP2

-y

MP3

-x

MP3

-y

MP4

-x

MP4

-y

MP5

-x

MP5

-y

MP6

-x

MP6

-y

MP7

-x

MP7

-y

(c) Stage 3




5 Conclusions



0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1


00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000






Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

005

01

015

02

025

0 200 400 600 800 1000

Fitn

ess

Cycle

120590 = 10

120590 = 1


00000

50000

100000

150000

00000 50000 100000 150000

Com

pute

d di

spla

cem

ent u

sing

FEM

bas

ed o

n LS

SVM

(mm

)


120590 = 10

120590 = 1

minus150000

minus100000

minus50000






Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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