analysis and structure optimization of radial...

Research ArticleAnalysis and Structure Optimization of Radial HalbachPermanent Magnet Couplings for Deep Sea Robots

Bo Cheng 12 and Guang Pan 12

1School of Marine Science and Technology Northwestern Polytechnical University Xirsquoan 710072 China2Key Laboratory for Unmanned Underwater Vehicle Northwestern Polytechnical University Xirsquoan 710072 China

Correspondence should be addressed to Guang Pan panguangnwpueducn

Received 5 April 2018 Accepted 27 August 2018 Published 12 September 2018

Academic Editor Yannis Dimakopoulos

Copyright copy 2018 Bo Cheng and Guang Pan This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Permanent magnet couplings (PMCs) can convert the dynamic seal of transmission shaft into a static seal which will significantlyimprove the transmission efficiency and reliability Therefore the radial Halbach PMC in this paper is suitable as the transmissionmechanism of deep sea robots A two-segment Halbach array is adopted in the radial PMC and the segment arc coefficient can beadjustable This paper presents the general analytical solutions of the distinctive Halbach PMCs based on scalar magnetic potentialand Maxwell stress tensor The analytical solutions of magnetic field are in good agreement with 2-D finite element analysis (FEA)results In addition an initial prototype of the radial Halbach PMC has been fabricated and the analytical solutions of magnetictorque are compared with 3-D FEA and experiment results This paper also establishes an optimization procedure for PMCs basedon the combination of 3-DFEA Back PropagationNeuralNetwork (BPNN) andGenetic Algorithm (GA) 3-DFEA is performed tocalculate the pull-out torque of the samples fromLatin hypercube sampling then BPNN is used to describe the relationship betweenthe optimization variables and pull-out torque Finally GA is applied to solve the optimization problem and the optimized schemeis proved to be more reasonable with the FEA method

1 Introduction

Deep sea robots have attracted more and more attentionwith the development of ocean explorations However as thesubmerged distance of deep sea robots increases the dynamicseal problem of transmission shaft appears especially inthe propulsion system and joint of deep sea robots Thetraditional dynamic seal will bring large friction loss whenworking in larger submerged distance Moreover the relia-bility of traditional seal is still a problem with the increaseof operating time The radial Halbach permanent magnetcoupling (PMC) in this research can convert the rotationallydynamic seal of transmission shaft into a static seal andwill significantly improve the sealing effect and reliabilityTherefore the research of PMCs is of great significance toenhance the dynamic performance of deep sea robots

The schematic of the radial PMC for deep sea robots isshown in Figure 1 The inner and outer permanent magnets(PMs) are respectively fixed on the inner and outer rotor

core The inner rotor core is integrated with driving shaftand the outer rotor core is integrated with driven shaft Theouter side of the isolation sleeve is seawater and the inside isfilled with oil to balance the pressure of the external seawaterTo avoid eddy current loss the isolation sleeve is made ofnonconductive or high resistivity materials such as highstrength engineering plastics or ceramics

The analytical methods of PMC analysis are based on theanalysis of forces between two magnets [1ndash8] or based on theanalysis of magnetic field [9ndash12]The calculated results of theanalytical methods were compared with the experimental orfinite element analysis (FEA) results Reference [13] designedand constructed an easy-to-use test-rig for static performancetest of PMCs and the coupling-performance shows a clearinfluence of end-effects for axially short couplings H B Kangand J Y Choi et al [14] presented a comparative study oftorque analysis for synchronous PMCs with parallel- andconventional Halbach-magnetized magnets using analyticalfield calculations based on the magnetic vector potential

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 7627326 11 pageshttpsdoiorg10115520187627326

2 Mathematical Problems in Engineering

outer rotor yoke (driven sha)

inner rotor yoke (driving sha)

isolation sleeveinner PMs

outer PMs

oil

seawater

Figure 1 Schematic of the radial PMC

and Maxwell stress tensor In this paper a two-segmentHalbach array is adopted in the radial PMC Differentfrom the conventional Halbach the segment arc coefficientcan be adjusted On the basis of scalar magnetic potentialand Maxwell stress tensor the general analytical solutionsof radial Halbach PMCs are presented An initial PMCprototype has been fabricated and the analytical solutionsof the prototype are compared with the FEA calculated andexperimental results

Although the analytical solution is very useful in analyz-ing the PMC parameters analytical methods and 2-D FEAare proved to be not accurate enough due to ignoring theend magnetic flux leakage effect The structure optimizationof PMCs should be based on 3-D FEA or experimentalresults Reference [15] implemented an optimizing strategyconsisting of the 3-D FEA the design of experiments (DOE)and an exhaust algorithm in a small range to obtain thenear-optimal parameter set of PMCs W Wu et al [16]demonstrated the optimization process of PMCs using 3-D finite element methods combined with computer searchtechniques this optimization process depends mostly onthe initial feasible design and ignores the weight factorof design variables and their interaction with each otherMoreover frequently used optimization procedures cannotbe well integrated in experiments or 3-D FEATherefore thispaper presents a newoptimization procedure for PMCs basedon the combination of 3-D FEA Back Propagation NeuralNetwork (BPNN) andGeneticAlgorithm (GA)Anobjectivefunction is proposed to evaluate the mixed goal of maximalpull-out torque and minimal PM material mass The BPNNpredictionmodel of pull-out torque is established and provedto be relatively accurate GA is applied to find the optimizedscheme of the radial Halbach PMC and the performance ofthe optimized scheme is compared with that of the initialdesign with the 2-D and 3-D FEA methods

2 Radial Halbach PMC Analysis

21 Structure and Analytical Model The cross section of theradial Halbach PMC in this paper is shown in Figure 2 Theinner and outer PMs are respectively fixed on the innerand outer rotor core with high permeability and the rotorcores will contribute to reducing the loss of flux Due to

La

hm1

g

hm3

hc1

hc3

O

into paper

I II III

R1oR1i R3o

R3i

r

Figure 2 The cross section and analytical model of the radialHalbach PMC

the space limitations of the transmission mechanism theexternal dimensions of the radial PMC will be constrainedso the outer rotor is expected to be as thin as possible Inaddition the air gap between inner and outer magnets shouldbe large enough so that it can allow an isolation sleeve withsufficient thickness Therefore a two-segment Halbach arrayis adopted in the outer PMs and the segment arc coefficientis adjustable The segment arc coefficient represents the ratioof the arc of a radially magnetized segment 120573119903 to the arc of amagnet pole 120573119901 and is defined as follows

120572119903119901 = 120573119903120573119901 =120573119903

120573119903 + 120573120579 =120573119903120587119901 (1)

where 119901 stands for the pole pair number In particular whenthe segment arc coefficient is 05 the outer PMs will be aconventional Halbach array

In addition the internal space of the radial PMC isrelatively sufficient so the inner core and the inner PMs canbe designed to be thick and the inner PMs adopt the ordinarytile-shaped

The magnetic torque between inner and outer PMs is animportant indicator to evaluate the performance of PMCsand it represents the ability of PMCs to transmit torque Basedon the analytical method of magnetic field the magnetictorque is calculated and analyzed as follows

22 Magnetic Field and Magnetic Torque In order to obtainthe PMC analytical solution we assume that rotor cores areinfinitely permeable and the end-effects are neglected InFigure 2 the analytical model is divided into three regions inwhich regions I and III are the PMs and region II is air In thepolar coordinates the radial magnetization of the Halbacharray outer PMs is an even function and the tangential

Mathematical Problems in Engineering 3

magnetization is an odd function and the magnetizationvectorM1 can be given by

M1 = 1198721119903r +1198721120579120579

=infin

sum119899=135sdotsdotsdot

1198721119903119899cos (119899119901120579) r

+infin


1198721120579119899sin (119899119901120579) 120579

(2)

where

1198721119903119899 = 41198611199031205830119899120587 sin (1198991205871205721199031199012 )

1198721120579119899 = 41198611199031205830119899120587 cos (1198991205871205721199031199012 )

(3)

When the relative angular displacement between theinner and outer PMs is 120575 the magnetization vectorM3 of theinner PMs can be written as follows

M3 = 1198723119903r +1198723120579120579 =infin


1198723119903119899cos [119899119901 (120579 minus 120575)] r (4)

where

1198723119903119899 = 41198611199031205830119899120587 sin (1198991205871205722 ) (5)

The magnetic flux density vectors B13 in the PM regionand B2 in the air regions can be expressed as follows

B13 = 1205830120583119903H13 + 1205830M13

B2 = 1205830H2(6)

where 120583119903 is the relative permeability H=-grad120593 and 120593 is thescalar magnetic potential

The scalar magnetic potential in the PM regions can begoverned by quasi-Poissonian equations while in air regionit can be governed by Laplacersquos equation The magnetic fieldproduced by the Halbach array PMs is described by the scalarmagnetic potential as shown in

nabla21205931 = 120597212059311205971199032 +

11199031205971205931120597119903 +

11199032120597212059311205971205792 =

1120583119903 divM1

nabla21205932 = 120597212059321205971199032 +

11199031205971205932120597119903 +

11199032120597212059321205971205792 = 0

nabla21205933 = 120597212059331205971199032 +

11199031205971205933120597119903 +

11199032120597212059331205971205792 =

1120583119903 divM3

(7)

The general solutions of the scalar magnetic potential inthe PMs and air regions are given by the following

1205931 (119903 120579) =infin


(1198861119899119903119899119901 + 1198871119899119903minus119899119901) cos (119899119901120579)

+infin


1198721119903119899 + 1198991199011198721120579119899120583119903 [1 minus (119899119901)2]

119903 cos (119899119901120579)(8)

1205932 (119903 120579) =infin


(1198861198992119903119899119901 + 1198871198992119903minus119899119901) cos (119899119901120579) (9)

1205933 (119903 120579) =infin


(1198861198993119903119899119901 + 1198871198993119903minus119899119901) cos (119899119901120579)

+infin


1198721199033119899120583119903 [1 minus (119899119901)2]

119903 cos [119899119901 (120579 minus 120575)](10)

where np =1The boundary conditions to be satisfied are shown as

follows11986711205791003816100381610038161003816119903=1198771119900 = 011986731205791003816100381610038161003816119903=1198773119894 = 011986711205791003816100381610038161003816119903=1198771119894 = 11986721205791003816100381610038161003816119903=119877111989411986111199031003816100381610038161003816119903=1198771119894 = 11986121199031003816100381610038161003816119903=119877111989411986721205791003816100381610038161003816119903=1198773119900 = 11986731205791003816100381610038161003816119903=119877311990011986121199031003816100381610038161003816119903=1198773119900 = 11986131199031003816100381610038161003816119903=1198773119900

(11)

The tangential force between the inner and outer PMs canbe derived by the Maxwell stress tensor and the magnetictorque of PMCs is produced by the tangential force If thetangential force is integrated along a circle of radius 119903 in airregion II the magnetic torque of the radial Halbach PMC canbe expressed as follows

119879119898 = 1198711198861205830 ∮119903211986121199031198612120579119889120579 (12)

3 Comparison with FEA and Experiment

31 Magnetic Field The initial design of the radial PMCadopts a conventional Halbach array with equal segmentand the initial design parameters are shown in Table 1 Thecomputational domain diameter of the 2-D and 3-D FEAmodel is 12 times that of the radial Halbach PMC The2-D FEA results of the PMC magnetic field distributionsin different relative angular displacements are illustrated inFigure 3

Figure 4 shows the comparison between the analyticalsolutions and 2-D FEA results of radial and circumferentialflux density in air region when 120575 is 0∘ 15∘ and 30∘We can seethat the analytical solutions of flux density in air region underdifferent relative angular displacement are in good agreementwith the 2-D FEA results


= 0∘ = 15∘ = 30∘

Figure 3 Magnetic field distributions from 2-D FEA

Table 1 Initial design parameters of the radial Halbach PMC

Parameters ValueAxial length 119871119886 50mmOuter radius of outer PMs 1198771119900 27mmInter radius of outer PMs 1198771119894 24mmOuter radius of inner PMs 1198773119900 208mmInter radius of inner PMs 1198773119894 168mmPole pair number 119901 6Segment arc coefficient 120572119903119901 05Pole-arc coefficient 120572 1Remanence 119861119903 139TRelative permeability 120583119903 103Rotor core material Steel 1010Outer rotor core thickness ℎ1198881 2mmInner rotor core thickness ℎ1198883 4mm

32 Magnetic Torque Figure 5 shows the experimental appa-ratus for measuring the PMC magnetic torque It mainlyconsists of the PMCprototype a torque sensor and a steppingmotor to rotate the PMC inner rotor The step angle of thestepping motors is 15∘ The maximal positioning error is 3and the stepping motor has no cumulative error precisionThe measurement error of the torque sensor is plusmn05 Ascan be seen from Figure 5 the PMC outer rotor is fixed onthe torque sensor and is supported by bearings During theexperiment the PMC outer rotor is not rotated When thePMC inner rotor rotates to a certain angle the correspondingmagnetic torque can be measured by the torque sensor

3-D FEA are also used to calculate the PMC magnetictorque in this paperThemagnetic torque from the analyticalsolution 3-D FEA and experiments with changes of therelative angular displacement are shown in Figure 6 Whenthe inner and outer PMs are aligned the PMC is at thezero torque position and when the angular displacementis 15∘ mechanical degree the magnetic torque reaches itspeak As the relative angular displacement further increasesthe magnetic torque will decrease to zero at 30∘ mechanicaldegreeTherefore the maximal magnetic torque is also calledpull-out torque T119898119886119909 which reflects the maximal ability ofPMCs to transmit torque

Table 2 Comparison with experimental pull-out torque

Analytical solution (error) 3-D FEA (error) Experiment259Nsdotm (116) 236Nsdotm (17) 232Nsdotm

The analytical solution and 3-D FEA results of pull-out torque are compared with the experimental results asshown in Table 2 We can see that the pull-out torquecalculated with the analytical method is 116 larger thanthe experimental result by contrast 3-D FEA result is closeto the experimental result This is due to the assumptionsof the analytical model for simplifying calculation and theanalytical solution ignores the end magnetic flux leakageeffects Despite consuming more computing resources andtime 3-D FEA has higher simulation accuracy

In this research considering the transmitted torque andsafety factor the allowed minimum pull-out torque of theradial Halbach PMC is 25Nsdotm We can see that the PMCinitial design cannot meet the needs of practical applicationsSince the analytical method and 2-D FEA are proved to benot accurate enough the analytical solution will only be usedto analyze the effect of PMC parameters For better accuracyan optimization procedure based on 3-D FEA is establishedto obtain optimized PMCs with larger pull-out torque withminimal PM material mass

4 Description of the Optimization Problem

41 Objective Function The structure optimization of PMCscan be achieved by defining the objective function optimiza-tion variables and constraint functions This optimizationproblem can be expressed as follows

min119891 (x) x isin 119863119863 = x | 119892119901 (x) le 0 119901 = 1 2 sdot sdot sdot 119902

(13)

where x is the optimization variables vector 119891(x) is theobjective function and 119892119901(x) are constraint functions

The purpose of this research is to design a radial Hal-bach PMC with maximal pull-out torque and minimal PMmaterial mass at the same external dimensions (1198771119900=27mm119871119886=50mm) and this is a multiple-objective optimizationproblem For the practical requirement a judgment criterionis proposed as the objective function in this research


analytical Br2D FEA Br

analytical Bc2D FEA Bc

10 20 30 40 50 600mechanical angle position (degree)

minus10

minus05

00

05

10flu

x de

nsity

(T)

(a)




minus10

minus05

00

05

10

flux

dens

ity (T

)

(b)



minus10

minus05

00

05

10

flux

dens

ity (T

)

15 30 45 600mechanical angle position (degree)

(c)

Figure 4 Analytical solutions of radial and circumferential flux density compared with 2-D FEA results in air region 119903=224mm (a) 120575=0∘(b) 120575=15∘ (c) 120575=30∘

The pull-out torque and the PM material mass of PMCshave different units so dimensionless method is used toconstruct the objective function Because maximizing thepull-out torque is a maximal optimization problem andminimizing the PMmaterial mass is a minimal optimizationproblem the objective function can be expressed as follows

119891 (119909) = 1205961 sdot 119886V119890119903119886119892119890 (119879max)119879max

+ 1205962 sdot 119898119886V119890119903119886119892119890 (119898) (14)

where the weight coefficients 1205961 and 1205962 (1205961+1205962=1) representthe important degree of the pull-out torque and the PMmaterial mass respectively and their value can be adjustedaccording to different application requirements In this paper

1205961 is set to 07 and 1205962 is set to 03The total PMmaterial massof the inner and outer rotor can be obtained from

119898 = 120588120587119871119886 [ℎ1198981 (21198771119900 minus ℎ1198981)+ 120572ℎ1198983 (21198771119900 minus 2ℎ1198981 minus 2119892 minus ℎ1198983)]

(15)

In this way the multiple-objective optimization problemis successfully translated into a solvable single objectiveoptimization This conversion will make it more efficient toimplement the optimal design

42 Optimization Variables The analytical method is usedto analyze the effect of PMC parameters on the magnetictorque As can been seen from the analytical solution theouter and inner PM thickness segment arc coefficient pole-arc coefficient and pole pair number will affect the pull-out


magnet coupling

torque sensor

stepping motor

sensor display

Figure 5 Photograph of the experimental apparatus for measuringthe magnetic torque

angular displacement (degree)

analytical3D FEAexperiment

3025201510500

5

10

15

20

25

30

mag

netic

torq

ue (N

middotm)

Figure 6 Calculated and experimental results of the magnetictorque with changes of angular displacement

torque of PMCs The range of these variables in this researchcan be listed as follows

2119898119898 le ℎ1198981 le 41198981198983119898119898 le ℎ1198983 le 511989811989805 le 120572119903119901 le 0907 le 120572 le 13 le 119901 le 8 119901 isin Nlowast

(16)

It should be noted that the pole pair number can only betaken as a positive integer whichmeans the pole pair numberis difficult to be optimizedThe analytical and 3-DFEA resultsof the pull-out torque with changes of pole pair numbers areshown in Figure 7We can see that pole pair number has greatinfluence on the pull-out torque For the radial Halbach PMCin this research when pole pair number is 6 the pull-outtorque reaches its maximumTherefore the pole pair numberwill be regarded as a constant in the optimization procedure

pole pair number p

analytical3D FEA

4 5 6 7 8318

20

22

24

26

28

pull-

out t

orqu

e (Nmiddotm

)

Figure 7 Pull-out torque with changes of pole pair numbers whenℎ1198981=3mm ℎ1198982=4mm 120572119903119901=07 120572=085

The final optimization variables including four parametersare as follows

x = (ℎ1198981 ℎ1198983 120572119903119901 120572) (17)

43 Optimization Procedure For the nonlinear function ofpull-out torque it is difficult to accurately find the extremumonly through a certain number of discrete input data andoutput data The nonlinear fitting ability of BPNN and thenonlinear optimization ability of GA can be used to findthe extremum BPNN and GA have been used at some ofthe nonlinear data to find the optimal solution [17 18] andthis method proved to be feasible This paper establishes anoptimization procedure based on the combination of 3-DFEA BPNN andGATheoptimization procedure is to obtainthe pull-out torque of the samples by 3-D FEA and use BPNNto establish the prediction model between the optimizationvariables and pull-out torque Then GA is applied to solvethe optimization model and the solution of the optimalproblem can be obtained The optimization procedure ofBPNN predication model and the numerical optimizationwith GA is shown in Figure 8

5 BPNN Prediction Model

BPNN is a multilayer feedforward network trained based onerror back propagation algorithm and it has a great learningability in training and mapping the relations between inputsand outputs BPNN can be accepted as an alternative toproviding solutions to complex and ambiguous problems[19] This paper adopts BPNN to construct the predictionmodel of the pull-out torque

The topology of the BPNN in this paper is shown inFigure 9 Each node of the structure represents a neuronThe network takes four optimization variables as the inputneurons and takes the pull-out torque as the output neuronso the structure has four input layer nodes and one output


System modeling

Constructing BP Neural Network

BP Neural Network initialization

BP Neural Network training

End of training

Test data

BP Neural Network prediction

Population initialization

Calculate the fitness value

select

cross

mutation

Meet the termination condition

end

Network construction

Network training

Network prediction

Neural network training

Optimization with Genetic

Algorithm

N

YN

Y

Figure 8 The optimization procedure of BPNN and GA

input layer

threshold aj

threshold bk

nodes n

x1

x3

h1

h2

h5

h3

h4

y1

x2

nodes l

nodes m

hidden layer output layer

x4

weight ij

weight jk

Figure 9 The topology of the BPNN

layer node After several attempts and contrasts the numberof hidden layer nodes is selected as 5 The weights andthresholds of neural network are adjusted according to theerror during the training so that the predicted output of thenetwork is constantly approximating the expected output

A certain number of samples are necessary in the networktraining process and they should be able to describe theentire design space Latin hypercube sampling can generatea certain number of near-random samples of design variables

from amultidimensional distribution This sampling methodis frequently used to construct computer experiments Inorder to ensure the prediction model is accurate enough 40samples are randomly selected from the optimization variableranges using the Latin hypercube sampling method in thisresearch The pull-out torque of the samples is calculated by3-DFEAThe samples and corresponding responses (pull-outtorque) are shown in Table 3

In order to obtain a better training effect the sample datashould be normalized in [-1 1] before training MATLABneural network toolbox is used to train the BPNNWe set thetarget error of the network to 0000004 the learning rate to01 and the number of training steps to 100 All the iterationprocesses of the BPNN have mean square errors (MSE) asshown in (18) We set MSE as a performance function of thetraining In addition the average accuracy of the prediction(R) is defined as (19) R represents the goodness of fit andis used to measure the correlation between predicted outputdata and training sample data The closer 119877 is to 1 the morepredictable the training network is

119872119878119864 = 1119898119901119901

sum119901=1

119898

sum119895=1

(119910119901119895 minus 119910119901119895)2

(18)

119877 = 1119901119901

sum119894=1

119910119901119895119910119901119895 (19)

According to the above settings the training of BPNNis carried out until the training network meets the intended


Table 3 40 samples and corresponding responses

No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 285 488 0719 0953 2565 21 205 322 0829 0901 212 308 451 0501 0707 2175 22 314 319 0813 0988 2473 354 404 0584 0863 2535 23 274 417 0728 0816 24154 242 472 0706 0727 2235 24 386 431 0536 0781 24555 257 338 0546 0835 214 25 398 428 0878 0932 26356 285 334 0767 0903 2415 26 253 459 0688 0756 2297 317 469 0836 0912 262 27 369 407 085 0714 24358 357 374 0554 0775 2385 28 323 388 0623 0968 2559 266 5 0841 0811 2445 29 365 358 0807 0787 24910 222 329 0798 0854 2165 30 237 422 0785 0805 229511 217 41 0642 0765 2085 31 342 492 057 0879 256512 334 345 0867 0936 252 32 297 31 0897 0797 22913 211 44 0739 074 12 33 291 35 0666 084 23814 302 481 0671 0972 259 34 375 361 0593 0943 25715 382 375 076 098 265 35 247 477 0653 0921 238516 234 447 074 0888 235 36 391 343 0579 0827 247517 376 437 0618 0734 2465 37 202 394 0514 0715 178518 229 313 0638 0958 2135 38 264 367 0528 0994 22519 325 305 077 0887 244 39 347 462 0881 0846 258520 277 396 0695 0869 244 40 336 384 0609 0749 2385

TrainValidationTest

BestGoal

2010 15 250 5Epochs

10minus6

10minus4

10minus2

100

Mea

n Sq

uare

d Er

ror (

mse

)

Figure 10 The validation performance of the BPNN

target Figure 10 shows that the validation performance of thetrained neural network is closest to the target error at epoch22 Figure 11 shows the regression analysis of the BPNN Ascan be seen from Figure 11 the training set the validationset and the test set of 119877 are all close to 1 It indicates that theconstructed BPNN has high prediction accuracy

10 samples are also taken from the variable ranges withthe Latin hypercube sampling method to test the predictionmodel as shown in Table 4 and the corresponding pull-out torque is calculated by 3-D FEA Figure 12 shows the

Table 4 10 samples used to test the trained BPNN

No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 364 358 0564 0791 2412 222 331 0746 0925 21853 338 489 0782 0996 26854 398 31 0687 0737 23755 32 446 0524 0845 2426 297 419 0589 0784 23557 341 48 0636 0719 24358 211 39 0725 094 229 28 44 0842 0891 251510 25 369 0865 0877 2345

comparison of the neural network predicted output withexpected output of the test samples It can be seen that thetraining network is close to the date the maximal relativeerror between the predicted output and expected output isonly about 09 and this error is acceptable in engineeringThe prediction model of BPNN is relatively accurate to beused in the optimization procedure of PMCs

6 Optimization Using GA

GA is a computational model of the biological evolution pro-cess of the simulation genetic mechanism It is a frequentlyused method to search the optimal solution for the feasiblesolution of the problem [20] In this paper GA is applied tosolve the optimization problem to obtain the design variableswith maximal pull-out torque and minimal PM materialmass


Training R=099995

DataFitY = T

0

Validation R=099928

Test R=099291 All R=099803

0 05 1minus05

Target0 1minus1

Target

0 05 1minus05

Target0 1minus1

Target

minus1

minus05

0

05

1

Out

put ∽

= 1lowast

Targ

et +

-00

0014

minus05

Out

put ∽

= 0

94lowast

Targ

et +

00

38

minus1

minus05

0

05

1

Out

put ∽

= 0

99lowast

Targ

et +

00

11

minus05

0

05

1

Out

put ∽

= 0

92lowast

Targ

et +

00

79

DataFitY = T

DataFitY = T

DataFitY = T

05

1

Figure 11 The regression analysis of the BPNN

predicted outputexpected output

4 6 8 102test samples

22

24

26

28

outp

ut

(a)


minus001

minus0005

0

0005

001

rela

tive e

rror

(b)

Figure 12 Comparison of predicted output and expected output


pop size 20

pop size 60pop size 40

0960

0962

0964

0966

0968

0970

fitne

ss

100 200 300 400 5000iteration

Figure 13 Fitness with changes of iterations under three differentpopulation sizes

Table 5 Comparison of optimized results and initial design

Parameters Initial design Optimized resultsℎ1198981mm 3 331ℎ1198983mm 4 390120572119903119901 05 0782120572 1 0856119879maxNsdotm 236 257119898g 3574 3435

This multiobjective optimization problems are solvedby the following GA parameters number of iterations-500crossover probability-07 andmutation probability-01Threedifferent population sizes of the GA have been executed andcompared The fitness varies with the number of iterationsunder different population sizes as shown in Figure 13 Theconvergent fitness is all 09611918 and the error is less than10-7 When the population size is 20 after several executionsof the optimization procedure the GA converges to anidentical solution The same applies to the case of populationsizes 40 and 60 Furthermore the optimal solutions underpopulation sizes of 20 40 and 60 are the same

3-D FEA is used to analyze the optimized scheme andthe optimized results are as depicted in Table 5 The pull-outtorque of the optimized scheme is 890 larger than that ofthe initial design and the PM material mass is 389 lessthan that of the initial design at the same time Thereforethis optimized scheme can be well suited to the needs ofpractical applications 2-D FEA is also used to evaluate theoptimized schemeThe flux line distribution of the optimizedscheme is plotted in Figure 14 and is compared with that ofthe initial design From the results of 3-D FEA the magneticdensity vector of the optimized scheme and the initial designare exhibited and compared in Figure 15 As we can see the

initial

optimized

Figure 14 Magnetic flux line distribution of the optimized schemeand the initial design

optimized scheme makes better use of the outer rotor coreand the flux density in the inner rotor core in the inner PMsand in the outer PMswill decrease slightlyThemagnetic fielddistribution of the optimized scheme is more reasonable

7 Conclusions

In this paper a two-segment Halbach array is adopted inthe outer PMs of the radial PMC and the segment arccoefficient can be adjusted to get larger magnetic torqueBased on analytical magnetic field and Maxwell stress tensorthe general analytical solutions of radial Halbach PMCs areobtained The analytical solution of magnetic field is in goodagreement with the 2-D FEA results In addition an initialPMC prototype has been fabricated The pull-out torquecalculated with the analytical method is 116 larger thanthe experimental result and the 3-D FEA result is closeto the experimental result Then this paper establishes anoptimization procedure for PMCs based on the combinationof 3-D FEA BPNN and GA BPNN is used to establish theprediction model of pull-out torque based on 40 samplesfrom Latin hypercube sampling The test of the BPNNprediction model shows the maximal relative error betweenthe predicted output and expected output is only about 09Finally GA is applied to achieve the optimized scheme of theradial Halbach PMC and the 3-D FEA results show that thepull-out torque of the optimized scheme is 890 larger thanthat of the initial design meanwhile the PMmaterial mass is389 less than that of the initial design

Data Availability

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

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper


16

0

120804

initial optimized

Figure 15 Magnetic density vector of the optimized scheme and the initial design

Acknowledgments

Thiswork was supported by the National Science Foundationof China (Grant No 51479170and No 11502210) and NationalKey RampD Plan of China (Grant No 2016YFC0301300)

References

[1] Q C Tan D Xin W Li and H Meng ldquoStudy on TransmittingTorque and Synchronism of Magnet Couplingsrdquo Proceedingsof the Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 206 no 6 pp 381ndash3841992

[2] V V Fufaev and A Y Krasilnikov ldquoTorque of a cylindricalmagnetic couplingrdquo Russian Electrical Engineering vol 65 no8 pp 51ndash53 1994

[3] J F Charpentier and G Lemarquand ldquoStudy of permanent-magnet couplings with progressive magnetization using ananalytical formulationrdquo IEEETransactions onMagnetics vol 35no 5 pp 4206ndash4217 1999

[4] J F Charpentier and G Lemarquand ldquoCalculation of ironlesspermanent magnet couplings using semi-numerical magneticpole theory methodrdquo COMPEL - The International Journalfor Computation and Mathematics in Electrical and ElectronicEngineering vol 20 no 1-4 pp 72ndash89 2001

[5] S M Huang and C K Sung ldquoAnalytical analysis of magneticcouplings with parallelepiped magnetsrdquo Journal of Magnetismand Magnetic Materials vol 239 no 1-3 pp 614ndash616 2002

[6] R Ravaud and G Lemarquand ldquoComparison of the coulom-bian and amperian currentmodels for calculating the magneticfield produced by radially magnetized arc-shaped permanentmagnetsrdquo Progress in Electromagnetics Research vol 95 pp309ndash327 2009

[7] R Ravaud G Lemarquand V Lemarquand and C DepollierldquoTorque in permanent magnet couplings Comparison of uni-form and radial magnetizationrdquo Journal of Applied Physics vol105 no 5 2009

[8] R Ravaud V Lemarquand and G Lemarquand ldquoAnalyticaldesign of permanent magnet radial couplingsrdquo IEEE Transac-tions on Magnetics vol 46 no 11 pp 3860ndash3865 2010

[9] E P Furlani ldquoAnalysis and optimization of synchronous mag-netic couplingsrdquo Journal of Applied Physics vol 79 no 8 pp4692ndash4694 1996

[10] T Lubin S Mezani and A Rezzoug ldquoSimple analytical expres-sions for the force and torque of axial magnetic couplingsrdquo IEEETransactions on Energy Conversion vol 27 no 2 pp 536ndash5462012

[11] H-B Kang and J-Y Choi ldquoParametric analysis and experimen-tal testing of radial flux type synchronous permanent magnetcoupling based on analytical torque calculationsrdquo Journal ofElectrical Engineering amp Technology vol 9 no 3 pp 926ndash9312014

[12] C-W Kim and J-Y Choi ldquoParametric analysis of tubular-typelinear magnetic couplings with halbach array magnetized per-manentmagnet by using analytical force calculationrdquo Journal ofMagnetics vol 21 no 1 pp 110ndash114 2016

[13] S Hogberg B B Jensen and F B Bendixen ldquoDesign anddemonstration of a test-rig for static performance-studies ofpermanentmagnet couplingsrdquo IEEE Transactions onMagneticsvol 49 no 12 pp 5664ndash5670 2013

[14] H-B Kang J-Y Choi H-W Cho and J-H Kim ldquoComparativestudy of torque analysis for synchronous permanent magnetcoupling with parallel and halbach magnetized magnets basedon analytical field calculationsrdquo IEEE Transactions on Magnet-ics vol 50 no 11 2014

[15] W Y Lin L P Kuan W Jun and D Han ldquoNear-optimaldesign and 3-D finite element analysis of multiple sets of radialmagnetic couplingsrdquo IEEE Transactions on Magnetics vol 44no 12 pp 4747ndash4753 2008

[16] WWuH Lovatt and J Dunlop ldquoAnalysis and design optimisa-tion of magnetic couplings using 3D finite element modellingrdquoIEEE Transactions on Magnetics vol 33 no 5 pp 4083ndash4094

[17] F YinHMao andLHua ldquoAhybridof backpropagationneuralnetwork and genetic algorithm for optimization of injectionmolding process parametersrdquo Materials amp Design vol 32 no6 pp 3457ndash3464 2011

[18] K Hu A Song M Xia X Ye and Y Dou ldquoAn adaptive filteringalgorithm based on genetic algorithm-backpropagation net-workrdquoMathematical Problems in Engineering vol 2013 ArticleID 573941 8 pages 2013

[19] J Li J H Cheng J Y Shi and F Huang ldquoBrief introductionof back propagation (BP) neural network algorithm and itsimprovementrdquo in Advances in Computer Science and Informa-tion Engineering D Jin and S Lin Eds vol 169 pp 553ndash558Springer Berlin Germany 2012

[20] C R Houck J Joines and M G Kay ldquoA genetic algorithm forfunction optimization amatlab implementationrdquoNCSU-IE TRvol 95 no 9 1995

Hindawiwwwhindawicom Volume 2018

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


outer rotor yoke (driven sha)

inner rotor yoke (driving sha)

isolation sleeveinner PMs

outer PMs

oil

seawater

Figure 1 Schematic of the radial PMC

and Maxwell stress tensor In this paper a two-segmentHalbach array is adopted in the radial PMC Differentfrom the conventional Halbach the segment arc coefficientcan be adjusted On the basis of scalar magnetic potentialand Maxwell stress tensor the general analytical solutionsof radial Halbach PMCs are presented An initial PMCprototype has been fabricated and the analytical solutionsof the prototype are compared with the FEA calculated andexperimental results

Although the analytical solution is very useful in analyz-ing the PMC parameters analytical methods and 2-D FEAare proved to be not accurate enough due to ignoring theend magnetic flux leakage effect The structure optimizationof PMCs should be based on 3-D FEA or experimentalresults Reference [15] implemented an optimizing strategyconsisting of the 3-D FEA the design of experiments (DOE)and an exhaust algorithm in a small range to obtain thenear-optimal parameter set of PMCs W Wu et al [16]demonstrated the optimization process of PMCs using 3-D finite element methods combined with computer searchtechniques this optimization process depends mostly onthe initial feasible design and ignores the weight factorof design variables and their interaction with each otherMoreover frequently used optimization procedures cannotbe well integrated in experiments or 3-D FEATherefore thispaper presents a newoptimization procedure for PMCs basedon the combination of 3-D FEA Back Propagation NeuralNetwork (BPNN) andGeneticAlgorithm (GA)Anobjectivefunction is proposed to evaluate the mixed goal of maximalpull-out torque and minimal PM material mass The BPNNpredictionmodel of pull-out torque is established and provedto be relatively accurate GA is applied to find the optimizedscheme of the radial Halbach PMC and the performance ofthe optimized scheme is compared with that of the initialdesign with the 2-D and 3-D FEA methods

2 Radial Halbach PMC Analysis

21 Structure and Analytical Model The cross section of theradial Halbach PMC in this paper is shown in Figure 2 Theinner and outer PMs are respectively fixed on the innerand outer rotor core with high permeability and the rotorcores will contribute to reducing the loss of flux Due to

La

hm1

g

hm3

hc1

hc3

O

into paper

I II III

R1oR1i R3o

R3i

r

Figure 2 The cross section and analytical model of the radialHalbach PMC

the space limitations of the transmission mechanism theexternal dimensions of the radial PMC will be constrainedso the outer rotor is expected to be as thin as possible Inaddition the air gap between inner and outer magnets shouldbe large enough so that it can allow an isolation sleeve withsufficient thickness Therefore a two-segment Halbach arrayis adopted in the outer PMs and the segment arc coefficientis adjustable The segment arc coefficient represents the ratioof the arc of a radially magnetized segment 120573119903 to the arc of amagnet pole 120573119901 and is defined as follows

120572119903119901 = 120573119903120573119901 =120573119903

120573119903 + 120573120579 =120573119903120587119901 (1)

where 119901 stands for the pole pair number In particular whenthe segment arc coefficient is 05 the outer PMs will be aconventional Halbach array

In addition the internal space of the radial PMC isrelatively sufficient so the inner core and the inner PMs canbe designed to be thick and the inner PMs adopt the ordinarytile-shaped

The magnetic torque between inner and outer PMs is animportant indicator to evaluate the performance of PMCsand it represents the ability of PMCs to transmit torque Basedon the analytical method of magnetic field the magnetictorque is calculated and analyzed as follows

22 Magnetic Field and Magnetic Torque In order to obtainthe PMC analytical solution we assume that rotor cores areinfinitely permeable and the end-effects are neglected InFigure 2 the analytical model is divided into three regions inwhich regions I and III are the PMs and region II is air In thepolar coordinates the radial magnetization of the Halbacharray outer PMs is an even function and the tangential



M1 = 1198721119903r +1198721120579120579

=infin


1198721119903119899cos (119899119901120579) r

+infin


1198721120579119899sin (119899119901120579) 120579

(2)

where

1198721119903119899 = 41198611199031205830119899120587 sin (1198991205871205721199031199012 )

1198721120579119899 = 41198611199031205830119899120587 cos (1198991205871205721199031199012 )

(3)


M3 = 1198723119903r +1198723120579120579 =infin


1198723119903119899cos [119899119901 (120579 minus 120575)] r (4)

where

1198723119903119899 = 41198611199031205830119899120587 sin (1198991205871205722 ) (5)


B13 = 1205830120583119903H13 + 1205830M13

B2 = 1205830H2(6)



nabla21205931 = 120597212059311205971199032 +

11199031205971205931120597119903 +

11199032120597212059311205971205792 =

1120583119903 divM1

nabla21205932 = 120597212059321205971199032 +

11199031205971205932120597119903 +

11199032120597212059321205971205792 = 0

nabla21205933 = 120597212059331205971199032 +

11199031205971205933120597119903 +

11199032120597212059331205971205792 =

1120583119903 divM3

(7)


1205931 (119903 120579) =infin


(1198861119899119903119899119901 + 1198871119899119903minus119899119901) cos (119899119901120579)

+infin


1198721119903119899 + 1198991199011198721120579119899120583119903 [1 minus (119899119901)2]

119903 cos (119899119901120579)(8)

1205932 (119903 120579) =infin


(1198861198992119903119899119901 + 1198871198992119903minus119899119901) cos (119899119901120579) (9)

1205933 (119903 120579) =infin


(1198861198993119903119899119901 + 1198871198993119903minus119899119901) cos (119899119901120579)

+infin


1198721199033119899120583119903 [1 minus (119899119901)2]

119903 cos [119899119901 (120579 minus 120575)](10)


follows11986711205791003816100381610038161003816119903=1198771119900 = 011986731205791003816100381610038161003816119903=1198773119894 = 011986711205791003816100381610038161003816119903=1198771119894 = 11986721205791003816100381610038161003816119903=119877111989411986111199031003816100381610038161003816119903=1198771119894 = 11986121199031003816100381610038161003816119903=119877111989411986721205791003816100381610038161003816119903=1198773119900 = 11986731205791003816100381610038161003816119903=119877311990011986121199031003816100381610038161003816119903=1198773119900 = 11986131199031003816100381610038161003816119903=1198773119900

(11)


119879119898 = 1198711198861205830 ∮119903211986121199031198612120579119889120579 (12)





= 0∘ = 15∘ = 30∘













(13)







minus10

minus05

00

05

10flu

x de

nsity

(T)

(a)




minus10

minus05

00

05

10

flux

dens

ity (T

)

(b)



minus10

minus05

00

05

10

flux

dens

ity (T

)


(c)



119891 (119909) = 1205961 sdot 119886V119890119903119886119892119890 (119879max)119879max

+ 1205962 sdot 119898119886V119890119903119886119892119890 (119898) (14)




(15)




magnet coupling

torque sensor

stepping motor

sensor display




3025201510500

5

10

15

20

25

30

mag

netic

torq

ue (N

middotm)




(16)


pole pair number p

analytical3D FEA

4 5 6 7 8318

20

22

24

26

28

pull-

out t

orqu

e (Nmiddotm

)



x = (ℎ1198981 ℎ1198983 120572119903119901 120572) (17)






System modeling




End of training

Test data




select

cross

mutation


end


Network training

Network prediction



Algorithm

N

YN

Y


input layer

threshold aj

threshold bk

nodes n

x1

x3

h1

h2

h5

h3

h4

y1

x2

nodes l

nodes m


x4

weight ij

weight jk






119872119878119864 = 1119898119901119901

sum119901=1

119898

sum119895=1

(119910119901119895 minus 119910119901119895)2

(18)

119877 = 1119901119901

sum119894=1

119910119901119895119910119901119895 (19)




No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 285 488 0719 0953 2565 21 205 322 0829 0901 212 308 451 0501 0707 2175 22 314 319 0813 0988 2473 354 404 0584 0863 2535 23 274 417 0728 0816 24154 242 472 0706 0727 2235 24 386 431 0536 0781 24555 257 338 0546 0835 214 25 398 428 0878 0932 26356 285 334 0767 0903 2415 26 253 459 0688 0756 2297 317 469 0836 0912 262 27 369 407 085 0714 24358 357 374 0554 0775 2385 28 323 388 0623 0968 2559 266 5 0841 0811 2445 29 365 358 0807 0787 24910 222 329 0798 0854 2165 30 237 422 0785 0805 229511 217 41 0642 0765 2085 31 342 492 057 0879 256512 334 345 0867 0936 252 32 297 31 0897 0797 22913 211 44 0739 074 12 33 291 35 0666 084 23814 302 481 0671 0972 259 34 375 361 0593 0943 25715 382 375 076 098 265 35 247 477 0653 0921 238516 234 447 074 0888 235 36 391 343 0579 0827 247517 376 437 0618 0734 2465 37 202 394 0514 0715 178518 229 313 0638 0958 2135 38 264 367 0528 0994 22519 325 305 077 0887 244 39 347 462 0881 0846 258520 277 396 0695 0869 244 40 336 384 0609 0749 2385

TrainValidationTest

BestGoal

2010 15 250 5Epochs

10minus6

10minus4

10minus2

100

Mea

n Sq

uare

d Er

ror (

mse

)





No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 364 358 0564 0791 2412 222 331 0746 0925 21853 338 489 0782 0996 26854 398 31 0687 0737 23755 32 446 0524 0845 2426 297 419 0589 0784 23557 341 48 0636 0719 24358 211 39 0725 094 229 28 44 0842 0891 251510 25 369 0865 0877 2345





Training R=099995

DataFitY = T

0

Validation R=099928

Test R=099291 All R=099803

0 05 1minus05

Target0 1minus1

Target

0 05 1minus05

Target0 1minus1

Target

minus1

minus05

0

05

1

Out

put ∽

= 1lowast

Targ

et +

-00

0014

minus05

Out

put ∽

= 0

94lowast

Targ

et +

00

38

minus1

minus05

0

05

1

Out

put ∽

= 0

99lowast

Targ

et +

00

11

minus05

0

05

1

Out

put ∽

= 0

92lowast

Targ

et +

00

79

DataFitY = T

DataFitY = T

DataFitY = T

05

1




22

24

26

28

outp

ut

(a)


minus001

minus0005

0

0005

001

rela

tive e

rror

(b)



pop size 20


0960

0962

0964

0966

0968

0970

fitne

ss

100 200 300 400 5000iteration






initial

optimized



7 Conclusions


Data Availability





16

0

120804

initial optimized


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018










M1 = 1198721119903r +1198721120579120579

=infin


1198721119903119899cos (119899119901120579) r

+infin


1198721120579119899sin (119899119901120579) 120579

(2)

where

1198721119903119899 = 41198611199031205830119899120587 sin (1198991205871205721199031199012 )

1198721120579119899 = 41198611199031205830119899120587 cos (1198991205871205721199031199012 )

(3)


M3 = 1198723119903r +1198723120579120579 =infin


1198723119903119899cos [119899119901 (120579 minus 120575)] r (4)

where

1198723119903119899 = 41198611199031205830119899120587 sin (1198991205871205722 ) (5)


B13 = 1205830120583119903H13 + 1205830M13

B2 = 1205830H2(6)



nabla21205931 = 120597212059311205971199032 +

11199031205971205931120597119903 +

11199032120597212059311205971205792 =

1120583119903 divM1

nabla21205932 = 120597212059321205971199032 +

11199031205971205932120597119903 +

11199032120597212059321205971205792 = 0

nabla21205933 = 120597212059331205971199032 +

11199031205971205933120597119903 +

11199032120597212059331205971205792 =

1120583119903 divM3

(7)


1205931 (119903 120579) =infin


(1198861119899119903119899119901 + 1198871119899119903minus119899119901) cos (119899119901120579)

+infin


1198721119903119899 + 1198991199011198721120579119899120583119903 [1 minus (119899119901)2]

119903 cos (119899119901120579)(8)

1205932 (119903 120579) =infin


(1198861198992119903119899119901 + 1198871198992119903minus119899119901) cos (119899119901120579) (9)

1205933 (119903 120579) =infin


(1198861198993119903119899119901 + 1198871198993119903minus119899119901) cos (119899119901120579)

+infin


1198721199033119899120583119903 [1 minus (119899119901)2]

119903 cos [119899119901 (120579 minus 120575)](10)


follows11986711205791003816100381610038161003816119903=1198771119900 = 011986731205791003816100381610038161003816119903=1198773119894 = 011986711205791003816100381610038161003816119903=1198771119894 = 11986721205791003816100381610038161003816119903=119877111989411986111199031003816100381610038161003816119903=1198771119894 = 11986121199031003816100381610038161003816119903=119877111989411986721205791003816100381610038161003816119903=1198773119900 = 11986731205791003816100381610038161003816119903=119877311990011986121199031003816100381610038161003816119903=1198773119900 = 11986131199031003816100381610038161003816119903=1198773119900

(11)


119879119898 = 1198711198861205830 ∮119903211986121199031198612120579119889120579 (12)





= 0∘ = 15∘ = 30∘













(13)







minus10

minus05

00

05

10flu

x de

nsity

(T)

(a)




minus10

minus05

00

05

10

flux

dens

ity (T

)

(b)



minus10

minus05

00

05

10

flux

dens

ity (T

)


(c)



119891 (119909) = 1205961 sdot 119886V119890119903119886119892119890 (119879max)119879max

+ 1205962 sdot 119898119886V119890119903119886119892119890 (119898) (14)




(15)




magnet coupling

torque sensor

stepping motor

sensor display




3025201510500

5

10

15

20

25

30

mag

netic

torq

ue (N

middotm)




(16)


pole pair number p

analytical3D FEA

4 5 6 7 8318

20

22

24

26

28

pull-

out t

orqu

e (Nmiddotm

)



x = (ℎ1198981 ℎ1198983 120572119903119901 120572) (17)






System modeling




End of training

Test data




select

cross

mutation


end


Network training

Network prediction



Algorithm

N

YN

Y


input layer

threshold aj

threshold bk

nodes n

x1

x3

h1

h2

h5

h3

h4

y1

x2

nodes l

nodes m


x4

weight ij

weight jk






119872119878119864 = 1119898119901119901

sum119901=1

119898

sum119895=1

(119910119901119895 minus 119910119901119895)2

(18)

119877 = 1119901119901

sum119894=1

119910119901119895119910119901119895 (19)




No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 285 488 0719 0953 2565 21 205 322 0829 0901 212 308 451 0501 0707 2175 22 314 319 0813 0988 2473 354 404 0584 0863 2535 23 274 417 0728 0816 24154 242 472 0706 0727 2235 24 386 431 0536 0781 24555 257 338 0546 0835 214 25 398 428 0878 0932 26356 285 334 0767 0903 2415 26 253 459 0688 0756 2297 317 469 0836 0912 262 27 369 407 085 0714 24358 357 374 0554 0775 2385 28 323 388 0623 0968 2559 266 5 0841 0811 2445 29 365 358 0807 0787 24910 222 329 0798 0854 2165 30 237 422 0785 0805 229511 217 41 0642 0765 2085 31 342 492 057 0879 256512 334 345 0867 0936 252 32 297 31 0897 0797 22913 211 44 0739 074 12 33 291 35 0666 084 23814 302 481 0671 0972 259 34 375 361 0593 0943 25715 382 375 076 098 265 35 247 477 0653 0921 238516 234 447 074 0888 235 36 391 343 0579 0827 247517 376 437 0618 0734 2465 37 202 394 0514 0715 178518 229 313 0638 0958 2135 38 264 367 0528 0994 22519 325 305 077 0887 244 39 347 462 0881 0846 258520 277 396 0695 0869 244 40 336 384 0609 0749 2385

TrainValidationTest

BestGoal

2010 15 250 5Epochs

10minus6

10minus4

10minus2

100

Mea

n Sq

uare

d Er

ror (

mse

)





No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 364 358 0564 0791 2412 222 331 0746 0925 21853 338 489 0782 0996 26854 398 31 0687 0737 23755 32 446 0524 0845 2426 297 419 0589 0784 23557 341 48 0636 0719 24358 211 39 0725 094 229 28 44 0842 0891 251510 25 369 0865 0877 2345





Training R=099995

DataFitY = T

0

Validation R=099928

Test R=099291 All R=099803

0 05 1minus05

Target0 1minus1

Target

0 05 1minus05

Target0 1minus1

Target

minus1

minus05

0

05

1

Out

put ∽

= 1lowast

Targ

et +

-00

0014

minus05

Out

put ∽

= 0

94lowast

Targ

et +

00

38

minus1

minus05

0

05

1

Out

put ∽

= 0

99lowast

Targ

et +

00

11

minus05

0

05

1

Out

put ∽

= 0

92lowast

Targ

et +

00

79

DataFitY = T

DataFitY = T

DataFitY = T

05

1




22

24

26

28

outp

ut

(a)


minus001

minus0005

0

0005

001

rela

tive e

rror

(b)



pop size 20


0960

0962

0964

0966

0968

0970

fitne

ss

100 200 300 400 5000iteration






initial

optimized



7 Conclusions


Data Availability





16

0

120804

initial optimized


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









= 0∘ = 15∘ = 30∘













(13)







minus10

minus05

00

05

10flu

x de

nsity

(T)

(a)




minus10

minus05

00

05

10

flux

dens

ity (T

)

(b)



minus10

minus05

00

05

10

flux

dens

ity (T

)


(c)



119891 (119909) = 1205961 sdot 119886V119890119903119886119892119890 (119879max)119879max

+ 1205962 sdot 119898119886V119890119903119886119892119890 (119898) (14)




(15)




magnet coupling

torque sensor

stepping motor

sensor display




3025201510500

5

10

15

20

25

30

mag

netic

torq

ue (N

middotm)




(16)


pole pair number p

analytical3D FEA

4 5 6 7 8318

20

22

24

26

28

pull-

out t

orqu

e (Nmiddotm

)



x = (ℎ1198981 ℎ1198983 120572119903119901 120572) (17)






System modeling




End of training

Test data




select

cross

mutation


end


Network training

Network prediction



Algorithm

N

YN

Y


input layer

threshold aj

threshold bk

nodes n

x1

x3

h1

h2

h5

h3

h4

y1

x2

nodes l

nodes m


x4

weight ij

weight jk






119872119878119864 = 1119898119901119901

sum119901=1

119898

sum119895=1

(119910119901119895 minus 119910119901119895)2

(18)

119877 = 1119901119901

sum119894=1

119910119901119895119910119901119895 (19)




No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 285 488 0719 0953 2565 21 205 322 0829 0901 212 308 451 0501 0707 2175 22 314 319 0813 0988 2473 354 404 0584 0863 2535 23 274 417 0728 0816 24154 242 472 0706 0727 2235 24 386 431 0536 0781 24555 257 338 0546 0835 214 25 398 428 0878 0932 26356 285 334 0767 0903 2415 26 253 459 0688 0756 2297 317 469 0836 0912 262 27 369 407 085 0714 24358 357 374 0554 0775 2385 28 323 388 0623 0968 2559 266 5 0841 0811 2445 29 365 358 0807 0787 24910 222 329 0798 0854 2165 30 237 422 0785 0805 229511 217 41 0642 0765 2085 31 342 492 057 0879 256512 334 345 0867 0936 252 32 297 31 0897 0797 22913 211 44 0739 074 12 33 291 35 0666 084 23814 302 481 0671 0972 259 34 375 361 0593 0943 25715 382 375 076 098 265 35 247 477 0653 0921 238516 234 447 074 0888 235 36 391 343 0579 0827 247517 376 437 0618 0734 2465 37 202 394 0514 0715 178518 229 313 0638 0958 2135 38 264 367 0528 0994 22519 325 305 077 0887 244 39 347 462 0881 0846 258520 277 396 0695 0869 244 40 336 384 0609 0749 2385

TrainValidationTest

BestGoal

2010 15 250 5Epochs

10minus6

10minus4

10minus2

100

Mea

n Sq

uare

d Er

ror (

mse

)





No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 364 358 0564 0791 2412 222 331 0746 0925 21853 338 489 0782 0996 26854 398 31 0687 0737 23755 32 446 0524 0845 2426 297 419 0589 0784 23557 341 48 0636 0719 24358 211 39 0725 094 229 28 44 0842 0891 251510 25 369 0865 0877 2345





Training R=099995

DataFitY = T

0

Validation R=099928

Test R=099291 All R=099803

0 05 1minus05

Target0 1minus1

Target

0 05 1minus05

Target0 1minus1

Target

minus1

minus05

0

05

1

Out

put ∽

= 1lowast

Targ

et +

-00

0014

minus05

Out

put ∽

= 0

94lowast

Targ

et +

00

38

minus1

minus05

0

05

1

Out

put ∽

= 0

99lowast

Targ

et +

00

11

minus05

0

05

1

Out

put ∽

= 0

92lowast

Targ

et +

00

79

DataFitY = T

DataFitY = T

DataFitY = T

05

1




22

24

26

28

outp

ut

(a)


minus001

minus0005

0

0005

001

rela

tive e

rror

(b)



pop size 20


0960

0962

0964

0966

0968

0970

fitne

ss

100 200 300 400 5000iteration






initial

optimized



7 Conclusions


Data Availability





16

0

120804

initial optimized


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018












minus10

minus05

00

05

10flu

x de

nsity

(T)

(a)




minus10

minus05

00

05

10

flux

dens

ity (T

)

(b)



minus10

minus05

00

05

10

flux

dens

ity (T

)


(c)



119891 (119909) = 1205961 sdot 119886V119890119903119886119892119890 (119879max)119879max

+ 1205962 sdot 119898119886V119890119903119886119892119890 (119898) (14)




(15)




magnet coupling

torque sensor

stepping motor

sensor display




3025201510500

5

10

15

20

25

30

mag

netic

torq

ue (N

middotm)




(16)


pole pair number p

analytical3D FEA

4 5 6 7 8318

20

22

24

26

28

pull-

out t

orqu

e (Nmiddotm

)



x = (ℎ1198981 ℎ1198983 120572119903119901 120572) (17)






System modeling




End of training

Test data




select

cross

mutation


end


Network training

Network prediction



Algorithm

N

YN

Y


input layer

threshold aj

threshold bk

nodes n

x1

x3

h1

h2

h5

h3

h4

y1

x2

nodes l

nodes m


x4

weight ij

weight jk






119872119878119864 = 1119898119901119901

sum119901=1

119898

sum119895=1

(119910119901119895 minus 119910119901119895)2

(18)

119877 = 1119901119901

sum119894=1

119910119901119895119910119901119895 (19)




No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 285 488 0719 0953 2565 21 205 322 0829 0901 212 308 451 0501 0707 2175 22 314 319 0813 0988 2473 354 404 0584 0863 2535 23 274 417 0728 0816 24154 242 472 0706 0727 2235 24 386 431 0536 0781 24555 257 338 0546 0835 214 25 398 428 0878 0932 26356 285 334 0767 0903 2415 26 253 459 0688 0756 2297 317 469 0836 0912 262 27 369 407 085 0714 24358 357 374 0554 0775 2385 28 323 388 0623 0968 2559 266 5 0841 0811 2445 29 365 358 0807 0787 24910 222 329 0798 0854 2165 30 237 422 0785 0805 229511 217 41 0642 0765 2085 31 342 492 057 0879 256512 334 345 0867 0936 252 32 297 31 0897 0797 22913 211 44 0739 074 12 33 291 35 0666 084 23814 302 481 0671 0972 259 34 375 361 0593 0943 25715 382 375 076 098 265 35 247 477 0653 0921 238516 234 447 074 0888 235 36 391 343 0579 0827 247517 376 437 0618 0734 2465 37 202 394 0514 0715 178518 229 313 0638 0958 2135 38 264 367 0528 0994 22519 325 305 077 0887 244 39 347 462 0881 0846 258520 277 396 0695 0869 244 40 336 384 0609 0749 2385

TrainValidationTest

BestGoal

2010 15 250 5Epochs

10minus6

10minus4

10minus2

100

Mea

n Sq

uare

d Er

ror (

mse

)





No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 364 358 0564 0791 2412 222 331 0746 0925 21853 338 489 0782 0996 26854 398 31 0687 0737 23755 32 446 0524 0845 2426 297 419 0589 0784 23557 341 48 0636 0719 24358 211 39 0725 094 229 28 44 0842 0891 251510 25 369 0865 0877 2345





Training R=099995

DataFitY = T

0

Validation R=099928

Test R=099291 All R=099803

0 05 1minus05

Target0 1minus1

Target

0 05 1minus05

Target0 1minus1

Target

minus1

minus05

0

05

1

Out

put ∽

= 1lowast

Targ

et +

-00

0014

minus05

Out

put ∽

= 0

94lowast

Targ

et +

00

38

minus1

minus05

0

05

1

Out

put ∽

= 0

99lowast

Targ

et +

00

11

minus05

0

05

1

Out

put ∽

= 0

92lowast

Targ

et +

00

79

DataFitY = T

DataFitY = T

DataFitY = T

05

1




22

24

26

28

outp

ut

(a)


minus001

minus0005

0

0005

001

rela

tive e

rror

(b)



pop size 20


0960

0962

0964

0966

0968

0970

fitne

ss

100 200 300 400 5000iteration






initial

optimized



7 Conclusions


Data Availability





16

0

120804

initial optimized


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









magnet coupling

torque sensor

stepping motor

sensor display




3025201510500

5

10

15

20

25

30

mag

netic

torq

ue (N

middotm)




(16)


pole pair number p

analytical3D FEA

4 5 6 7 8318

20

22

24

26

28

pull-

out t

orqu

e (Nmiddotm

)



x = (ℎ1198981 ℎ1198983 120572119903119901 120572) (17)






System modeling




End of training

Test data




select

cross

mutation


end


Network training

Network prediction



Algorithm

N

YN

Y


input layer

threshold aj

threshold bk

nodes n

x1

x3

h1

h2

h5

h3

h4

y1

x2

nodes l

nodes m


x4

weight ij

weight jk






119872119878119864 = 1119898119901119901

sum119901=1

119898

sum119895=1

(119910119901119895 minus 119910119901119895)2

(18)

119877 = 1119901119901

sum119894=1

119910119901119895119910119901119895 (19)




No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 285 488 0719 0953 2565 21 205 322 0829 0901 212 308 451 0501 0707 2175 22 314 319 0813 0988 2473 354 404 0584 0863 2535 23 274 417 0728 0816 24154 242 472 0706 0727 2235 24 386 431 0536 0781 24555 257 338 0546 0835 214 25 398 428 0878 0932 26356 285 334 0767 0903 2415 26 253 459 0688 0756 2297 317 469 0836 0912 262 27 369 407 085 0714 24358 357 374 0554 0775 2385 28 323 388 0623 0968 2559 266 5 0841 0811 2445 29 365 358 0807 0787 24910 222 329 0798 0854 2165 30 237 422 0785 0805 229511 217 41 0642 0765 2085 31 342 492 057 0879 256512 334 345 0867 0936 252 32 297 31 0897 0797 22913 211 44 0739 074 12 33 291 35 0666 084 23814 302 481 0671 0972 259 34 375 361 0593 0943 25715 382 375 076 098 265 35 247 477 0653 0921 238516 234 447 074 0888 235 36 391 343 0579 0827 247517 376 437 0618 0734 2465 37 202 394 0514 0715 178518 229 313 0638 0958 2135 38 264 367 0528 0994 22519 325 305 077 0887 244 39 347 462 0881 0846 258520 277 396 0695 0869 244 40 336 384 0609 0749 2385

TrainValidationTest

BestGoal

2010 15 250 5Epochs

10minus6

10minus4

10minus2

100

Mea

n Sq

uare

d Er

ror (

mse

)





No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 364 358 0564 0791 2412 222 331 0746 0925 21853 338 489 0782 0996 26854 398 31 0687 0737 23755 32 446 0524 0845 2426 297 419 0589 0784 23557 341 48 0636 0719 24358 211 39 0725 094 229 28 44 0842 0891 251510 25 369 0865 0877 2345





Training R=099995

DataFitY = T

0

Validation R=099928

Test R=099291 All R=099803

0 05 1minus05

Target0 1minus1

Target

0 05 1minus05

Target0 1minus1

Target

minus1

minus05

0

05

1

Out

put ∽

= 1lowast

Targ

et +

-00

0014

minus05

Out

put ∽

= 0

94lowast

Targ

et +

00

38

minus1

minus05

0

05

1

Out

put ∽

= 0

99lowast

Targ

et +

00

11

minus05

0

05

1

Out

put ∽

= 0

92lowast

Targ

et +

00

79

DataFitY = T

DataFitY = T

DataFitY = T

05

1




22

24

26

28

outp

ut

(a)


minus001

minus0005

0

0005

001

rela

tive e

rror

(b)



pop size 20


0960

0962

0964

0966

0968

0970

fitne

ss

100 200 300 400 5000iteration






initial

optimized



7 Conclusions


Data Availability





16

0

120804

initial optimized


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









System modeling




End of training

Test data




select

cross

mutation


end


Network training

Network prediction



Algorithm

N

YN

Y


input layer

threshold aj

threshold bk

nodes n

x1

x3

h1

h2

h5

h3

h4

y1

x2

nodes l

nodes m


x4

weight ij

weight jk






119872119878119864 = 1119898119901119901

sum119901=1

119898

sum119895=1

(119910119901119895 minus 119910119901119895)2

(18)

119877 = 1119901119901

sum119894=1

119910119901119895119910119901119895 (19)




No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 285 488 0719 0953 2565 21 205 322 0829 0901 212 308 451 0501 0707 2175 22 314 319 0813 0988 2473 354 404 0584 0863 2535 23 274 417 0728 0816 24154 242 472 0706 0727 2235 24 386 431 0536 0781 24555 257 338 0546 0835 214 25 398 428 0878 0932 26356 285 334 0767 0903 2415 26 253 459 0688 0756 2297 317 469 0836 0912 262 27 369 407 085 0714 24358 357 374 0554 0775 2385 28 323 388 0623 0968 2559 266 5 0841 0811 2445 29 365 358 0807 0787 24910 222 329 0798 0854 2165 30 237 422 0785 0805 229511 217 41 0642 0765 2085 31 342 492 057 0879 256512 334 345 0867 0936 252 32 297 31 0897 0797 22913 211 44 0739 074 12 33 291 35 0666 084 23814 302 481 0671 0972 259 34 375 361 0593 0943 25715 382 375 076 098 265 35 247 477 0653 0921 238516 234 447 074 0888 235 36 391 343 0579 0827 247517 376 437 0618 0734 2465 37 202 394 0514 0715 178518 229 313 0638 0958 2135 38 264 367 0528 0994 22519 325 305 077 0887 244 39 347 462 0881 0846 258520 277 396 0695 0869 244 40 336 384 0609 0749 2385

TrainValidationTest

BestGoal

2010 15 250 5Epochs

10minus6

10minus4

10minus2

100

Mea

n Sq

uare

d Er

ror (

mse

)





No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 364 358 0564 0791 2412 222 331 0746 0925 21853 338 489 0782 0996 26854 398 31 0687 0737 23755 32 446 0524 0845 2426 297 419 0589 0784 23557 341 48 0636 0719 24358 211 39 0725 094 229 28 44 0842 0891 251510 25 369 0865 0877 2345





Training R=099995

DataFitY = T

0

Validation R=099928

Test R=099291 All R=099803

0 05 1minus05

Target0 1minus1

Target

0 05 1minus05

Target0 1minus1

Target

minus1

minus05

0

05

1

Out

put ∽

= 1lowast

Targ

et +

-00

0014

minus05

Out

put ∽

= 0

94lowast

Targ

et +

00

38

minus1

minus05

0

05

1

Out

put ∽

= 0

99lowast

Targ

et +

00

11

minus05

0

05

1

Out

put ∽

= 0

92lowast

Targ

et +

00

79

DataFitY = T

DataFitY = T

DataFitY = T

05

1




22

24

26

28

outp

ut

(a)


minus001

minus0005

0

0005

001

rela

tive e

rror

(b)



pop size 20


0960

0962

0964

0966

0968

0970

fitne

ss

100 200 300 400 5000iteration






initial

optimized



7 Conclusions


Data Availability





16

0

120804

initial optimized


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018










No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 285 488 0719 0953 2565 21 205 322 0829 0901 212 308 451 0501 0707 2175 22 314 319 0813 0988 2473 354 404 0584 0863 2535 23 274 417 0728 0816 24154 242 472 0706 0727 2235 24 386 431 0536 0781 24555 257 338 0546 0835 214 25 398 428 0878 0932 26356 285 334 0767 0903 2415 26 253 459 0688 0756 2297 317 469 0836 0912 262 27 369 407 085 0714 24358 357 374 0554 0775 2385 28 323 388 0623 0968 2559 266 5 0841 0811 2445 29 365 358 0807 0787 24910 222 329 0798 0854 2165 30 237 422 0785 0805 229511 217 41 0642 0765 2085 31 342 492 057 0879 256512 334 345 0867 0936 252 32 297 31 0897 0797 22913 211 44 0739 074 12 33 291 35 0666 084 23814 302 481 0671 0972 259 34 375 361 0593 0943 25715 382 375 076 098 265 35 247 477 0653 0921 238516 234 447 074 0888 235 36 391 343 0579 0827 247517 376 437 0618 0734 2465 37 202 394 0514 0715 178518 229 313 0638 0958 2135 38 264 367 0528 0994 22519 325 305 077 0887 244 39 347 462 0881 0846 258520 277 396 0695 0869 244 40 336 384 0609 0749 2385

TrainValidationTest

BestGoal

2010 15 250 5Epochs

10minus6

10minus4

10minus2

100

Mea

n Sq

uare

d Er

ror (

mse

)





No ℎ1198981 ℎ1198983 120572119903119901 120572 119879max

1 364 358 0564 0791 2412 222 331 0746 0925 21853 338 489 0782 0996 26854 398 31 0687 0737 23755 32 446 0524 0845 2426 297 419 0589 0784 23557 341 48 0636 0719 24358 211 39 0725 094 229 28 44 0842 0891 251510 25 369 0865 0877 2345





Training R=099995

DataFitY = T

0

Validation R=099928

Test R=099291 All R=099803

0 05 1minus05

Target0 1minus1

Target

0 05 1minus05

Target0 1minus1

Target

minus1

minus05

0

05

1

Out

put ∽

= 1lowast

Targ

et +

-00

0014

minus05

Out

put ∽

= 0

94lowast

Targ

et +

00

38

minus1

minus05

0

05

1

Out

put ∽

= 0

99lowast

Targ

et +

00

11

minus05

0

05

1

Out

put ∽

= 0

92lowast

Targ

et +

00

79

DataFitY = T

DataFitY = T

DataFitY = T

05

1




22

24

26

28

outp

ut

(a)


minus001

minus0005

0

0005

001

rela

tive e

rror

(b)



pop size 20


0960

0962

0964

0966

0968

0970

fitne

ss

100 200 300 400 5000iteration






initial

optimized



7 Conclusions


Data Availability





16

0

120804

initial optimized


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









Training R=099995

DataFitY = T

0

Validation R=099928

Test R=099291 All R=099803

0 05 1minus05

Target0 1minus1

Target

0 05 1minus05

Target0 1minus1

Target

minus1

minus05

0

05

1

Out

put ∽

= 1lowast

Targ

et +

-00

0014

minus05

Out

put ∽

= 0

94lowast

Targ

et +

00

38

minus1

minus05

0

05

1

Out

put ∽

= 0

99lowast

Targ

et +

00

11

minus05

0

05

1

Out

put ∽

= 0

92lowast

Targ

et +

00

79

DataFitY = T

DataFitY = T

DataFitY = T

05

1




22

24

26

28

outp

ut

(a)


minus001

minus0005

0

0005

001

rela

tive e

rror

(b)



pop size 20


0960

0962

0964

0966

0968

0970

fitne

ss

100 200 300 400 5000iteration






initial

optimized



7 Conclusions


Data Availability





16

0

120804

initial optimized


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









pop size 20


0960

0962

0964

0966

0968

0970

fitne

ss

100 200 300 400 5000iteration






initial

optimized



7 Conclusions


Data Availability





16

0

120804

initial optimized


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









16

0

120804

initial optimized


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








analysis and structure optimization of radial...

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