美赛a题文档格式.docx

美赛a题文档格式.docx

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II.Models……………………………………………………5

2.1problemone……………………………….………….5

2.2Problemtwo…………..……………………………11

2.3Problemthree……………………………………….13

III.Conclusions…………………………………………….14

IV.FutureWork……………………………………………15

V.Modelevaluation……………………………….……….15

5.1 

Model 

Advantages………………...…………………15

5.2 

disadvantages…………………..…………….16

VI.References……………………………………..………..17

I.Introduction

Snowboardingisapopularpoolgamewiththeworldofsports.TheU-Snowboard’lengthisgenerally100-140m,U-typewithawidthof14-18m,U-typeDepthof3-4.5m.theslopeis14°

-18°

.IncompetitionU-athletesSkatewithinthetaxirampedgemakingtheuseofslidetodoallsortsofspinsandjumpsaction.Therefereesscoreaccordingtotheathletes’performanceastheVerticalairandthedifficultyandeffectivenessofaction.TheactionsConsistmainlyoftheleapinggrabtheboard,leapingcatchofnon-board,rotatingleapingupsidedownandsoon.

1.1Halfpipestructure

Halfpipestructurecontains:

steelbodyframe,slideboard,stepstohelpslideandrails.

1.2Backgroundproblem

Inordertoimprovethemovementofthewatch,itcanbeimprovedfromtwoaspects:

orbitandtheathletesthemselves.Nowaccordingtotheproblemtheorbitcanbedesignedasacurve.onthecurvetheathletescangetamaximumspeed.Thedesignoforbitincludesawiderangeofcontent,suchastheshapeofU-groovedesign,trackgradient,widthandlengthdesignedtohelpthedesignofslidingsection,andsoon.Therationaldesignofhalfpipecanbeachievedtotransformtheenergytoefficiencypower,maketheathletesachievethebestperformanceintheinitialstateoftheair.Thispaperdiscussestherationaldesignofhalfpipetotheseissues.

1.3Athletesaerials

Athletesonthehillsidecoveringwiththicksnowskilldownwiththeinertiaoftheplatform,jumpintotheair,andcompleteavarietyoftwistsorsomersault.Ratingcriteria:

vacated,takeoff,heightanddistanceaccountedfor20%;

bodypostureandthelevelofskillaccountedfor50%;

landing30%.Accordingtotheprovisionsthedifficultyofmovementsarerangedintosmall,mediumandlarge.Theathletesoptiontheactions.However,thegroundmusthaveaslopeofabout37°

and60cmabovethesoftsnowlayer.

1.4Assume

Inordertosimplifythemodelandcancometoafeasiblesolution,makingthefollowingassumptions:

1•theshapeofasnowboardcoursehasalowestpoint,thewideandthelengthofthesnowboardcourse.

2•airresistancecanbenegligible.

3•itisassumedthattheathletesthemselveshavenoinfluence.

II.TheDescriptionoftheProblem

Thisproblemisatypicalengineeringdesign,involvingalotofdisciplines,suchasadvancedmathematics,engineeringmathematics,mechanicaldynamicsandbiologicaldynamics,aswellastherelevantprovisionsofsportscompetitionandjudgingstandards,andsoon.

Accordingtotherequirementoftheproblem,determinetheshapeofasnowboardcoursetomaximizetheproductionof“verticalair”byaskilledsnowboarder.Forthisproblem,wecanusetheboundaryconditionsandsiteproperties（e.g.symmetry）andotherrequirementstoestablishfunctionalcombiningwiththevariationprincipleEulerequations.Theoriginalequationcanbechangedintoafunctionalextremumproblem.

Secondly,weoptimizethemodelboundaryanddeterminetheappropriatesnowboardcourse’sslopetoetomaketheathletesperformmaximumtwistordomoredifficultaction.

Finallyapracticalmodelshouldmeettherequirementsofsafety,sustainabilityandeconomic.Accordingtothehighdegreeofhumansecurity,thesourceofthesnowandthetopographythemodelwillbeoptimizedmorereasonable.

II.Models

2.1problemone

Asshown（3.1）Aisthelowestpointofthesnowboardcourse.

FromAtoBwewanttofindacurvetomaketheathletesgetthemaximumverticaldistanceabovetheedgeofthesnowboardcourse.

Figure1Half-pipe

SetAasoriginofcoordinate.

Awingofconservationenergyandneglectingairresistance,themathematicalfunctionis,

（1）

Where

v0istheinitialvelocity（m/h）,

visthevelocitytowardsdestination（m/h）,

misthemassofanathlete（kg）,

wistheenergywhichismadebytheathlete（J）,

histheVerticalheight（m）,

Afisthefrictionwork（J），

Accordingto 

mechanicalanalysis：

（2）

Where

θistheangletheanglebetweenthetangentandthehorizontalline,

Nisthepressureontheobject,

ristheradiusofcurvature,

Accordingtofrictionformula：

（3）

To 

（3） 

into 

equation 

（1）,combinedwithcalculus：

（4）

Friction 

acting 

Af：

（5）

Usehigher 

mathematics：

（6）

（7）

Accordingtothenatureofthecurve,thespeedcanbeassumedtosatisfythisexpression：

（8）

Putalltheseformulasinorderandsupposetheexpressionforthefunctional：

（9）

Set

（10）

Reference 

Euler 

equation：

（11）

Obtained：

（12）

（13）

（14）

To（12）-（14）into 

equation（11），

（15）

Integrateit：

（16）

Simplified：

（17）

（18）

Suppose：

So：

Substitutedinto 

theaboveequation：

（19）

（20）

（21）

Obtainedthefinalresults：

（22）

Forthedifficultequation,weobtainnumericalsolutionsbynumericaldifferentiation,andthenobtainedfunctionequationbynumericalfitting

method：

Discreteinterval[0,8],

takesteps：

h=1.

Eachpointxi,i=0,1,……8.

Everyinterval[xi,xi+1],

theboundaryconditions：

y（0）=0,y'

（0）=0.

Intothe 

formula（22）for 

the 

boundaryconditions：

C=0

Put

intoformula（22）：

NumericalSolutionof 

eachpointis 

obtained 

inturn：

x

1

2

3

4

5

6

7

8

y

0.0069

0.0094

0.0336

0.1329

0.3669

0.7872

1.4127

4.0382

Functions 

imagesand 

functionequationareobtainedby 

numericalfitting 

onExcel：

Figure2Theresultsofthenumericalsolutionofthefittingimage

After 

fitting 

y=0.0003x6-0.0063x5+0.0435x4-0.1289x3+0.1594x2-0.0571x-0.0008（23）

Thentheentireimagecanbegotbysymmetryalongthey-axis.Thismodelsofproblemonecanbesolved.

2.2Problemtwo

Onquestion2,itsmainpurposeistoimprovethemodelinproblemoneundertheconditionofmeetingtherequirementsofotherpossiblecases.Analyzeotherpossiblerequirementswhichincludeanumberofaspects,suchasthemaximumtwistintheair,players’safetywhentheyleavethegroundandthestabilityofathleteswhentheyland.Amongthem,wemainlyconsiderthemaximumtwistofsnowboarderintheair.

Whenplayersleavetheground,theyareonlyaffectedbygravityandairresistance.Weignoretheplayers’adjustmentintheair.Aftertheprojectflyingoutoftheground,inordertoanalyzingsimplyandthinkingclearly,thevelocityoftheobjectisdividedintolinesvelocityandangularvelocity.Velocitycontainscomponentsofthreedifferentdirections:

horizontal,verticalandlongitudinal.Angularvelocityconsistsofsomersaultangularvelocityandtwistangularvelocity.

Figure3Flip 

velocityanalysis

Afterathletesflyingoutofthecourse,thevelocityoflongitudinaldependsonarationalallocationoftheirownenergywhentheyski,sothedesignofcoursecannotbeconsidered.Verticalspeeddeterminesthemaximumheightwithwhicha