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美赛a题
PerfectHalf-pipe:
TheThinkofSnowboardCourse
Abstract
Withthecontinuousprogressanddevelopment,Peopleareactivelyinvolvedinsportsandexploringinitcontinually.Skiingispopularwiththemajorityofsportsfansgraduallyunderthiscondition.Especially,Snowboardingwithgoodview,challengeandthebasisofthemassesdevelopsrapidlyandhasbecomeamajorOlympicprojects.Inthispaper,howtodesignandoptimizethesnowboardcourseofhalfpipeisdiscussedindetail.Westrivetogettheperfectcoursesothatsnowboarderscanachievethebestmotionstateintheestablishedphysicalconditions.What’smore,itmaypromotethedevelopmentofthesport.
Thisproblemcanbedividedintothreemodulestodiscussandsolve.Forthefirstproblemofthedesignofhalfpipe,itcanbebasedonthepointoftheenergyconservationlaw.Themethodoffunctionalanalysis(VariationprincipleandEulerdifferentialequation)isusedtosetupequations,whenthesecondarycauseisignoredandtheboundaryconditionsaretakenintoconsideration.Thecurveequationisobtainedbytheaboveequation,thatis,askilledsnowboardercanmakethemaximumproductionof“verticalair”.Forthesecondquestion,athletes’maximumtwistintheairandsomeotherfactorsneedtobetakenintoaccountwhentooptimizethepreviousmodel,sothatcurvecanmeettheactualgameconditionsandappraisalrequirementsasmuchaspossible.Ultimately,asatisfyingcurvewillbegot.Thethirdproblemisaproblemrelativelyclosecontactwiththeactual,whichistosettingdownaseriesoftradeoffsthatmayberequiredtodevelopa“practical”course.Inthispaper,fortheformulationofthesefactors,themaindiscussionsarethethicknessofsnowonhalfpipeandtheaspectofeconomyfortheconstruction.
Afterdiscussingthesethreeaspects,thepaperfinallysummarizesaconstructionprogramandevaluationcriteriaofthecourseincurrentconditions.Finally,byevaluatingtheadvantagesanddisadvantagesofthewholemodel,weputforwardtheadvancednatureofthemodel,butalsopointoutsomelimitationsofthemodel.
Keywords:
Snowboardcourse,Half-pipe,Functional,Eulerequation,Fittedcurve,Numericaldifferentiation
Tableofcontents
Tableofcontents……………………………………………1
I.Introduction……………………………….…….………..2
1.1Halfpipestructure…………………….……………..2
1.2Backgroundproblem……..………………………….3
1.3Athletesaerials….………………………..…………..3
1.4Assume………………………………………………..3
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 Model 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),
Obtained:
(15)
Integrateit:
(16)
Simplified:
(17)
(18)
Suppose:
So:
Substitutedinto theaboveequation:
(19)
(20)
Integrateit:
(21)
Obtainedthefinalresults:
(22)
Forthedifficultequation,weobtainnumericalsolutionsbynumericaldifferentiation,andthenobtainedfunctionequationbynumericalfitting
method:
Discreteinterval[0,8],
Where
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
0
1
2
3
4
5
6
7
8
y
0
0.0069
0.0094
0.0336
0.1329
0.3669
0.7872
1.4127
4.0382
Functions imagesand functionequationareobtainedby numericalfitting onExcel:
Figure2Theresultsofthenumericalsolutionofthefittingimage
After fitting the equation:
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