2011美国数学建模竞赛A题优秀获奖论文.pdf
A MathematicalModel in EvaluatingtheSnowboardHalf-pipeDesignTeam#9253February 14,20111Team#9253Page 2 of 19Contents1Introduction31.1Backgrounds.31.2Problem Restatement.42Notationsand Assumptions52.1Basic Terms and Variables.52.2ModelAssumptions.63The Model73.1Sliding Route.73.2ModelEquationsand Demonstration.74Discussionof the ModelResult114.1ConstraintConditions.114.2Vertical Air.124.3Twists:the MaximumAngle of Rotation.164.4Applicationin Practice.175ModelEvaluating17References19Team#9253Page 3 of 191Introduction1.1BackgroundsEven never paid any attentionto the snowboardsport before,people wouldbeshocked by the extreme attractive?ip tricks and spin maneuversperformedbyShaun White at the?rst sight.As one of the most famous talented snowboarderand WinterOlympichalf-pipechampions,Shaun Whiteis a legendin snow-board historynot just in name only.Apartfrom the superiornaturalgifts,per-fect techniquesand the large quantityof exercises,are there any else elementsmake Shaun White among the top snowboarders?“It sa mix of factors.Ultimately,though,White ssoaringperfor-mances all come down to raw energy.”replied by Louis Bloom?eld1Seeing from the angle of physics,the main laws and factors can be summa-rized as follows:?NewtonsLawsof Motion2:When snowboardersslidingon the half-pipe,these are the basic principlesthey are satis?ed with.?Gravity:The gravityacting on the centre of mass of snowboardershelps toslide downthe slope as well as to drag them back into groundafter takingoff from the lip.?Friction:There are two essential kinds of resistingforces,kineticfrictionand air resistance acting on snowboardsand players in the motion.Snow-boarders can wax the board to decrease the force of friction,while in othercondition,the frictioncan be used to help snowboardersto speed down.?Energy:The conservationof energy allows players to store potentialgrav-ity energy by adjust ramp in height accordingto differentsituations.Analyzingin detail,we foundthat these factors can all be in?uencedby theshape design of a half-pipeif ignoringthe effects by qualitiesof snowboardandsnowboarders.When we start to search for informationand data,we foundthatthe originallyhalf-pipeswere simplya half sliced large pipe.Since the1980s,half-pipeshave had extendedwith a?at ground(the?at bottom)addedbetween the quarter-pipes3.Team#9253Page 4 of 19Then,there appearmanydesigns of half-pipessuch as the mini-pipe,theverthalf-pipeand super-pipe.Today s half-pipesused in snowboardingaregenerallymade in the shape of as the FederationInternationalede Ski(FIS)recommended4.Althoughit is widelybelievedthat the characterof a half pipe mainlyde-pends on the relationshipbetweenfour qualities:the transitionradiusand theheightof vert,the widthof?at bottomand slope angle of the pipe,we couldhardly?nd well organizedphysicaland mathematicalanalysis or comparisonson the differentshape of half-pipedesigns.1.2ProblemRestatementWith a?rst observationon the most popularhalf-pipesused nowadays,we pre-sume that there exits some connectionsamongthe designof a half-pipe,theverticalair,which refers to the maximumverticaldistance above the edge of thehalf-pipe,and the maximumturns can be achieved by a skilledsnowboarderinthe air.Actually,the theoreticaldemonstrationis rather simple.As every one cansee,the fact is obviousto be foundupon?rst observationthat the highera s-nowboarderjump,the more air time he willgain to accomplishhis moves inthe air.Therefore,once we take the movementprocess startingfrom the time asnowboarderset out from the drop-inramp to the momenthe/her?ying out ofthe half-pipeat the?st time as a whole,the?yingout speed,which has a crucialin?uenceon the verticalair one athletecould achieve,dependingon the me-chanicalenergy one had in the initialstate and the heat energy one lost duringthe sliding,accordingto the law of the conservationof energy.Our objective is to optimizethe shape of the half-pipeby discussingand an-alyzingthe?nal results proposedby our modelwithspeci?c constraints.We llbegin our mathematicalanalysis and developa detailedmodelto simulatetheslidingprocess in half-pipeusing physicaland mathematicaltheoryin the fol-lowingsections.Team#9253Page 5 of 19Figure 1:The Cross-Section of Half-pipe2Notationsand Assumptions2.1Basic Terms and VariablesFirst,wed like to explainsome basic terms whichare widelyused to describethe shape of the half-pipein our paper on(?g.1).And the followingpart de?nes importantvariables which are widelyused inthis paper while some additionalparametersonly con?ned to particularsectionsmay be de?ned later.the slope angle of a half-pipe the intersectionangle of the directionof v0and X-axisthe intersectionangle of the normaldirectionand Y-axis when slidingin thegroovel the length of a half-pipeh the height of the vertR the radius of the piped the widthof the?at bottomof the pipeHthe height from the?oor to crown of a pipev0the entry velocityof the snowboarderTeam#9253Page 6 of 19v2the projectionof v0to the Y-Z planevBthe instantvelocityof point BvCthe instantvelocityof point CvDthe taking-offvelocityfthe force of frictionfrthe projectionof f to the Y-Z planeNthe normalforce acting on the snowboarder the coef?cient of frictionwfthe workof frictionforce in slidingvmaxthe maximumvelocityin the process of slidingG the gravitationalaccelerationacting on snowboardersGmaxThe maximumgravitationalacceleration that snowboarderscan bear witha good control of balance2.2ModelAssumptionsAs our observationis not focusingon the snowboardersbut concentratingonanalyzingthe characteristicsof the half-pipe,we considerthe snowboardersata giventechnicallevel to simplifyour model.The followinglist containsthespeci?c informationand statement of the technicallevel.?The weightof the snowboarder/boardis uniformdistributed,meanwhilethe board and playerare alwaysconglutinatetogetherin the simulatedprocess.?If ignoringthe twistand positionchanges in centre of gravity(COG),wecan assume the two as one mass point,whichcoincides withthe COG.?Snowboardersenter a U-shapedpipe witha given speed v0as long as theentry ramp is well designed.Besides,in our analysis,we also made several key assumptionson half-pipesfor simplicity:Team#9253Page 7 of 19?The half-pipeis coveredwith?rmsnowand the surface of the pipeisplane.?Generallythe snowboardersstart from the same height relative to the pipein competition.At this point,all their energy is in the form of gravitationalpotentialenergy.?The effects of environmentalconditionssuch as the geographylocationand climate whichcan not be in?uencedby the shape design of the half-pipe are out of our consideration.?we assume that the half-pipeis set in a vacuumspace.by neglectingtheminorcontributionsof the air drag either.?We willonlyinvestigatethe semi-circularconcave half-pipein our mod-el.The purposebuiltramphalf-pipewillbe discussed in the discussionsection of our paper if time allows.3The Model3.1SlidingRouteBased on the analysis on how slide skills on the“U”course affects taking-offheightby XiaojianTian and WeiguangChen from Dept.of PE in HarbinPhys-ical EducationInstitute5,we assume that the snowboarderenteringinto thegroove at a speed of v0and slidingwitha given route(?g.2),whichis widelybelievedto lead to minimumenergy loss,in the pipe before the?rst taking-offhappens.In addition,consideringthe loss in energyand the deteriorationinvigor in the followingtime,the air height and twists afterwardsis generallyde-creasing every time the player?ies,we will demonstratethe process from whenthe snowboarder?rst entering into the half-pipegroove to the moment when heis takingoff the lip.3.2ModelEquationsand DemonstrationAs the picture(?g.2)shows,we builda path based on X-Y-Z coordinatesystemby de?ningthe cross-section of the half-pipeasY-Z plane and locating the originof the system at the point O.In order to simplifyour analysis,we also de?neda 2-dimX-Y coordinatesystem as a correspondenceto the X-Y-Z coordinatesystem,the originof which is positionedon point A.Presume that the directionTeam#9253Page 8 of 19Figure 2:The X-Y-Z CoordinateSystem and Corresponding2-dim X-Y Coordi-nate Systemof v0satis?ed withthe conditionthat it is always hold a same angle withX-axis,which is to say,we wouldget a straight-linepath if the concave is unfoldedinto a plane(?g.2).The movingpath,whichis de?ned in the assumptionpart of the paper,canbe dividedinto three sections:the slidingdownroute AB in the concave ramp,the straight-linemovingroute BC on the?at track and the slidingup route CD.This allows a relativelyindependentobservationon each part.On our investigationsabout the curve route AB and CD,the motionof thesnowboarderis separated into two easier formulatedkinds of movementusingmethodsprovidedby the classical mechanics:a straight-linemove along the X-axis and circularmotionwithradiusof R in the Y-Z plane.Thus,we can writeX=X,Y=X tan,=Y/R.The numeraland analyzingsolutiondevelopedby ShijunXu and XiaolingRen 6 is stimulatingand has been widelyused to analyze the frictionon circu-lar orbit.However,as the solutionthey proposedis a 2-dimensionsituation,itTeam#9253Page 9 of 19requiredampli?esto adjust to our analysis.Figure 3:Forces Actingon the Snowboarderin the X-Y Cross-sectionFirstly,we decomposedv0(the velocityof snowboarder)withrespect to theX-axis and Y-Z plane.Then makingdecompositionsto the force of frictioncaus-ing by the snowboardersslidingdownmovementon the groove,and notingthe dividedfrictionin the Y-Z plane as fr,we can obtain the whole forces actingon the snowboarderin the X-Y cross-section(?g.3).Accordingto the conserva-tion of energy,we can describethe physicalprocess as equationsin tangentialand normaldirectionseparately withinitialconditionv0f2cos-fr=mdv2dt(1)N-f2=mv2r(2)Where fris de?ned by fr=Nsin and f2is a projected force of gravityin Y-Zplane,whichcan be given by f2=mg cos.Furthermore,we can obtaindNdt-f2cosddt=2mv2dv2dtwhichis a result of makingpartialderivationwithrespect to the variablet onboth sides of the equation(2).Puttingv2=rd/dtand equation(1)into the former equation,we havedNdt-f2cosddt=2(f2cos-Nsin )ddtwhichcan be rewroteasdN+2Nsin=3f2cos(3)Team#9253Page 10 of 19As we can see,it is a?rst-orderlinear non-homogeneousdifferentialequation.Using the methodof Variationof ArbitraryConstant,we can obtainN=3mg cos1+42(sin )2exp(2 cos sin+sin )+c exp(-2 sin )(4)Moreover,the slidingfrictionbetween the snowboarderand?at is noted as f=N,then we havef=3mg cos1+42(sin )2exp(2 cos sin+sin )+c exp(-2 sin)(5)Taking the initialconditionthat the slidingin speed v0of the snowboarderatpoint A into account and let=0,N=m(v0sin )2Rwe can getc1=m(v0sin )2R-3mg cos1+42(sin )22m sin(6)Thus,the functionof frictionon route AB isf1=3mg cos1+42(sin )2exp(2 cos sin+sin )+c1exp(-2 sin )(7)Due to the Conservationof Energy12mv20+mg x sin+mgR sin cos-wf=12mv2(8)where wfis the powergenerated by the frictionto till.With equations(1)to(8),we can calculate the velocityof the snowboarderatevery pointwe wantto knowon the curve route AB,whichcertainlyincludesthe speed of the player at point B.Note it as vB,then we can discuss the routeBC.Duringthis process,thesnowboarderis makingstraight-linemovementand the forces acting on the s-nowboarderare the weightW,the normalforce N and the frictionalforce f2,due to the snowboardersmotion.Besides,the directionof W is perpendicularto the?at and f2is contraryto the movingdirectionwhileN is along withtheX-axis.Obviouslywe can writef2=mgsin(9)Accordingto the Conservationof Energy,we can obtain12mv2C+mg xsin-wf=12mv2(10)Team#9253Page 11 of 19Whichgives us the instantspeed vCat pointC.Whenit comes to the routeCD,similaranalysis can be used as the symmetryqualityof the half-pipe.Let=/2,N=mvCsin R+mg cosCombiningwiththe equation(4),we obtainc2=exp(sin )mg cos+m(v2sin )2R-3mg1+(4sin )2(11)Applyingthis into equation(5),we havef3=3mg cos1+42(sin )2exp(2 cos sin+sin )+c2exp(-2 sin )(12)fv(y)is denoted as the numeralsolutionfunctionof speed v.However,whentakinginto practise,we only calculatethe instantvelocityon pointsneeded ofcurve AD by MATLABprogram.v=fv(y)(13)4Discussionof the ModelResult4.1ConstraintConditionsThe characteristicswhichdecidethe shape of a half-pipeare the slope angle,the lengthof the vert,the transitionradius,the widthof the?at,and so on.Given that the snowboarderslidingout of the concave at a?xed direction,themaximumverticaldistance above the edge only relates to the rate of takingoffspeed.Taking the functionof the snowboarderlike pumpingskillout of consider-ation,the vert wallof the pipe wont help to increase the energy of the slidingprocess.Instead,the vert part even can decrease speed slidingout of the con-cave ramp.Therefore,in order to gain verticalair,we set the heightof vert wallh as zero in the model.Then we take the slope angle into consideration.As we mentionedin the in-troductionsection,the FIS providesplayers a range of recommendangle with-out detailedexplanation.Now,we ll offer a formulato estimate the slope angelmg sincos=mgcosTeam#9253Page 12 of 19We obtainedthis formulafroman uniquesituationthatthe snowboarderslidingat an constant speed on the?at bottomwitha path angle .As long as and are given,we can calculates the slope angle out.Set=0.1,=70?,we can get the results=16.3?and while other param-eters are given,we can easily get some conclusionsabout the widthof the?atd.?When 70?,the rider is deceleratingin the area of?at bottom,and vDincreases while d goes down.Only withthese simplesituations,obviouslywe can not get the values of dor other parameters.However,withsuch analysis,we can at least give the con-straintconditionsthat offered below(supposethe rider is in the same technicallevel):-Constraint1:The maximumof the slidingspeed vmaxis limitedcause s-nowboarderswouldlose control at a certain speed.-Constraint2:The maximumgravitationalacceleration(G)acting on snow-boardersis limitedfor playercan only keep balance and move their bodyat easeunder a certain amountof presure.-Constraint3:The reacting time providedby the?at area of the pipe shouldbe adequate for snowboardersto adjust their sliding.4.2VerticalAirTo illustratethe use of the above model,we willdo a sample calculation?rst.Table 1:Values of Parametersm(kg)g(m/s2)v0(m/s)d(m)R(m)609.816.3708830.1Team#9253Page 13 of 19For example,letssay at a given instantv0=8 m/s,g=9.8 m/s2,=16.3?,=70?,d=8 m,=0.1,m=60 kg and the transitionradiusR=3 m.The generaltrend of snowboardersspeed in the whole process of movementis proposed by(?g.4).Figure 4:The General Trend of SnowboardersSpeedis the angle that enables the riderslidingat a constantvelocitywe havecalculatedin the former part.As(?g.4)shows,the snowboardersslidingspeedcurves coincides very well withthe analysis we made before.Therefore,we cancon?rmthat our model has a good simulationresults.Figure 5:The ChangingLine of Taking-offSpeed vDas R VariesThen,keep the other parametersstay the same and let R values from2.5 to5.5 m at an intervalof 0.2 m,we achieve a serial of results.With these results,we manufacturehow the taking-offs
