美国数学建模竞赛论文.docx
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美国数学建模竞赛论文.docx
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美国数学建模竞赛论文
Perfectsnowboardcourse
Summary
Nowadays,snowboardinghasbecomeaworldwidepopularsports,moreandmoreskifieldprovidessnowboardcourse.Howtobuildingasuitablesnowboardcourseisaprojectworthstudying
Inthispaper,threemodelsareconstructed.Inthefirstbasicmodel,weonlytake“verticalair”intoconsideration.Supposingsnowboardcourseisflat.Thesnowboarderhastoflyoutofthehalf-pipeverticallyifhewantstomaximize“verticalair”.Itisprovedinthemodelthattheworkoffrictionalforceisaconstantvalueifthecoefficientofkineticfrictionisaconstantvalue.Itisalsoprovedthatifthesnowboarderstandsupontheflatbottomandbendsonthevertside,thesnowboarder’senergyisincreasedbythedifferenceheightofcenterofgravity.Theequationcanbeestablishedbasedonlawofconversationofenergy,theobjectivefunctionconcerning“verticalair”isestablished.Wedevelopanonlinearoptimizationtofindthebestshapeofthesnowboardcourse.Themaximum“verticalair”
m;Theshapeofthesnowboardcoursewillbederived:
thelengthofflatbottom
;Thetransitionshapeslikeellipticalarc,thelengthofmajoraxis
;Thelengthofminoraxis
;Thelengthofvertis
;Thelengthofhalf-pipe
;Thewidthofthehalf-pipe
.
Intheseconddevelopedmodel,snowboarder’stwististakenintoconsideration.Wealsosupposethesnowboardcourseisflat.Thesnowboarderfliesoutofthesnowboardcourseatanangle.Thesnowboarderhasvelocityalongyaxisandzaxis.Supposingthevelocityalongyaxisistransformedtotwistangularatacertainefficiencyη.wefindthattheworkoffrictionalforceisalsoaconstantvalueandthetrackofthesnowboarderisanarcofcircle.Thenumberoftwististheobjectivefunction,wedevelopanonlinearoptimizationtofindthemaximumtwist.Themaximumroundnumberoftwist
.Andtheshapeofthesnowboardcoursewillbederived:
thelengthofvert
;Thelengthoftheflatbottom
;Thewidthofsnowboardcourse
;Thelengthofmajoraxis
;Thelengthofminoraxis
;Thelengthofhalf-pipe
.
Inthethirdadvancedmodel,consideringthepracticalsnowboardcourseisinclined,wemaketradeoffstodesignapracticalcourse.Themaximumroundnumberoftwist
.Themaximum“verticalair”
m;Theshapeofthesnowboardcoursewillbederived:
thelengthofflatbottom
;Thetransitionshapeslikeellipticalarc,thelengthofmajoraxis
;Thelengthofminoraxis
;Thelengthofvertis
;Thelengthofhalf-pipe
;Thewidthofthehalf-pipe
.
Afterbuildingthemodel,weanalyzethemodelonthecostofmanufacture,safetyanduniversalitytomakesurethatthemodelhaspracticalvalue.
Keywords:
Nonlinearoptimization,snowboard,theworkoffrictionalforce
1Introduction
Half-pipesnowboardingisanaerialacrobaticroutineperformedbysnowboardathletesonahalf-pipeconstructedofsnow.Themainaimtotherecreationalandcompetitivehalf-pipesnowboardingistoperformwellexecutedandstylishroutinesconsistingofcomplexaerialacrobaticmaneuversthatareexecutedashighabovetheedgeofthehalf-pipeaspossible[1].
Theperformanceassessmentmeasuretakesintoaccountalargenumberofsport-specificcomponents,suchas:
airtime,degreeofrotation,difficulty,style,andexecutionassociatedwitheachaerialacrobaticmaneuver;thesequenceandcombinationofaerialacrobaticmaneuvers;theamountofriskintheroutine;theoveralluseofthehalf-pipe,includingthelinetakenthroughthecourse;andhowtherunprogressesandflows[2].
Aboveallthefactorsmentionedindetermininggrade,theairtimeanddegreeofrotationareofgreatimportanceinasnowboardgame.Theshapeofsnowboardcourse
(currentlyknownasahalf-pipe)canaffecttheverticalair(verticalairistheverticaldistanceabovethehalf-pipe).Theshapecanalsoinfluencetheotherfactors,suchasthetwistintheair,thesafetyofplayersandsoon.
2Goals
Snowboardingisasportthathastakenvariousaspectsofothersportsandadaptedthemtoitsownpurpose.Whileitisstronglyrelatedtoskiing,italsoevolvedinpartfromsurfingandslalomskiingonwaterskis,hasborrowedmuchintechniquefromskateboarding,andhassimilaritiestowindsurfing.[3]
Inthemodelsconstructed,wemainlyaimtosolvethreeproblems:
●Findabestshapeofthesnowboardcourse(currentlyknownashalf-pipe)tomaximizetheproductionof“verticalair”byaskilledsnowboarder.“Verticalair”isthemaximumverticaldistanceabovetheedgeofthehalf-pipe.
●Tailortheshapetooptimizeotherpossiblerequirements,suchasmaximumtwistintheair.
●Maketradeoffsoftheshapetodesignapracticalcourse.
3Assumptions
Asourgoalisdesigningabestshapeofthesnowboardcourse.Forsolvethisproblemwellweconstructmodelswiththeassumptionsbelow:
●Ignoreairresistanceintheprocessofsnowboarding.
●Thecoefficientofkineticfrictionisaconstantvalue.
●Inthebasicmodel,thehalf-pipeandgrounddon’thaveangleofslope.
●WethinkaskilldsnowboardercanridethesnowboardsmoothlyIntheprocessofsnowboarding.sothereisnoenergylostofcollision.
●Thequalityofsnowinthehalf-pipedon’tchangeinourmodel.
●Theskillofsnowboarderisperfect,whichdoesn’taffectourmodel.
4SymbolsDescriptionforthewholeessay
Table.1Symbolsdescription
Symbols
Description
Unit
μ
Thecoefficientofkineticfriction
g
Accelerationofgravity
kg﹒m/s2
l
Thelengthofflatbottomoftheflatbottom
m
v0
Initialvelocity
m/s
v
Velocitytheplayerfliesoutofthehalf-pipe
m/s
h0
Depthofhalf-pipe
m
h
Thehighestheighttheplayerreaches
m
Δh
Verticalair
m
lw
Thewidthofthehalf-pipe
m
hv
Thelengthofvert
m
n
Thenumberoftwist
nf
Thefigurethesnowboarderfly
t
Thetimethesnowboarderspendsontheflatbottom
s
tf
Thetimebetweenthesnowboarderfliesoutofthecourseandlandsonthecourse
s
5BasicModel
briefintroduction
Firstly,wechoosetobuildupabasicmodel.Inthismodel,weignoretheinclinedslopeofthesnowboardcourse.AsFigure.1showsthehalf-pipeincludesvert,transition,flatbottomanddeck.Thesnowboardersflyoutofthehalf-pipefromvertvertically,sothathecanreachthemaximumheight.
Figure.1Snowboardcourse
Figure.2isapictorialaboutinternationalrulesofthesnowboardcourse.Ithasdescribedfourbasicessentialfactorofasnowboardcourse:
thelengthofhalf-pipeisbetween120and140m;thewidthisbetween16and18m;theheightisbetween3.5and4.5m.Intheend,itcomeswithaninclinedfrom16to18degrees.
Figure.2Theinternationalrulesofsnowboardcourse.[4]
Inthesnowboardinggame,thecompetitorsperformtomusicbetween5and8figureswhicharemarkedbyjudges,5judgesgivemarksoutof10formoveseachskierexecutesanddependingondifficulty.Thetotalofthosemarksareaddedforafinalscoreforthatround.Thereare3phrasestothecompetition.Qualifying,semi-finalsandfinal.Foreachroundtheskierhastworunsandkeepsonlyhisbestscore.Aqualifyingroundintwogroupsforthemen.Thetopthreefromeachgroupgodirectlytothefinal.Thesixotherscompeteinasemi-final.Aqualifyingroundwithroundwithonegroupforwomen.Thetopsixadvancedirectlytothefinal.Thenext12competeinthesemi-finals.[4]
Figure.3Asnowboarder’smotiontrackinahalf-pipe.[4]
5.1Analysis
Theverticalairinthebasicmodeldependsontheworkofthefrictionalforce,theinitialenergy,andtheheightofthesnowboarder’scenterofgravity.
Wedemonstratethattheworkofthefrictionalforceisaconstantvalueonconditionthatthecoefficientofkineticfrictionisaconstantvalue.Ithasnorelationwiththeshapeofthehalf-pipeifhefliesverticallyoutofthehalf-pipe.
Wealsodemonstratethatifthesnowboarderstandsupattheflatbottomofthehalf-pipe,andbendsatthevert,thatistosaythesnowboarder’scenterofgravityrisesattheflatbottom,andthecenterofgravitydropsdownatthevert,theenergycanincrease.
Asthesnowboarderneedstimetostandupattheflatbottom,sowedesignthelengthoftheflatbottomtoletthesnowboardertohaveenoughtimetostandup.Theshapeoftransitionpartisrandomintheoryonconditionthatitcanconnectthevertpartandflatbottompartsmoothly.Vertpartisnecessary,becauseitcanguaranteethesnowboardersflyoutofthehalf-pipevertically,onlyinthatway,canthesnowboardersreachthemaximumverticalair.
Formodelingthebasicmodelwemustdefinesomeothersymbols.
●Symbolsdescriptioninaddition
Table.2Symbolsdescriptioninaddition
Symbol
Description
Unit
Q
Theworkoffrictionalforce
J
θ
Includedangle
rad
r1
Pendulumlengthontherightside
m
r2
Pendulumlengthontheleftside
m
α1
Pivotangleontherightside
rad
α2
Pivotangleontheleft
side
rad
v1
Initialvelocity
m/s
v2
Finalvelocity
m/s
5.2Modeling
Forceanalysisofsnowboardersontransitionpart
Inthismodel,wedemonstratethattheworkoffrictionalforceisaconstantvalueandtheenergyincreaseasthesnowboarder’scenterofgravityrisesattheflatbottomandthecenterofgravitydropsdownatvert.
Figure.4Forceanalysisofsnowboardersontransitionpart
Theworkofthefrictionalforce:
(1)
(2)
(3)
WecanseefromFigure.2
(4)
(5)
Throughtheequation
(1),
(2),(3),(4),(5),wecanderive:
(6)
Aswingmodel
Figure.5Aswingmodel
AsFigure.3shows,tosimplytheprocessofanalysis,wechooseaswingmodel,whoseresultalsoappliestothehalf-pipe.
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