2015年第四届认证杯数学建模国际赛优秀1584a

第四届“认证杯”数学中国

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Theslimlineseatsontheairne

:

Someairlinesareintroducingnew“slimline”seatsineconomyclass,whichtheoreticallyallowairlinestoincreasecapacity.However,manypassengershaveexpresseddispleasurewiththeseseats.Becausethecommonpointofthemisathinnerbackteandlesspaddingwhichsignificantlyaffectingpassengers’backprpersupport,andcause fortduringthejourney.

Inordertomaketheseatmorecomfortableandimproveservicequality,herewe plishtwotasks:

·Task1:Optimizethebestseatbacktecurve

Whenthebodybackvertebraisinanaturalform,thebacktensionisthelowest,sothatpassengerswillbemorecomfortableduringthevoyage.Consideringhumanbodyindifferentpositions,thesizeofthecompressive,shear

andbendingmomentthattheeverywhereofthevertebrabear,wecandeterminethebendingstateofthevertebra.Respectivelymakingmechanicalysisofrigidcylindervertebraeandflexiblecylinderintervertebraldisc,thenusingrecursivemethod,thentheshear,bendingmomentandcompressiveoneachvertebraandintervertebraldiskareobtained,respectivelysummingthreephysicaltiesastotalshear,totalbendingmomentandtotalcompressive.Thenwewillsummingthetotalshear,totalbendingmomentandtotalcompressive asthevalueof fortatthattime.Sothatwecangetamathematicalmodeltoassesspassenger fort,atthistimethemodelisanoptimizationproblem,wecansolvingthisproblembysoftwareLingo,togetaseriesoftheanglevaluewhenbackismostcomfortable.

Then,weuse,Fourierfittingandcubicsplineinterpolationtofittheseriesofanglevalue,sowecandrawanoptimalcurve,thiscurveisnotonlythevertebra’scurvewhenbodyfeelmostcomfortable,butalsoistheoptimalseatbacktecurvethatcanmakepassengersmostcomfortable.

·Task2:Designawell-lookingadvertisingmaterial

ShowingthebestseatbacktecurveobtainedinTask1、thedesignofthe“slimline”seats,intheme introducingourfeaturesandadvantagesofthe"slimline"seats.

Whileconsideringtheaestheticsofadvertising",thereareillustrationsandtextdescriptioninthematerial,theexpressionofproductisvivid,youcanintuitivelyaccesstotheinformation.

:Ergonomics.Biomechanics.Optimizationproblem,Lingo,Fourierfitting,Cubicsplineinterpolation,

Introduction

Whysomeairlinesareintroducing"slimline"seats

Withthedevelopmentofscienceandtechnology,Aviationtechnologyalsogotgreatdevelopment,therefore,relevantairneseatstechnologyresearchhas ethefocusofattention.Fortherequirementoftheairneseats,firstofallshouldbesafe,thesecondistoletthepassengerssittingcomfortable,ithasastricttechnicalrequirementsandspecifications.Thusforairlines,airpassengerseat’sselectionandmaintenanceareparticularlyimportant.Generally,theairlineswhenchoosingseats,inadditiontoconsidersafety,comfort,diversity,therearesomewilltendtochooselight,easytomaintain,theseat-"slimline"seatstoreducestheorderofthepassengerseatcostsandmaintenancecosts.

Thecharacteristicsofthe"slimline"seats

"Slimline"seats,inadditiontoweighingless,theoreticallyallowairlinestoincreasecapacitywithoutsignificantlyaffectingpassengercomfort.Theseseatsmayormaynotfeaturemoveableheadrests,andgenerallydonotfeatureadjustablelumbarsupport.Thecommonpointofthemisathinnerbackteandlesspadding.

Theapplicationofhumanbodyengineeringinthedesignoftheseatbackte

Thekeytothedesignofseatbackteistheseatbacktecurve.Whendesigningbackte,weusuallyusetheposturethatthevertebrainanaturalextensionofthestateasabasic.Notonlydoesseatbacktesupportthelumbar,butalsosupportthebackandhead,otherwiseitcan’tmaintainanaturalposture.Itstiltangleshallnotbelessthan90°inanycase,withtheimprovementofitstherestfunctionwillincrease,usually95°~120°isappropriate.Relationshipbetweentiltangleandsupportingpointcanbesummarizedintothefollowingpoints:

Whenthetiltangleissmall,selectsupportingpointinthesecondtothethirdlumbarvertebra;

Whenthetiltangleislarge,Supportingpointmovestothelowerpartofthethoracicspine;

Whenthetiltangleexceeds114°,itisnecessaryforthelumbarandlowerthoracicandheadthreepointssupport.

Accordingtotheabovepoints,supportpositionisdetermined,sothatwecandesignthegeometryoftheseatbackte,wecanalsoadjustthehardnessofthebackte,suchasmakethesupportingpointlargerhardnessthansomeotherparts,sowecanmaketheseatbacktemorecomfortable.

Theapplicationofbiomechanicsinthedesignoftheseatbackte

Biomechanicalysisshows,whentheanglebetweentheseatandthebackrestisbetween90°-120°,theshearbetweenseatandIschialtuberositycanbecompleyeliminated.Atthistimebuttockdon’thasslidingtrend.Inthiscase,wedoysisofvertebraeandintervertebraldiscwillconsiderthreemain,pressureonvertebraandintervertebraldisc,shearandthestresswhenintervertebraldiscbending.Accordingtotheexperience,whenthevertebrasufferedsmallerandbendingmoment,peoplewillfeelmorecomfortable.

TheDescriptionofProblem

Afterresearchingontheproblem,webelievethatthekeytosolvingproblemsisobtainingtheoptimalcurveoftheseatbackte,inordertoachievetheairne’sgoalsthattheycantakeadvantageofthe"slimline"seatswithoutchangingthemaininternalstructureandatthesametimemaketheseatmorecomfortable.Secondly,wewillwriteanadvertisingmaterial,tointroducethefeaturesandadvantagesofourdesignofthis"slimline"seats.

Theoptimalseatbacktecurve

Setanappropriateseatbacktecurvecanreducebacktension,sothatpassengerswillbemorecomfortableduringthevoyage.

Thebodywillfeelmostcomfortablewhenthebodybackvertebraisinanaturalform,toachievethis,theshapeoftheseatshouldbeconsistentwiththenaturalcurvatureofthevertebra.So,iftheseatbacktecurveisclosetothehumanecologicalcurve,itwillmaketheridecomfortgreatlyimproved.

Tomeasurecomfort，weconsiderthehumanbodyindifferentpositions,thesizeofthecompressive,shear andbendingmomentthattheeverywhereofthevertebrabear,establishedamathematicalmodelabouttheresultingphysical fort.Thefocusistoyzetheconditionsofthe24vertebraandIntervertebraldisc(inthiscasetheseriesofanglebetweenthemandverticaldirectionaredifferent):Respectivelystressysisfortherigidcylindervertebraandelasticdisc,atthesametimeusetherecursivemethod,thentheshear,bendingmomentandcompressiveoneachvertebraandintervertebraldiskareobtained,thenrespectivelysummingthreephysicaltiesastotalshear,totalbendingmomentandtotalcompressive.Thenwewillsummingthetotalshear,totalbendingmomentandtotalcompressiveasthevalueof fortatthattime.Whilethelowestvalueof fortcorrespondingtoasetofanglewhichcanbeuniquelydeterminedbythebendingstatuswherethecompositionofsonthevertebraissmallest.

Throughfindingoptimalsolutionandfittingthedata,wecanfindasmoothcurve.Thiscurveisnotonlythecurveofvertebrawhenitismostcomfortable,butalsocanbeseenastheseatbacktecurvethatcanmakepassengersmostcomfortable.Asaresult,wecangetthe

completemathematicalmodelwhichnotonlycanassessmentthe fortvalueofvertebrabutalsocanobtaintheoptimalseatbacktecurve.

Advertisingmaterial

Theadvertisingmaterialswillincludetheoptimalseatbacktecurvewhichwegotthroughthemathematicalmodeling,thedesigndrawingsofthe"slimline"seatsandourfeaturesandadvantagesofthe"slimline"seats

FundamentalAssumption

Eachsectionvertebraearerigid,equallength;Intervertebraldisciselasticmaterial.

Radiusofthespineisthesameeverywhere.

Ignoretheweightofintervertebraldisc,andthemassofthevertebraareuniformlydispersed.

Vertebraestressdeformationdoesnotoccur;Thedeformationofintervertebraldiscispurebending,don’tconsideringthetransversetwist,etc.

Allpassengersarenormaladults(24blockvertebrae).

Passengersposeisstilldependsonthebackofthechairforalongtime.

Symbols

Symbol Meaning

𝑁𝑢𝑖 Thecompressive ontheuppersurfaceofvertebra

𝑁𝑑𝑖 Thecompressive onthelowersurfaceofvertebraTheshearparalleledtotheuppersurfaceonthe

𝑄𝑢𝑖 vertebra

𝑄𝑑𝑖 Theshear paralleledtothelowersurfaceonthevertebra

Theanglebetweentheverticaldirectionofthevertebraand

α𝑖(𝑖=1,2,……,24)

𝐹𝑥1

Theactualverticaldirection

Theresultantonthevertebrainthehorizontal

direction

𝐹𝑦1 Theresultant onthevertebraintheverticaldirection

* densityofbone

a Thelengthofeachvertebra

g Accelerationofgravity

Theresultantontheintervertebraldiskinthe

𝐹𝑥2

horizontaldirection

𝐹𝑦2 Theresultant ontheintervertebraldiskinthevertical

direction

R Theradiusoftheintervertebraldisk’scrosssection

M Thebendingmomentactingontheintervertebraldisc

Modeling

Basedontheassumption,whenwedomechanicalysisofvertebraeandintervertebraldiscwillconsiderthreemain , pressureonvertebraandintervertebraldisc,shearandthestresswhenintervertebraldiscbending.Accordingtothebiomechanicsandhumanbodyengineering,whenthevertebrasufferedsmaller andbendingmoment,peoplewillfeelmorecomfortable.Wecalculatethebendingmoment,compressive andshearonthevertebraandintervertebralineachunit,thenwewillsummingthetotalshear ,

totalbendingmomentandtotalcompressiveasthevalueof fortatthattime.Sothevalueof fortwillincreasewiththeincreaseofthesumoftotalcompressive,totalshearandtotalbendingmoment.

Firstlywedomechanicalysisforvertebraandintervertebraldiskrespectively,providingconstraintsforseekinganoptimalcurveoftheseatbackte.

ModelBasic

Mechanical ysisofvertebra

(figure1:Mechanical ysisofavertebra）

Asshowninfigure,makemechanical ysisofvertebra.

Accordingtotheassumption,vertebraearerigidandstressdeformationdoesnotoccur.Sothereare4onthevertebra:

Thecompressive ontheuppersurface𝑁𝑢𝑖

Thecompressive onthelowersurface𝑁𝑑𝑖

Theshear paralleledtotheuppersurface𝑄𝑢𝑖

Theshear paralleledtothelowersurface𝑄𝑑𝑖

Accordingtotheassumption,passengersarestilldependsonthebackofthechairforalongtime.Sotheonthevertebraisbalanced.Atthistime,theanglebetweentheverticaldirectionofthevertebraandtheactualverticaldirectionisα𝑖(𝑖=1,2,……,24).

Supposetheresultantonthevertebrainthehorizontaldirectionis𝐹𝑥1,whilethatinverticaldirectionis𝐹𝑦1,sothat:

{𝐹𝑥1𝑖=0

𝐹𝑦1𝑖=0

24

(𝑖=1,2,……,24)

∑𝐹𝑥1𝑖=0

𝑖=124

∑𝐹𝑦1𝑖=0

{𝑖=1

Getthefirstpartoftheconstraintconditionsoftheoptimizationproblem:

(𝑄𝑢𝑖−𝑄𝑑𝑖)cos𝛼𝑖−(𝑁𝑢𝑖−𝑁𝑑𝑖)sin𝛼𝑖=0

{

(𝑄𝑢𝑖−𝑄𝑑𝑖)sin𝛼𝑖+(𝑁𝑢𝑖−𝑁𝑑𝑖)cos𝛼𝑖+𝜆ag=0

(𝑖=1,2,……,24)

Aftersimplification:

{𝑁𝑑𝑖−𝑁𝑢𝑖=𝜆ag·cos𝛼𝑖

𝑄𝑢𝑖−𝑄𝑑𝑖=𝜆ag·sin𝛼𝑖

Andaccordingtotheassumption:

(𝑖=1,2,……,24)

𝛼3=𝛼12=𝛼21=0.

Mechanical ysisofintervertebraldisk

(figure2:Mechanical ysisofanintervertebraldisk）

Asshowninfigure,makemechanical ysisofintervertebraldisk.

Accordingtotheassumption,intervertebraldiskisrigidandstressdeformationdoesnotoccur.

Thecompressive ontheuppersurface𝑁𝑑𝑖

Thecompressive onthelowersurface𝑁𝑢(𝑖+1)

Theshear paralleledtotheuppersurface𝑄𝑑𝑖

Theshear paralleledtothelo rsurface𝑄𝑢(𝑖+1)

Accordingtotheassumption,passengersarestilldependsonthebackofthechairforalongtime.Sotheontheintervertebraldiskisbalanced.Atthistime,theanglebetweentheverticaldirectionoftheuppersurfaceandlowersurfaceoftheintervertebraldiskandtheactualverticaldirectionarerespectivelyα𝑖and𝛼i+1(𝑖=1,2,……,23).

Supposetheresultantontheintervertebraldiskinthehorizontaldirectionis𝐹𝑥2,whilethatinverticaldirectionis𝐹𝑦2,sothat:

{𝐹𝑥2𝑖=0

𝐹𝑦2𝑖=0

23

(𝑖=1,2,……,23)

∑𝐹𝑥2𝑖=0

𝑖=123

∑𝐹𝑦2𝑖=0

{𝑖=1

Getthesecondpartoftheconstraintconditionsoftheoptimizationproblem:

𝑄𝑑𝑖·cos𝛼𝑖+𝑁𝑢(𝑖+1)·sin𝛼i+1−𝑁𝑑𝑖·sin𝛼i+1−𝑄𝑢(𝑖+1)·cos𝛼𝑖=0

{

𝑁𝑑𝑖·cos𝛼𝑖+𝑄𝑑𝑖·sin𝛼i−𝑁𝑢(𝑖+1)·cos𝛼𝑖+1−𝑄𝑢(𝑖+1)·sin𝛼i+1−𝜆ag=0

(𝑖=1,2,……,23)

Accordingtotheassumption,thedeformationofintervertebraldiscispurebending.

SupposeRistheradiusofthecrosssection,Misthebendingmomentactingontheintervertebraldisc.

2 3

𝑀𝑖=3𝐸𝛼𝑖𝑅

Buildingoptimizationproblem

Bythistime,webuildanoptimizedmodeloftheangle’ssizeoftheseatbacktethroughout:

Theobjectivefunction:

=∑ +∑ +∑ +∑ +∑

Constraintcondition

𝛼𝑖∈[0,25°]

𝑁𝑑𝑖−𝑁𝑢𝑖=𝜆ag·cos𝛼𝑖

𝑄𝑢𝑖−𝑄𝑑𝑖=𝜆ag·sin𝛼𝑖

𝑄𝑑𝑖·cos𝛼𝑖+𝑁𝑢(𝑖+1)·sin𝛼i+1−𝑁𝑑𝑖·sin𝛼i+1−𝑄𝑢(𝑖+1)·cos𝛼𝑖=0

𝑁𝑑𝑖·cos𝛼𝑖+𝑄𝑑𝑖·sin𝛼i−𝑁𝑢(𝑖+1)·cos𝛼𝑖+1−𝑄𝑢(𝑖+1)·sin𝛼i+1−𝜆ag=0

{ 𝛼3=𝛼12=𝛼21=0

(𝑖=1,2,……,24)

UsingoptimizationsoftwareLingotoruntheprogram,wegotAsetofoptimalvalueα(usinganglesystem)(SeeAppendix-Annex1)

𝑖

𝑓`(x)=tan𝛼𝑖，(𝑖=1,2,……,24)

Establishtwo-dimensionalrectangularcoordinatesystemonthebasisofsetingthetopofthevertebtraascoordinateoriginandthethedirectionofverticaldownwardasx-axis’positivedirection.

Leteachvertebralengthis25,sowecangettheobjectivefunction’sderivativedistributionat25points.(SeeAppendix-Annex2)

fitting

WecanuseFourierfunctionbysoftware :

methodtoobtaintheobjectivefunction’scurveofderivative

𝟖

f(x)＝∑𝐚·𝐬（𝐛ｘ+𝐜）

=

Thecoefficientamongthema,b,cis:

Thegraphicsis:

(figure2:ThederivativefunctioncurveofobjectivefunctionobtainedbyFourierfittingmethod)

Obtainedtheobjectivefunctionbyintegratingthefunction:

𝑥

F(x)=∫𝑓(𝑥)dx+C

0

(Cisaconstant)

CalculatedvalueofCbyundeterminedcoefficientmethodandthenweobtainedtheexpressionofobjectivefunctionF(x)is:

8 𝑎𝑖

𝑓(𝑥)=∑− ·cos(𝑏∗𝑥+𝑐)

𝑏𝑖 𝑖 𝑖

𝑖=1

Thevalueofcoefficientis:

Drawobjectivefunctioncurvebysoftware (ThealgorithmisintheAppendix–Annex3)

(figure3:Theobjectivefunction’scurveobtainedbyFourierfittingmethod)

Error ysis:

Sumofsquaresduetoerror、R-square、AdjustedR-square、RMSEare:SSE:0.000874

R-square:0.9974

AdjustedR-square:0.991

RMSE:0.01117

ThemoreSSEisclosetozerotheerrorissmaller,themoreR-squareiscloseto1thecurveisbetterfitting,themoreRMSSEiscloseto0,thecurveisbetterfitting.

ThusSSE>0.0001,RMSE>0.0１theerrorisrelativelylarge.Theerrorrangecorrespondingtothecurveisshownbelow:

(figure5:Theobjectivefunctionandthebandedareaformedbyerrorrange)

SoweneedtofindanoptimizationschemetomakeSSE<0.0001、RMSE<0.0.

ModelOptimization

OuroptimizationpurposesistofindabetteroptimizationmethodtomakeRMSE<0.0１、SSE<0.0001、AdjustedR-squaresufficientlyclose1,throughyzingandfindingrelevantinformation,wechoosesmoothingsplinefittingmethodtooptimizederivativefunctioncurve.

（ThealgorithmofisintheAppendix–Annex4）

“coefs”representativesinterpolatedpiecewisefunctioncoefficientmatrix,“breaks”representativespiecewiseintervalnodematrix,thefittingcurveimagesis:

(figure6:Thederivativefunctioncurveofobjectivefunctionfittedoutbysmoothingsplinemethod)

Its’SSE、R-square、AdjustedR-square、RMSEare:

SSE:8.409e-005

R-square:0.9997

AdjustedR-square:0.9939

RMSE:0.00917

Atthistime,thecasewhereRMSE<0.0１、SSE<0.0001satisfyerrorcontrollablerangeandR2greaterthantheprevious,thefittingdegreeisbetter.（ThealgorithmisintheAppendix–Annex5）

A=sp.coefsA2=sp.breaks

WeobtainedcoefficientmatrixBcorrespondingtotheprimitivefunctionwhichthefunctionCorrespondingtothecoefficientmatrixAcorrespondingto:(ThealgorithmandmatrixBareintheAppendix–Annex6)

Sotheobjectivefunctioncanberepresentedasapiecewisepolynomialfunction,Andthe

coefficientmatrixofinterpolationpolynomialoneachpiecewiseintervalcoefs=B,piecewiseintervalnodematrixA2,numberofsegmentsis25,theorderofpolynomialis5,wedrawtheobjectivefunctionimageswithasfollows:

(figure7:Theobjectivefunction’scurveobtainedbysmoothingsplinemethod)

ComparingoptimizedModelcurvewithpreviousmodelcurve:

(figure8:previousmodelcurveandoptimizedModelcurve)

Therefore,wefinallyobtainedagraphasfollows(aftercoordinatetransformedinequalproportion):

(figure9:Theoptimalcurveincoordinatethattransformedinequalproportions)

Thefunctionexpressionisapolynomialfunctionwhichcoefficientmatrixcoefs=B,piecewiseintervalnodematrixA2，numberofsegmentsis25,theorderofpolynomialis5.

economyclass.

Ourairlinesarenowintroducinganew"slimline"seatsin

Theseatback tecurvewhichnotsuitableforthevertebramakespassengersfeeltiredand

fortable

Whileournewslimlineseatswillliberatingmorepassengerspace,andmakeyoumorecomefortable.

Forithastheoptimalseatback tecurve,andnewtypeoffillmaterial.

Theapplicationofbiomechanicsinthedesignoftheseatback te

Wesummingthetotalshear ,totalbendingmomentandtotalcompressive asthevalueof

fortatthattime.

Whilethelowestvalueof

fort

correspondingtoasetofanglewhich

canbeuniquelydeterminedbythebendingstatuswherethecomposition

of sonthevertebraissmallest.

Throughfindingoptimalsolutionandfittingthedata,wecanfindasmoothcurve.ThiscurveIsnotonlythecurveofvertebrawhenitismostcomfortable,butalsocanbeseenastheseatback tecurvethatcan

makepassengersmostcomfortable.

Appendix：

Annex1

Columns1through8

9.3841 0.2142 -8.5110 -14.9356 -17.9228 -17.2622 -13.5392 -7.8644

Columns9through16

-1.5402 4.2764

8.8241

11.9555

14.6021

19.2318

27.9011

36.2675

Columns17through24

32.1341 6.9702

-26.0330

-42.4099

-39.3301

-28.4607

-19.9661

-15.6969

Column25

-13.4032

Annex2

Columns1through8

0.0522 0.0012 -0.0473 -0.0832 -0.0999 -0.0962 -0.0754 -0.0437

Columns9through16

-0.0086 0.0238 0.0491 0.0665 0.0813 0.1073 0.1563 0.2043

Columns17through24

0.1804 0.0387 -0.1456 -0.2401 -0.2220 -0.1594 -0.1114 -0.0874

Column25

-0.0746

Annex3

>>t=1:600;

>>

gx=-((0.07859./0.02094).*cos(0.02094.*t+0.883)+(0.1386./0.01043).*cos(0.01043.*t-1.978)+(0.07079.

/0.03142).*cos(0.03142.*t+2.002)+(0.07358./0.005429).*cos(0.005429.*t+2.026)+(0.03204./0.04189).

*cos(0.04189.*t-2.539)+(0.01488./0.05236).*cos(0.05236.*t-0.4748)+(0.008279/0.06283).*cos(0.006283.*t+1.212)+(0.002986./0.0733).*cos(0.0733.*t+2.156));

>>plot(t,gx)

Annex4

>>x=0:25:600;

>>sp=spline(x,k)sp=

form:'pp'

breaks:[1x25double]coefs:[24x4double]pieces:24

order:4

dim:1

Annex

5

A=

0.0000

-0.0000

-0.0019

0.0522

0.0000

0.0000

-0.0021

0.0012

0.0000

0.0000

-0.0017

-0.0473

0.0000