Research Article Reliability Analysis-Based …downloads.hindawi.com/journals/mpe/2013/260976.pdf[] N.Saravanan,V.N.S.K.Siddabattuni,andK.I.Ramachan dran, Fault diagnosis of spur bevel

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 260976 5 pageshttpdxdoiorg1011552013260976

Research ArticleReliability Analysis-Based Numerical Calculation ofMetal Structure of Bridge Crane

Wenjun Meng1 Zhengmao Yang1 Xiaolong Qi2 and Jianghui Cai1

1 Taiyuan University of Science and Technology Taiyuan 030024 China2 Tongji University Shanghai 200092 China

Correspondence should be addressed to Wenjun Meng tyustmwjtyusteducn

Received 17 March 2013 Accepted 21 September 2013

Academic Editor Zhichun Yang

Copyright copy 2013 Wenjun Meng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The study introduced a finite element model of DQ75t-28m bridge crane metal structure and made finite element static analysis toobtain the stress response of the dangerous point of metal structure in the most extreme condition The simulated samples of therandom variable and the stress of the dangerous point were successfully obtained through the orthogonal designThen we utilizedBP neural network nonlinear mapping function trains to get the explicit expression of stress in response to the random variableCombined with random perturbation theory and first-order second-moment (FOSM) method the study analyzed the reliabilityand its sensitivity of metal structure In conclusion we established a novel method for accurately quantitative analysis and designof bridge crane metal structure

1 Introduction

Bridge crane metal structure is significant in the bridge craneto host tract walk and brake which is the most critical partof the performance to determine its safety As the stabilityand reliability of metal structure is the guarantee of theentire system its corresponding analysis therefore is crucialbefore the application [1 2] Currently there are variousuncertainties in the metal structure of bridge crane suchas size parameters material properties and external loadThese uncertainties have directly influenced the safety andreliability of the metal structure Therefore studies using thereliability analysis theory to ensure the stability are encourag-ing In addition due to the different influence degrees of theserandom parameters the reliability sensitivity analysis is alsoessential to obtain dependence of metal structure reliabilityto each random parameter which will instruct the optimizeddesign and applicable conditions of the metal structure

The study introduced a finite element model of bridgecrane metal structure and then applied finite element sim-ulation and orthogonal experimental design to obtain thedangerous parts and stress response in themost extreme con-dition We utilized BP neural network fitting technology toget an explicit expression of the stress on the design variables

Besides random perturbation theory and FOSM methodwere used for the reliability analysis The matrix differen-tial technology further contributed to deduce the influencedegree of various random parameters on the reliability of thebridge crane metal structure

2 Finite Element Analysis of Bridge CraneMetal Structure

21 Finite Element Analysis ANSYS parametric design lan-guage (APDL) was used to establish the finite element modelof bridge crane metal structure (Figure 1) Box-type bridgestructure model selected shell elements SHELL63 (4-NodeElastic Shell UX UY UZ ROTX ROTY ROTZ) becausethe unit had capacity to deal with stress stiffening and largedeflection [3] Car trackmodel selected beam element BEAM189 (3 Node 3-D Quadratic Finite Strain Beam UX UY UZROTX ROTY ROTZ) which is based on the Timoshenkobeam analysis theory with default of shear effects and largedeformation effects [3 4]

22 Stress Response Analysis We applied finite element anal-ysis software ANSYS to simulate and analyze the metal struc-ture of bridge crane The stress response in the most severe

2 Mathematical Problems in Engineering

Finite element analysis of structure of bridge crane

Figure 1 Finite element model of bridge crane metal structure

operating conditions was obtained (Figure 2) With full loadthe maximum stress (110MPa) was observed at the midspanof metal structure and the maximum deformation occurredat the midspan of top flange plate Meanwhile the biggeststress existed at the intersection of the lower flange plate themain web and the diaphragms of box structure (Figure 2)Although the intersection presented the potential fractureand failure the value did not exceed the material yieldstrength and conformed to the requirements of static strengthand static stiffness Therefore it can be concluded that themetal structure will not permanently deform and can meetthe needs of safe carrying and running The intermetalstructure von-Mises stress nephogram in the condition of fullload was shown in Figure 2

3 BP Neural Network Fitting Stress Response

Since the metal structure of bridge machine is an extremelycomplex box structure the function of stress and randomdesign variables is a highly nonlinear and implicitThereforewe utilized the orthogonal experimental design methodand finite element simulation to get sample data Then wecombined the BP neural network technology to map therelationship between structure design variables and stressvalues in dangerous parts fitting to acquire their explicitexpression [5ndash7]

31 Neural Network Training Samples The study appliedorthogonal experimental design method to establish the rea-sonable quantity and distribution of neural network trainingsamples to accurately express the mapping relationship ofneural networkmodel According to the reliability theory weselected the simulated full-load condition as the object andobtained the data of random variable as well as the stressresponse (Table 1)

The output samples of stress response of dangerousparts could be gained through the experimental design Itwould be further used as the training samples of BP neuralnetwork to eventually fit required explicit expressions 119878 =

(119882119867 119861 119864 119890 119884119884119861 119890 119865119861)Training samples of the neural network model was

obtained by using the orthogonal table 11987125

(56) that is

3113


0123E+08

0246E+08

0368E+08

0491E+08

0614E+08

0737E+08

0859E+08

0982E+08

0110E+09

Figure 2 Stress nephogram of metal structure under full-loadcondition

Table 1 Random variable and its statistical properties of bridgecrane metal structure

Random variable(factors) Meaning Average Standard

deviationW106 N Lifting load 250 20Hmm Main beam height 8155 366Bmm Main beam width 4108 207EMPa Elastic modulus 206 10

e YYBmm Upper and lower flangeplate thickness 14 42

e FBmm Main vice web thickness 12 25

Table 2 Orthogonal factors and the level

Level W H B E e YYB e FB1 230 8000 3800 196 11 62 240 8100 3900 201 12 73 250 8200 4000 206 13 84 260 8300 4100 211 14 95 270 8400 4200 216 15 10

the total number of samples was 25 the factors were 6each factor had 5 levels [4] (Table 2) The stress response ofits corresponding 25 samples as output variables of neuralnetwork model was calculated by the finite element model

32 Neural Network Model For the 6 random variables ofbridge cranemetal structure system BP neural network inputlayer had 6 neurons and the hidden layer had 15 neuronsEach neuron of the output layer was described as the stressresponse of the bridge crane metal structure

To get the stress response neural network model ofthe global significance each random variable in its feasibleregion took 5 discrete values (horizontal) and generated 25design samples as the training set by the combination of

Mathematical Problems in Engineering 3

orthogonal test Then finite element simulation experimentswere carried out on all samples by ANSYS software

33 Neural Network Learning After the model structure ofBP neural network was determined the toolbox (NNETToolbox) of MATLB was used in training the network usingthe input sample set and the output sample set to achieve agiven input-output mapping relationship and further correctthe thresholds and weights of the network [8]

Some neurons reached saturation due to the large differ-ence in the magnitude of each variable in the original sampleAs a result the input samples should be normalized first andthen selected the neurons pass for hidden layer and outputlayer as tansig() and purelin() respectively [9] Meanwhilethe training function trainlm in LM (Levenberg-Marquardt)algorithm was selected for training network when conver-gence speed and accuracy were taken into consideration

34 Neural Network Generalization Test Typically the train-ing patterns included in the training set are only part of thesource data set Even if the network was trained by all thepatterns within the training set it also could not guaranteethat the test by another mode could give satisfactory resultsTherefore the generalization ability of fitting functions of 25new samples as a test set is still needed to be examined afterthe network learning From the result of test error (Figure 3)the fitted model met the requirement of accuracy and couldreplace the finite element simulation for reliability analysis

In addition the convergence of the neural networktraining was demonstrated (Figure 4)

4 Reliability Analysis of BridgeCrane Metal Structure

A set of basic random variables 119883 = (1198831 1198832 119883

119899)119879

features the design dimensions the material properties theload that are normally distributed Its joint probability densityfunction is119891

119883(119883)The limit state function of the bridge crane

metal structure is expressed as119892 (119883) = 119892 (119883

1 1198832 119883

119899) (1)

119892(119883) lt 0 represents the failure state 119892(119883) = 0 is limit stateand 119892(119883) gt 0 is the security state Based on the probabilitytheory the reliability is calculated using multiple integrals inreliable domain when 119892(119883) gt 0 namely

119877 = int119892(119883)gt0

sdot sdot sdot int 119868 [119892 (119909)] 119891 (1199091 1199092 119909

119899) 11988911990911198891199092sdot sdot sdot 119889119909

119899

(2)

when 119892(119883) gt 0 119868[119892(119883) gt 0] = 1 when 119892(119883) lt 0 119868[119892(119883) lt

0] = 0Based on the perturbation theory 119883 and 119892(119883) can be

expanded as follows119883 = 119883

119889+ Δ119883119901

119892 (119883) = 119892119889(119883) + Δ119892

119901(119883)

(3)

where subscript 119889 and 119901 separately denote the determiningpart and random part of random variable and have zero

0 5 10 15 20 25 30minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10BP neural network stress fitting error of change

Rela

tive e

rror

The number of samples tested n

Figure 3 Relative error of BP neural network fitted model

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Mea

n sq

uare

d er

ror (

mse

)

4882 epochs

TrainBestGoal

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

Best training performance is 99979e minus 006 at epoch 4882

Figure 4 Grid training convergence curve

mean Δ denotes higher-order infinitesimal Based on therandom analysis theory the mean and variance of formula(3) are as follows

120583119892= 119864 [119892 (119883)] = 119864 [119892

119889(119883)] + Δ119864 [119892

119901(119883)] = 119892

119889(119883)

1205902

119892= Var [119892 (119883)] = Δ

2119864[(

120597119892119889(119883)

120597119883119879)Δ119883119901]

= (120597119892119889(119883)

120597119883119879)Var (119883)

(4)

Based on the FOSMmethod take a linear term of the Taylorexpansion of limit state equation and calculate the probabilityof the function greater than zero by basic random variables of


first moment (mean) and second moment (variance) Relia-bility is expressed as 119877

119877 = Φ (120573) = Φ(120583119892

120590119892

) = Φ(119864 [119892 (119883)]

radicVar [119892 (119883)]

) (5)

Set 120590 for the corresponding stress intensity of the dangerouspoint of bridge machine metal structureThen the limit statefunction is written as

119892 (119883) = 120590 minus 119878 (119882119867 119861 119864 119890 119884119884119861 119890 119865119861) (6)

5 Reliability Sensitivity Analysis of BridgeCrane Metal Structure

Based on the FOSMmethod andmatrix differential themeanand variance sensitivity of the metal structure reliability toeach random variable are expressed as follows

119889119877119894

119889119883119879=120597119877119894

120597120573119894

120597120573119894

120597120583119892119894

120597120583119892119894

120597119883119879 (119894 = 1 2 119899)

119889119877119894

119889Var119883=120597119877119894

120597120573119894

120597120573119894

120597120590119892119894

120597120590119892119894

120597Var119883 (119894 = 1 2 119899)

(7)

where

120597119877119894

120597120573119894

= 120593 (120573119894)

120597120573119894

120597120583119892119894

=1

120590119892119894

120597120583119892119894

120597119883119879= [

120597119892119894

1205971198831

120597119892119894

1205971198832

sdot sdot sdot120597119892119894

120597119883119899

]

120597120573119894

120597120590119892119894

= minus120583119892119894

1205902

119892119894

120597120590119892119894

120597Var119883=

1

2120590119892119894

[120597119892119894

120597119883otimes120597119892119894

120597119883]

(8)

6 Results and Analysis

According to formula (5) the reliability of themetal structureis119877 = 0999866 UseMonte Carlo simulation for test to calcu-late 106 times The reliability is 119877MCS = 0999764 Relativeerror is as follows

120576119877=

10038161003816100381610038161003816100381610038161003816

119877 minus 119877MSC119877MCS

10038161003816100381610038161003816100381610038161003816

= 1306 times 10minus4 (9)

Based on the known conditions and reliability calculationresults the reliability sensitivity of random variables includ-ing 119889119877

119894119889119883119879 and 119889119877

119894119889Var119883 can be calculated according to

formula (7) [10] The mean sensitivity of the bridge machinemetal structure is as follows

119889119877

119889119883119879=

[[[[[[[

[

119877 (119882)

119877 (119867)

119877 (119861)

119877 (119864)

119877 (119890 119884119884119861)

119877 (119890 119865119861)

]]]]]]]

]

=

[[[[[[[

[

4043 times 10minus3

1692 times 10minus4

1432 times 10minus4

3453 times 10minus5

3308 times 10minus7

4682 times 10minus8

]]]]]]]

]

(10)

Variance sensitivity is as follows

119889119877

119889Var119883=

[[[[[[[

[

119877Var (119882)

119877Var (119867)

119877Var (119861)119877Var (119864)

119877Var (119890 119884119884119861)

119877Var (119890 119865119861)

]]]]]]]

]

=

[[[[[[[

[

minus6385 times 10minus10






]]]]]]]

]

(11)

It can be drawn from the sensitivity matrix of 119889119877119894119889119883119879

that the reliability sensitivity of bridge crane metal structurecan be influenced by lifting load 119882 main beam height 119867main beam width 119861 and elastic modulus 119864 [10] Moreoverthe influence degree of 6 variables on structural reliabilityin decreasing order are lifting load 119882 main beam height119867main beam width 119861 elastic modulus 119864 flange plate thickness119890 119884119884119861 and web thickness 119890 119865119861

7 Conclusions

(i) The study applied the neural network technologyand orthogonal experimental design method to solvethe reliability calculation with the implicit limit statefunction and to obtain the explicit expression of therandom variable and the stress response of bridgecrane metal structure

(ii) The integrated use of orthogonal experimental designmethod and finite element simulation test in the relia-bility analysis engineering can significantly reduce thedesign costs and markedly shorten the cycle

(iii) Based on the theory of reliability design and sensi-tivity we made a derivation analysis of the reliabilityand its sensitivity and then established a basis foraccurately quantitative analysis of the reliability ofbridge crane metal structure

Acknowledgments

The project is supported by the Natural Science FoundationCommittee (51075289) National Natural Science FoundationInternational Cooperation and Exchanges Project(51110105011) 2012 Higher School Specialized ResearchFund for the Doctoral Program Joint Funding Issues(20121415110004) Research Foundation for ReturningScholar in 2009 of Shanxi province (20091074) InternationalScientific and Technological Cooperation Projects in 2010 ofShanxi province (2010081039) Shanxi Natural Science Foun-dation (2011011019-3) 2010 Scientific Star Project in Taiyuan


city (2010011605) 2009 Shanxi province Graduatesrsquo Excellentand Innovative Project (20093099) Shanxi province 2011UIT Item TYUST 2010 UIT Item (201008X) Doctor Startup Item and Characteristics amp Leading Academic DisciplineProject of Universities of Shanxi Province

References

[1] D NThatoi H C Das and D R Parhi ldquoReview of techniquesfor fault diagnosis in damaged structure and engineering sys-temrdquo Advances in Mechanical Engineering vol 2012 Article ID327569 11 pages 2012

[2] N Saravanan V N S K Siddabattuni and K I Ramachan-dran ldquoFault diagnosis of spur bevel gear box using artificialneural network (ANN) and proximal support vector machine(PSVM)rdquoApplied SoftComputing Journal vol 10 no 1 pp 344ndash360 2010

[3] P Zeng L Lei andG FangFinite ElementAnalysis GuideMod-eling and Analysis of Structure China Machine Press BeijingChina 2013 (Chinese)

[4] XHe andY Zhang ldquoReliability sensitivity design ofmechanicalsystemsrdquo Machinery Manufacturing vol 5 pp 26ndash28 2009(Chinese)

[5] M Hohenbichler and R Rackwitz ldquoSensitivity and importancemeasures in structural reliabilityrdquoCivil Engineering Systems vol3 no 4 pp 203ndash209 1986

[6] K Xu M Xie L C Tang and S L Ho ldquoApplication of neuralnetworks in forecasting engine systems reliabilityrdquo Applied SoftComputing Journal vol 2 no 4 pp 255ndash268 2003

[7] T Chen H Chen and R-W Liu ldquoApproximation capability inC(Rn) by multilayer feedforward networks and related prob-lemsrdquo IEEE Transactions on Neural Networks vol 6 no 1 pp25ndash30 1995 (Chinese)

[8] K Zhou and Y KangThe Neural Network Model and Its MAT-LAB Simulation Program Designed Tsinghua University PressBeijing China 2005

[9] G I Schueller ldquoOn the treatment of uncertainties in structuralmechanics and analysisrdquo Computational Stochastic Mechanicsvol 85 no 5-6 pp 235ndash243 2007

[10] H Xiangdong Z Yimin and W Bangchun ldquoReliability sensi-tivity design ofmechanical systemsrdquoMachineryManufacturingvol 47 no 5 pp 26ndash28 2009

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of



Figure 1 Finite element model of bridge crane metal structure

operating conditions was obtained (Figure 2) With full loadthe maximum stress (110MPa) was observed at the midspanof metal structure and the maximum deformation occurredat the midspan of top flange plate Meanwhile the biggeststress existed at the intersection of the lower flange plate themain web and the diaphragms of box structure (Figure 2)Although the intersection presented the potential fractureand failure the value did not exceed the material yieldstrength and conformed to the requirements of static strengthand static stiffness Therefore it can be concluded that themetal structure will not permanently deform and can meetthe needs of safe carrying and running The intermetalstructure von-Mises stress nephogram in the condition of fullload was shown in Figure 2

3 BP Neural Network Fitting Stress Response

Since the metal structure of bridge machine is an extremelycomplex box structure the function of stress and randomdesign variables is a highly nonlinear and implicitThereforewe utilized the orthogonal experimental design methodand finite element simulation to get sample data Then wecombined the BP neural network technology to map therelationship between structure design variables and stressvalues in dangerous parts fitting to acquire their explicitexpression [5ndash7]

31 Neural Network Training Samples The study appliedorthogonal experimental design method to establish the rea-sonable quantity and distribution of neural network trainingsamples to accurately express the mapping relationship ofneural networkmodel According to the reliability theory weselected the simulated full-load condition as the object andobtained the data of random variable as well as the stressresponse (Table 1)

The output samples of stress response of dangerousparts could be gained through the experimental design Itwould be further used as the training samples of BP neuralnetwork to eventually fit required explicit expressions 119878 =

(119882119867 119861 119864 119890 119884119884119861 119890 119865119861)Training samples of the neural network model was

obtained by using the orthogonal table 11987125

(56) that is

3113


0123E+08

0246E+08

0368E+08

0491E+08

0614E+08

0737E+08

0859E+08

0982E+08

0110E+09

Figure 2 Stress nephogram of metal structure under full-loadcondition

Table 1 Random variable and its statistical properties of bridgecrane metal structure

Random variable(factors) Meaning Average Standard

deviationW106 N Lifting load 250 20Hmm Main beam height 8155 366Bmm Main beam width 4108 207EMPa Elastic modulus 206 10

e YYBmm Upper and lower flangeplate thickness 14 42

e FBmm Main vice web thickness 12 25

Table 2 Orthogonal factors and the level

Level W H B E e YYB e FB1 230 8000 3800 196 11 62 240 8100 3900 201 12 73 250 8200 4000 206 13 84 260 8300 4100 211 14 95 270 8400 4200 216 15 10

the total number of samples was 25 the factors were 6each factor had 5 levels [4] (Table 2) The stress response ofits corresponding 25 samples as output variables of neuralnetwork model was calculated by the finite element model

32 Neural Network Model For the 6 random variables ofbridge cranemetal structure system BP neural network inputlayer had 6 neurons and the hidden layer had 15 neuronsEach neuron of the output layer was described as the stressresponse of the bridge crane metal structure

To get the stress response neural network model ofthe global significance each random variable in its feasibleregion took 5 discrete values (horizontal) and generated 25design samples as the training set by the combination of









119899)119879




1 1198832 119883

119899) (1)


119877 = int119892(119883)gt0


119899) 11988911990911198891199092sdot sdot sdot 119889119909

119899

(2)




119889+ Δ119883119901

119892 (119883) = 119892119889(119883) + Δ119892

119901(119883)

(3)


0 5 10 15 20 25 30minus10

minus8

minus6

minus4

minus2

0

2

4

6

8


Rela

tive e

rror



0 500 1000 1500 2000 2500 3000 3500 4000 4500

Mea

n sq

uare

d er

ror (

mse

)

4882 epochs

TrainBestGoal

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6




120583119892= 119864 [119892 (119883)] = 119864 [119892

119889(119883)] + Δ119864 [119892

119901(119883)] = 119892

119889(119883)

1205902

119892= Var [119892 (119883)] = Δ

2119864[(

120597119892119889(119883)

120597119883119879)Δ119883119901]

= (120597119892119889(119883)

120597119883119879)Var (119883)

(4)




119877 = Φ (120573) = Φ(120583119892

120590119892

) = Φ(119864 [119892 (119883)]

radicVar [119892 (119883)]

) (5)


119892 (119883) = 120590 minus 119878 (119882119867 119861 119864 119890 119884119884119861 119890 119865119861) (6)



119889119877119894

119889119883119879=120597119877119894

120597120573119894

120597120573119894

120597120583119892119894

120597120583119892119894

120597119883119879 (119894 = 1 2 119899)

119889119877119894

119889Var119883=120597119877119894

120597120573119894

120597120573119894

120597120590119892119894

120597120590119892119894

120597Var119883 (119894 = 1 2 119899)

(7)

where

120597119877119894

120597120573119894

= 120593 (120573119894)

120597120573119894

120597120583119892119894

=1

120590119892119894

120597120583119892119894

120597119883119879= [

120597119892119894

1205971198831

120597119892119894

1205971198832

sdot sdot sdot120597119892119894

120597119883119899

]

120597120573119894

120597120590119892119894

= minus120583119892119894

1205902

119892119894

120597120590119892119894

120597Var119883=

1

2120590119892119894

[120597119892119894

120597119883otimes120597119892119894

120597119883]

(8)



120576119877=

10038161003816100381610038161003816100381610038161003816

119877 minus 119877MSC119877MCS

10038161003816100381610038161003816100381610038161003816



119894119889119883119879 and 119889119877



119889119877

119889119883119879=

[[[[[[[

[

119877 (119882)

119877 (119867)

119877 (119861)

119877 (119864)

119877 (119890 119884119884119861)

119877 (119890 119865119861)

]]]]]]]

]

=

[[[[[[[

[

4043 times 10minus3

1692 times 10minus4

1432 times 10minus4

3453 times 10minus5

3308 times 10minus7

4682 times 10minus8

]]]]]]]

]

(10)


119889119877

119889Var119883=

[[[[[[[

[

119877Var (119882)

119877Var (119867)

119877Var (119861)119877Var (119864)

119877Var (119890 119884119884119861)

119877Var (119890 119865119861)

]]]]]]]

]

=

[[[[[[[

[







]]]]]]]

]

(11)



7 Conclusions




Acknowledgments




References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















119899)119879




1 1198832 119883

119899) (1)


119877 = int119892(119883)gt0


119899) 11988911990911198891199092sdot sdot sdot 119889119909

119899

(2)




119889+ Δ119883119901

119892 (119883) = 119892119889(119883) + Δ119892

119901(119883)

(3)


0 5 10 15 20 25 30minus10

minus8

minus6

minus4

minus2

0

2

4

6

8


Rela

tive e

rror



0 500 1000 1500 2000 2500 3000 3500 4000 4500

Mea

n sq

uare

d er

ror (

mse

)

4882 epochs

TrainBestGoal

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6




120583119892= 119864 [119892 (119883)] = 119864 [119892

119889(119883)] + Δ119864 [119892

119901(119883)] = 119892

119889(119883)

1205902

119892= Var [119892 (119883)] = Δ

2119864[(

120597119892119889(119883)

120597119883119879)Δ119883119901]

= (120597119892119889(119883)

120597119883119879)Var (119883)

(4)




119877 = Φ (120573) = Φ(120583119892

120590119892

) = Φ(119864 [119892 (119883)]

radicVar [119892 (119883)]

) (5)


119892 (119883) = 120590 minus 119878 (119882119867 119861 119864 119890 119884119884119861 119890 119865119861) (6)



119889119877119894

119889119883119879=120597119877119894

120597120573119894

120597120573119894

120597120583119892119894

120597120583119892119894

120597119883119879 (119894 = 1 2 119899)

119889119877119894

119889Var119883=120597119877119894

120597120573119894

120597120573119894

120597120590119892119894

120597120590119892119894

120597Var119883 (119894 = 1 2 119899)

(7)

where

120597119877119894

120597120573119894

= 120593 (120573119894)

120597120573119894

120597120583119892119894

=1

120590119892119894

120597120583119892119894

120597119883119879= [

120597119892119894

1205971198831

120597119892119894

1205971198832

sdot sdot sdot120597119892119894

120597119883119899

]

120597120573119894

120597120590119892119894

= minus120583119892119894

1205902

119892119894

120597120590119892119894

120597Var119883=

1

2120590119892119894

[120597119892119894

120597119883otimes120597119892119894

120597119883]

(8)



120576119877=

10038161003816100381610038161003816100381610038161003816

119877 minus 119877MSC119877MCS

10038161003816100381610038161003816100381610038161003816



119894119889119883119879 and 119889119877



119889119877

119889119883119879=

[[[[[[[

[

119877 (119882)

119877 (119867)

119877 (119861)

119877 (119864)

119877 (119890 119884119884119861)

119877 (119890 119865119861)

]]]]]]]

]

=

[[[[[[[

[

4043 times 10minus3

1692 times 10minus4

1432 times 10minus4

3453 times 10minus5

3308 times 10minus7

4682 times 10minus8

]]]]]]]

]

(10)


119889119877

119889Var119883=

[[[[[[[

[

119877Var (119882)

119877Var (119867)

119877Var (119861)119877Var (119864)

119877Var (119890 119884119884119861)

119877Var (119890 119865119861)

]]]]]]]

]

=

[[[[[[[

[







]]]]]]]

]

(11)



7 Conclusions




Acknowledgments




References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877 = Φ (120573) = Φ(120583119892

120590119892

) = Φ(119864 [119892 (119883)]

radicVar [119892 (119883)]

) (5)


119892 (119883) = 120590 minus 119878 (119882119867 119861 119864 119890 119884119884119861 119890 119865119861) (6)



119889119877119894

119889119883119879=120597119877119894

120597120573119894

120597120573119894

120597120583119892119894

120597120583119892119894

120597119883119879 (119894 = 1 2 119899)

119889119877119894

119889Var119883=120597119877119894

120597120573119894

120597120573119894

120597120590119892119894

120597120590119892119894

120597Var119883 (119894 = 1 2 119899)

(7)

where

120597119877119894

120597120573119894

= 120593 (120573119894)

120597120573119894

120597120583119892119894

=1

120590119892119894

120597120583119892119894

120597119883119879= [

120597119892119894

1205971198831

120597119892119894

1205971198832

sdot sdot sdot120597119892119894

120597119883119899

]

120597120573119894

120597120590119892119894

= minus120583119892119894

1205902

119892119894

120597120590119892119894

120597Var119883=

1

2120590119892119894

[120597119892119894

120597119883otimes120597119892119894

120597119883]

(8)



120576119877=

10038161003816100381610038161003816100381610038161003816

119877 minus 119877MSC119877MCS

10038161003816100381610038161003816100381610038161003816



119894119889119883119879 and 119889119877



119889119877

119889119883119879=

[[[[[[[

[

119877 (119882)

119877 (119867)

119877 (119861)

119877 (119864)

119877 (119890 119884119884119861)

119877 (119890 119865119861)

]]]]]]]

]

=

[[[[[[[

[

4043 times 10minus3

1692 times 10minus4

1432 times 10minus4

3453 times 10minus5

3308 times 10minus7

4682 times 10minus8

]]]]]]]

]

(10)


119889119877

119889Var119883=

[[[[[[[

[

119877Var (119882)

119877Var (119867)

119877Var (119861)119877Var (119864)

119877Var (119890 119884119884119861)

119877Var (119890 119865119861)

]]]]]]]

]

=

[[[[[[[

[







]]]]]]]

]

(11)



7 Conclusions




Acknowledgments




References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

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