第四屆“認證杯”數(shù)學中國

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1848

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競賽評閱編號（由競賽評委團評閱前進行編號）：

Theassessmentofseatbackcomfortbasedon

yticHierarchyProcess

:

Theseatwasoneoftheairne’scriticalfacilities,itsdesigndegreehadthedirectrelationshipwiththecomfortableperformanceoftheflight.Butsomeairlinesarenowintroducingnew"slimline"seatwhichhasthinnerbackteandlesspaddingineconomyclass.Thereisnodoubtmanypassengershaveexpresseddispleasurewiththeseseats.Thecomfortdependsnotonlyonthematerialselectionofseat,butalsocloselyrelatedtotheinclinationoftheseat,whetherequippedwithlumbarsupportandadjustability.Inordertoprovidetravelswithacomfortableexperience,itisessentialtodesignaseatbackbyusingtheergonomic.

Inthedesignoftraditionalseats,peopleareaccustomedtocreatingtheframefirstandthenputtingonasuitablefoamlayertomakethetravelsfeelcomfortable.Butinthedesignofthenew"slimline"seat,thedesignconceptisopposite.Weneedtocreateasuitablecurvethatfitthebodyperfectlyandthendesigntheseat.

Wefirststudytheergonomicandchoosefivepointsasthefeaturepointsofthespine.Afterestimatingtheparametersinhumandimensionsofadults,wefitacurveofthespineroughly.ThenwebuildacomfortmodelbyusingtheyticHierarchyProcesstogetagoodcomfortevaluation.Thecomfortmodelcanevaluatethedegreeofseatbackcomfortinabetterway.Butsubjectively,differentpeoplehavedifferentevaluations.Sothismodelcannotavoidthedependenceonsubjectiveevaluationtest.Thus,weonlyconsidertheabilitytoendurepressureinfourmainregionsofthehumanbackwhenweusethehierarchy.Thismethodobjectivelyevaluatethedegreeofthecomfort,andachieveagoodresult.Finally,wetestthemodelandgetaconclusionthatthebodywillfeelthemostcomfortablewhentheangleisabout112o.Themodeltakeaguidingroleinmaterialselection,angleselection,aswellascostestimationin"slimline"seat.

: Ergonomic;CurveFitting; yticHierarchyProcess

Introduction

Theseatwasoneoftheairne’scriticalfacilities,itsdesigndegreehadthedirectrelationshipwiththecomfortableperformanceoftheflight.Thecontemporarytraveler’sevaluationtotheairlineseatwasnotonlysatisfiedwithbeautifulshape,butalsovaluedthe ’sfactor,forexamplethecomfortoftheposture,theeasyandconvenientbodyoperation,thedistributionofreasonablebodypressureetc[1].Butsomeairlinesarenowintroducingnew"slimline"seatwhichhasthinnerbackteandlesspaddingineconomyclass.Slimlineseatsarebeingfurtherrefined,liberatingmorepassengerspace.Thereisnodoubtmanypassengershaveexpresseddispleasurewiththeseseats.

Inordertoimprovethepassengerscomfort,thehumanengineeringwasusedintheairlineseat‘sysis.Themostcomfortablesittingpostureshouldbethattheseatbackhadalittleincline.Thethighandtheupperpartofthebodykeepaintersectionanglebetween90oand120o.Wefirstfittedaseatbackcurvethathadthesimilarbackcurvewiththespinebyyzingsomeoftheanthropometricdata.Underthiscircumstance,webuildacomfortmodelbyusingtheyticHierarchyProcesstogetagoodcomfortevaluation.Thus,wedividedthisproblemintothefollowingtwosub-problems:

ProblemI:Designthecurve.ProblemII:Buildthecomfortmodel.

DefiningtheProblem

The"slimline"seats,inadditiontoweightingless,theoreticallyallowairlinestoincreasecapacitywithoutsignificantaffectingpassengercomfort.Theseseatsmayormaynotfeaturemoveableheadrests,andgenerallydonotfeatureadjustablelumbarsupport.Astoacomfortableseat,theessentialwasthatcouldproducethemostappropriatepressuretodistributetotheeachintervertebaldisc.Atpresent,mostofourcivilianaircraftareAirbusandserieswhicharecreatedabroad.ThesizeoftheseatisbasedonthedimensionoftheEuropeanandAmericanwhichislargerthan.Thismayresultinasenseofpressureduetothehighseatback.Forthisreason,weshouldtakethedimensionsofintoaccount.Theairlineseatshouldhaveanadjustabledesign.Weestablishaseatbackcurvebyyzingreliabledataonthespinemodelandanthropometry.Considerthemostcomfortableangleofhumanspine.Thenadjustthepaddingandcurvetooptimizetheseat.

Models

TheDesignofCurve

Thedesignoftheairlineseatbasedonanthropometricdata.Thedataaredividedintotwocategories,staticsizeanddynamicsize.Staticsizeisasizethattheexperimentalsubjectisinastationarystate.Thedesignrelatedtobodystructureismainlybasedonthestaticsizewhichcanbeusedintheseat’sheightanddepthaswellastheseat

back’swidthandheight.Meanwhile,thedynamicbodysizecanreflecttherelationsbetweenfunctionandstructureofthebody.Thesetwodimensionsareboththesignificantbasesforcivilaviation.Table1isthehumandimensionsofadults.ItisastandardcalledGB10000-88.Thisstandardthatbasedonergonomicsrequirementsprovidethefundamentaldataofanadulthumanbody.

Table1:Humandimensionsofadults units:[mm]

Male(18-60)

Female(18-60)

Percentiles

5

50

95

5

50

95

Sittingheight

858

908

958

809

855

901

Sittingcervicalheight

615

657

701

579

617

657

Sittingeyeheight

749

798

847

695

739

738

Sittingshoulderheight

557

598

641

518

556

594

Sittingelbowheight

228

263

298

215

251

284

Assumptions:

Thedataintable1areallnakedmeasurements.

Bodydimensionsarequitedifferentindifferentregions.

Thefifthpercentilerepresentsthat5%oftheheightdimensionaresmallerthanthissize.

Terms,DefinitionsandSymbols

Thehumanvertebralcolumnisthebackboneofthehumanskeleton,consistingof33vertebrae.Individualvertebraearenamedaccordingtotheirregionandposition.Fromtoptobottom,thevertebraeare:

Cervicalspine:7vertebrae(C1-C7)

Thoracicspine:12vertebrae(T1-T12)

Lumbarspine:5vertebrae(L1-L5)

Sacrum:5vertebrae(S1-S5)

Coccyx:4vertebrae

Accordingtotheergonomics,themain pointsareonthelumberspine,shoulderbladeandcervicalspinewhenthebodyinthecomfortablestate.Inordertofitabettercurve,wechoosefivefeaturepointsincludingthreeofthemain points,thebeginningandtheendoftheentirespine.Table2isthedefinitionofthefivepoints.

Table2:Definitionofthefeaturepoint

P1 ThecentralpointofS1

P2 ThecentralpointofthecombinationofL1andT12P3 Themainpointofshoulder

P4 ThecentralpointofthecombinationofT1andC7P5 ThecentralpointofC1

Themethodofleastsquaresisastandardapproachinregressionysistotheapproximatesolutionofoverdeterminedsystem."Leastsquares"meansthattheoverallsolutionminimizesthesumofthesquaresoftheerrorsmadeintheresultsof

everysingleequation.Themostimportantapplicationisindatafit.Thebestfitintheleast-squaressenseminimizesthesumofsquaredresiduals,aresidualbeingthedifferencebetweenanobservedvalueandthefittedvalueprovidedbyamodel.Polynomialleastsquaresdescribesthevarianceinapredictionofthedependentvariableasafunctionoftheindependentvariableandthedeviationsfromthefittedcurve.

SolutionandResult

Insolvingthisproblem,weusetheleastsquares[2]tofitthecurveonthebasisofthedataabove.Thepolynomialmodelandcubicsplineinterpolationmodelarebothusedinthefitting.Whenfittingwiththepolynomialinterpolation,low-orderpolynomialsmakethedata"smoothing"butthehigh-orderpolynomialshaslittledeviation.ThefivefeaturepointsP1-P5extractedaboveareusedtofitthespinecurve.Undertheabovecircumstance,webuildarectangularcoordinatesystem.Theoriginofthecoordinatesisthecenterofthejunctionoftheseatbackandcushion,theverticaldirectionisthex-axisandthehorizontaldirectionisy-axis.AccordingtotheparametersintheTable1wecangetthecoordinatesofP1-P5.Table3isthefeaturepoints’coordinates.

Table3:Coordinatesofthefeaturepoints units:[mm]

Coordinates

P1

(0,99)

P2

(263,148)

P3

(598,121)

P4

(657,135)

P5

(798,125)

Thestepstofitthefeaturepointsbyusingthe

andtheleastsquaresareas

follows:First,wechoosethethird-orderpolynomialtodocurvefitting.Thenormalequationofthethird-orderpolynomialis:

0 1 2 3

yaaxax2ax3.

ThenwegettheresultoffittingasFigure1.Figure1showsthecurve.

200

P2

y

100P1

P3 P4 P5

0

0 100 200 300 400 500 600 700 800

x

Figure1

Byyzingthecurvewecanobtainaequation:

y6.2184107x39.5808104x20.39504x100.19

Apparently,thefeaturepointP2hasadecentresultthatfeaturelumbarsupport.ButthefeaturepointP3andP4donothavetheperfectsupport.Sowetrytousethefifth-orderpolynomialtodocurvefitting.Thenormalequationofthethird-orderpolynomialis:

0 1 2 3 4 5

yaaxax2ax3ax4ax5

ThenwegettheresultoffittingasFigure2.Figure2showsthecurve.

200

P2

y

100P1

P3 P4 P5

0

0 100 200 300 400 500 600 700 800

x

Figure2

Byyzingthecurvewecanobtainaequation:

y1.52111011x52.8727108x41.7841105x33.7047103x2100

InFigure2,wegetanexpectedresultthatthefeaturepointsP1-P5areallfitthecurveperfectly.TotestwhetherthecurveisagoodresultweconductaresidualysisasFigure3shows.

x10-11 residuals

5thdegree:normofresiduals=1.554e-12

5thdegree

1

0

-1

0 100 200 300 400 500 600 700 800

Figure3

ModelI

Regressionysisisastatisticalprocessforestimatingtherelationshipsamongvariables[5].WeusethismethodtobuildthemodelI.Butthismethodneedsalotofobservationdata,weonlyyzefiveimportantfactorsduetothelimitedtime.Someresearchersusethepressuredistributiondataandsubjectiveevaluationofthetest

resultstobuildtheregressionequation[5].Theyevaluatedtheoverallcomfortinfiveaspects:appearance,adjustablesupport,staticcomfort,etc.Theoverallcomfortismainlyinfluencedbystaticcomfort,blazonryandadjustablesupport.Thestaticcomfortismainlyinfluencedbypaddingmaterial,sittingcomfortandskin.ThemodelstructureisshowedinFigure4.

Figure4

ModelII

Introduction

Thepressureoftheseatbackismainlyfromthepartofthelumbarregionandtheshoulderblade.Andthepressuredistributionoftheseatbackwillchangewiththeangle.Buttheangleformedbyseatbackandhorizontalneisnotunchangeable[6].Inthespine,themostbrawnypartislumbarvertebrabuttheshoulderbladecanalsobearhugepressures.Sothepressuredistributionoftheidealseatisthattheumpressureshouldappearinthejunctionofthethoracicvertebraandthelumbarvertebra.Behindtheumpressureistheinferiorshoulderblade,andtherestofthespine’spressuredescendformthecentertotheperiphery.seatbackcomfortevaluationisacomplexissue.Throughthedistributionofseatbackpressure,wecancomprehensiveyzetheimpactaboutmaterial,skin,shapeandadjustmentmechanismoftheseatbackoncomfort.Afterconductingextensiveexperiments,thescholarssummedupaidealcomfortabledistributionpatternwhichcanmeettheneedsofthemajority.Therefore,wecanassumethatthemoreidealtheseatbackpressuredistributionis,themorecomforttheseatbackis.Withthisevaluationmethod,wecangetridofthedependenceonevaluationofthesubjectivetestandassesstheseatbackcomfortobjectively.However,theabilitytoendurepressureisdifferentindifferentpartsofthebody.Sowhenthedifferentregionsoftheseatbackpressuredistributionpatternchanges,thedegreeofinfluenceinseatbackcomfortisdifferent.

Onthebasisofthebasicmodel,webuildanoptimizedseatbackmodel.Inthismodelweusetheytichierarchyprocess[4](AHP)tocreateacomfortableseatbackmodel.AHPcansystematicyzethecomplexproblems.It posetheproblemtosub-problemsofsinglerulestepbystepinthelightofthenatureoftheissue.Comparedpairwisewiththeeachfactorofthesamelevelcanwegetaweightofevaluationresult,thenmakeanaccurateassessmentoftheproblem.RelyontheidealpressuredistributionpatternandAHP,thenyzethedifferentregionoftheseatbackandbuildtheimprovedmodel.

Symbols

A Judgmentmatrixofcriterionlevel

B Judgmentmatrixofschemelevel

C.I Consistencyindex

R.I TheaverageofConsistencyindex

C.R Consistencyratio

W FeaturevectorofA

V FeaturevectorofB

xi Themeanofdeviationsetinregioni

yi Thevarianceofdeviationsetinregioni

zi Thesumofsquaresofdeviationsetinregioniwi1 Theweightofxi

wi2 Theweightofyi

wi3 Theweightofzi

TheStructureofModelII

Themodelisdividedintothreelevels:

Thehighestlevel:Thecomfortdegreeofseatback.

Thecriterionlevel:Thedegreeofcomfortinfluenceindifferentregion.Accordingtothetestofresistcompression,wechoosefourregionsofhumanback(cervicalspine,shoulderblade,thoracicspine,lumbarspine)tobuildthejudgmentmatrixA.

Theschemelevel:Thedegreeofdifferenceinactualpressuredistributionandidealpressuredistribution.Weusemean,variance,sumofsquaresofdeviationsofdatasettodescribethislevel.Thecloserto0,themoreidealthepressuredistributionis.

ThestructureoftheAHPmodelisshowedinFigure5.

Figure5

TheJudgmentMatrix

Throughthedataofpressuredistribution[6],wegettheconclusionthatthepressureontheshoulderbladeandlumbarspineishigherthantherests.Andtheshoulderbladehashighanti-pressstrength.Accordingtothedegreeofresistcompression,werankthefourregionsasfollows:lumbarspine,cervicalspine,shoulderblade,thecentralofthoracicspine.Thenumber1-9indicatethecriterionincrementally.

Accordingtotheysisofthedataabove,wegetthejudgmentmatrixasfollows:

Table4:TheJudgmentMatrixofCriterionLevel

Cervicalspine

Shoulderblade

Thoracicspine

ThecentralofLumbarspine

Cervicalspine

1

3

0.3333

5

Shoulderblade

0.3333

1

0.2

3

Thoracicspine

3

5

1

7

ThecentralofLumbarspine

0.2

0.3333

0.1429

1

1 3 0.3333 5

AccordingtotheTable4,

A0.3333 1

0.2 3

3 5 1 7

0.2

0.3333

0.1429

1

BycalculatingwegettheC.R=C.I/R.I=0.04<0.1,whentheconsistencyratioC.R≤0.1,webelievethejudgmentmatrixhasgoodconsistency.

Table5:TheJudgmentMatrixofSchemeLevel

Mean

Variance

Sumof

squaresofdeviations

Mean

1

5

7

Variance

0.2

1

3

Sumofsquaresofdeviations

0.1429

0.3333

1

1 5 7

AccordingtotheTable5,

B

0.2

1 3

1

0.1429 0.3333

BycalculatingwegettheC.R=C.I/R.I=0.06<0.1,whentheconsistencyratioC.R≤0.1,webelievethejudgmentmatrixhasgoodconsistency.

3.2.4SolutionsandResults

1)ParameterCalculation

Tocalculatetheweight,wedefinetheparametersasfollows:

nAijj1

n

Wi (1) W(W1,,Wn) (2)

n W W

NWi (3) W'(1,, n) (4)

i1 N N

nBijj1

n

V (5) V(V1,,Vn) (6)

n V V

EVi (7) V'(1,,n) (8)

i1 E E

AndaftercalculatingW'V'T,wegettheweightofevaluationinTable6.

Table6:TheWeightofEvaluation

Scheme

Weight

Scheme

Weight

Meanofcervicalspine

0.1924

Meanofshoulderblade

0.0861

Varianceofcervicalspine

0.0496

Varianceofshoulderblade

0.0222

sumofsquaresofdeviations

ofcervicalspine

0.0213

sumofsquaresofdeviations

ofshoulderblade

0.0095

Meanoflumbarspine

0.4119

Meanofthoracicspine

0.0402

Varianceoflumbarspine

0.1062

Varianceofthoracicspine

0.0104

sumofsquaresofdeviations

oflumbarspine

0.0456

sumofsquaresofdeviations

ofthoracicspine

0.0045

2)SimulationAlgorithm

Step1:Buildtheapproporiatehierarchyby yzingvariousfactors.

Step2:Buildthejudgmentmatrixbycomparingthefactorsinthesamelevel.

Step3:Evaluatethereasonablenessofthematrixbycalculatingtheeigenvalues,eigenvectorsandC.R.

Step4:Gettheweightofthesystemevaluation

3)Thecomfortmodel

Accordingtothealgorithmabove,wecangetthefollowingmodel:

n

Comfort[xiwi1yiwi2ziwi3]

i1

(9)

ModelTesting

Weusethecomfortdegreeofseatbacktotestmodelbyngthecorrelationysis.Theangleofinclinationoftheseatbackinhorizontaldirectionisdifferent,sothepressureonthefourregionsaredifferent.Weneedtoconsiderthedegreeofseatbackindifferentcircumstances.Generally,weconsidertheanglerange100o-120o.ThedegreeofseatbackisshowedinTable7.

Table7:TheDegreeofSeatBack

Number

Theangleofinclination

Thedegreeofseatback

1

100o

0.2279

2

102o

0.2263

3

104o

0.2210

4

106o

0.2116

5

108o

0.2103

6

110o

0.2005

7

112o

0.1921

8

114o

0.1977

9

116o

0.1983

10

118o

0.1988

11

120o

0.1992

Thedegreeofseatbackindicatethesenseofcomfort.Thelowerthedegreeis,themorecomfortabletheseatbackis.FromtheresultsonTable7,wecancometoaconclusionthattheangleofinclinationabout112oisthemostcomfortableposture.Besidestheseatbackisthinenough,weshouldensuretheseatbackfitthecurvewedesign.Andweshouldaddpaddingatthebottomoftheseatbacktoensuretheangleofseatbackandhorizontaldirectionatabout112o.It’sbettertohaveadjustablesupportofthecervicalspinefordifferentpassengers.

Conclusions

First,wechoosefivepointsasthefeaturepointsofthespine.Afterestimatingtheparametersinhumandimensionsofadults,wefitacurvebyusingthefifth-orderpolynomial.

Second,webuildacomfortmodelbyusingtheyticHierarchyProcesstogetagoodcomfortevaluation.Thecomfortmodelcanevaluatethedegreeofseatbackcomfortinabetterway.Andthedegreeofseatbackindicatethesenseofcomfort.Thelowerthedegreeis,themorecomfortabletheseatbackis.Finally,wetestthemodelandgetaconclusionthatthebodywillfeelthemostcomfortablewhentheangleisabout112o.Themodelshouldaddpaddingatthebottomoftheseatbacktoensuretheangleofseatbackandhorizontaldirectionatabout112o.

AdvertisingSheet

Sittingisthemostfrequentbodyposture:wesitatwork,atschool,inthecar,onthebus,onthetrain,athomeinfrontofTV,toeat,torestandsoon.Youareprobablysittingdownrightnow.Generally,seatsshouldallowyourbodytobecomfortableandnotrestricted.Inaddition,achairshouldenableyoutochangepostureatintervals,ensuringthatdifferentgroupsofmusclescanbeusedforsupport,andthatnoparticulargroupofmusclesgetstired.

But,doyouknowwhatisthemostcomfortablechair?

Thehumanvertebralcolumnisthebackboneofthehumanskeleton,consistingofcervicalspine,thoracicspine,lumbarspine,sacrumandcoccyx.Accordingtotheergonomics,themain pointsareonthelumberspine,shoulderbladeandcervicalspinewhenthebodyinthecomfortablestate.Ifyouchoosetotravelbyne,youneedtohaveadjustablesupportofyourcervicalspineandlumbarspinetoenjoyacomfortableandpleasantjourney.Wecandothisinourairline.

Whycanwedoit?Atpresent,mostofthecivilianaircraftareAirbusandserieswhicharecreatedabroad.ThesizeoftheseatisbasedonthedimensionoftheEuropeanandAmericanwhichislargerthan.Thismayresultinasenseofpressureduetothehighseatback.Forthisreason,wetookthedimensionsofintoaccount.Weestablishedaseatbackcurvebyyzingreliabledataonthespinemodelandanthropometry.Andweconsideredthemostcomfortableangleofhumanspine,thenadjustedthepaddingandcurvetooptimizetheseat.Wedesignedasampleandwedidalotoft