数学建模美赛优秀论文重点Word格式文档下载.docx
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数学建模美赛优秀论文重点Word格式文档下载.docx
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L.V.Smith:
Two-springModel…………………….…6
2.Convention………………………………………………………………………….6
2.1Terminology……………………………………………………………….…..6
2.2Variables…………………………………………………………………….…7
2.3Assumption………………………………………………………………….…8
3.COPModelasSweetSpot……………………………………………………….....8
3.1WhyisCOP……………………………………………………………………8
3.2ToFindtheCOP………………………………………………………………8
3.2.1AssumptionsabouttheCOPmodel……………………………………..8
3.2.2TheCOPModel…………………………………………………………8
3.3ModificationoftheCOPModel………………………………………….…..11
3.3.1TheSpinoftheBall……………………………………………………11
3.3.2TheAnglebetweenBallSpeedandtheBat……………………………12
3.3.3MotionofInertiaModification………………….…….……………….12
3.4DisadvantagesoftheCOP……………………………………………………13
3.4.1COPisChanging……………………………………………………….13
3.4.2COPisnotthelocationwheremaximizetheballspeed………………14
4.TheNOVModel………………..…………………………………………………14.
4.1TheadvantageoftheNOVModel……………………………………………14
4.1.1MoreconditionintheNOVModel……………………………………..14
4.1.2WhyisNOV…………………………………………………………….15
4.2TheNOVModel………………………………………………………………15
4.2.1VibrationEnergyoftheBat…………………………………………….15
4.2.2VibrationStateoftheBat………………………………………………18
4.2.3VibrationWaveformsandstandingwavenode………………………...19
4.3CoefficientofRestitution……………………………………………………..22
4.3.1TheDefinitionofeandtheDifferencebetweeneande……………22
4.3.2Thedynamicstiffnesskandhowitworks……………………………...23
5.FurtherDiscussion…………………………………………………………………24
5.1SweetZone—TheCombinationofCOPandNOV…………………………..25
5.2Maxperformancepoint………………………………………………………25
6.ApplicationoftheModel………………………………………………………….25
6.1CorkedBat……………………………………………………………………25
6.1.1changeofbatmass……………………………………………………...25
6.1.2changeofelasticenergy………………………………………………..26
6.2MetalBat……………………………………………………………………..27
6.2.2CharacteristicsofMetalBat……………………………………………27.
6.2.3FasterSwingSpeed……………………………………………………..29
7.Conclusion…………………………………………………………………………30
8.Reference…………………………………………………………………………..32
1Introduction
Everysporthashadtoevolvewiththetimetomeetthedemandsandinterestsofplayersandfans.Baseballevolutionbeginswiththedevelopmentofbaseballbat.
Baseballbatscomeinvariouslengths,diameters,materialsandcolors.Undoubtedly,aheavierbatcouldexertmoremomentumontheball.Butwithunchangedimpulse,aheavierbatalsomeanslessspeedintheswing.Alsothecoefficientofrestitutionofabatcontributestohowfarabatted-ballcouldtravel.Andthematerialchangedasinvolvementoftechnology:
●WoodBat(Ashisusedbecauseofitshardness,durability,strength,weightandfeel.
●CorkedBat(acorkedbatisaspeciallymodifiedbaseballbatthathasbeenfilledwithcorkorsimilarlight,lessdensesubstances
●MetalAlloyBat(Aluminumspecificallyinthe1970'
s
Inthemid-1980s,theNationalCollegiateAthleticAssociation(NCAAbeganconsideringguidelinesforlimitingbatperformance,andin1988,thefirstrulewasestablished.Thatrule,knownasthe“minus5”rule,statedthatabatcouldnotweigh(inounceslessthan5unitsfromitslength(ininches.
MajorLeagueBaseball(MLBhasstrictregulationsforbaseballbats:
●Thebatshallbeasmooth,roundsticknotmorethan23/4inchesindiameteratthethickestpartandnotmorethan42inchesinlength.Thebatshallbeonepieceofsolidwood.
●CuppedBats.Anindentationintheendofthebatuptooneinchindepthispermittedandmaybenowiderthantwoinchesandnolessthanoneinchindiameter.Theindentationmustbecurvedwithnoforeignsubstanceadded.
●Thebathandle,fornotmorethan18inchesfromitsend,maybecoveredortreatedwithanymaterialorsubstancetoimprovethegrip.
●NocoloredbatmaybeusedinaprofessionalgameunlessapprovedbytheRulesCommittee.
Alltheseendeavorsplayersandmanufacturedoistoimprovethegameperformance.Andempirically,apointoranarrowzoneonthebarrelofabat,called
the“sweetspot”isbestforhittingtheball.What’smore,someplayerswanttoimprovetheperformanceofthebaseballbatbycorkingitorusethealuminumbat.
1.1RestatementoftheProblem
Whatweneedtodo?
It’snotasimplecalculationofthelocationofthesweetspot,buttohaveaideatofindthereasonwhythereisasweetspotandusetheseknowledgetoexplaintheprobleminreality.
Becauseofthevariousdefinitionsabout“sweetspot”,wewillattempttoexaminethesweetspotintermsoftwoseparatecriteria.Onewillbethelocationwherethehandsensationisminimized,andtheotherwillbethelocationwherethemeasuredperformanceofthebatismaximized.Onthisbasisofthesetwocriteriaweputforwardtwodifferentmodels.The“CenterofPercussion”(COP,the“NodeofVibration”(NOV.Fromthetwocandidatesabove,wewillfigureoutwhichonestandfortherealsweetspot,orneitherofthemmatchestheempiricalsweetspotwhenhittingaball.
Somequestionstoconsider:
●WhatarethefactorsthataffectthelocationofthesweetspotinCOPmodelandNOVmodel?
●Howdothetwomodelscorrespondtotheexperimentdata?
Shouldwemodifythemodeltomakeitmoreexact?
●Canthesetwomodelsexplaintheassumptionthatacorkedoranaluminumbatmighthaveabetterperformance?
1.2SurveyofPreviousResearch
1.2.1RodCross:
FreeBatVibrationModel
Thismodelsimplifythebaseballbatasafreelysuspendonewhentheballimpactsit.
Themodelconcludesthatthesweetspotofawoodbaseballbatwasfoundtorepresentanimpactzoneofwidthabout3cmwheretheforceandtheimpulsetransmittedtothehandsarebothminimized.Theexistenceofthiszonecanbeattributedtoacombinationofthetwofactors.Thedominantvibrationsinducedinahand-heldbatareheavilydampedversionsofthefundamentalandsecondmodesofa
freebat.Theothermainfactorcontributingtothesweetspotzoneconcernsrotationofthebatinthehands.
Inthemodel,themeasurementsreportedinthispapercouldeasilyberepeatedinanundergraduatelaboratoryatlowcost.Andresultsindicatethatthefreelysuspendedbatexperimentsareparticularlysuitableforthispurposesincethewaveformsareespeciallyclearandinformative.
1.2.2A.M.Nathan&
Two-springModel
TheTwo-springModelisamodelofball-batcollisionthatconsiderstheballandbatastwolinearlossysprings.Thetwospringsmutuallycompresseachother,convertingtheinitialcenter-of-mass(CMkineticenergyentirelyintocompressionpotentialenergy.
Themodelexplicitlydemonstratesthedependenceoftheball-batCoefficient-of-Restitutioneanddynamicstiffnesskoftheball.ItalsohasexplainedthattheBatPerformanceFactor(BPFnormalizationmethodfailsforthenon-woodbattested.Andfinallyithasderivedanapproximatenormalizationexpressionwhichisvalidforlow-performingbats.
2Convention
Thissectiondefinesthebasictermsusedinthispaper.
2.1Terminology
●Barrel:
isthethickpartofthebat,wherethebatismeanttohittheball
●Handle:
isthethinpartwherethebatterhitstheball
●Node:
Whenthebatvibratesinitslowestfrequencymode,knownasthefundamentalfrequencymode,thereareplaceswherethebatdoesnotvibrateatall,namednode.
●Pivot:
thelocationbatterusuallyholdsthebatinthehands,theaxiswherethebatrevolvesabout.
2.2Variables
Wewilldefinethefollowingvariableshereastheyareusedwidelythroughoutourpaper.Additionalvariablesmightbedefinedlater,butwillbeconfinedtoaparticularsection.MMassofbatmMassofballzLocationofCOP
LocationofcenterofmasskDynamicstiffnessCollisionefficiencyeCoefficientofrestitutionMomentofinertiaoncertainaxisMomentofinertiaoncenterofmassdDistancefromapointoncertainaxis
WthequantitativeenergyYYoung’smodulusHDistancefromCOPIncomingvelocityOutgoingvelocityVelocityofbatPMomentumofballThevelocityofangle
Transformofcontact
′Thevelocityofcontact
U
Thedisplacementofthebat
2.3Assumption
Weonlymakesomegeneralassumptionbeforethespecificmodelisestablished.●Themassofthebatiswell-distributed.●Thesurfaceofthebatissmooth.
●Thebatandtheballwillretaintheirshapeafterthecollision
●Whenwemodel,themassandlengthofthebatandtheballwillnotchange.
3COPModelasSweetSpot3.1WhyisCOP
The"
centerofpercussion"
(COP,knownalsoas“centerofoscillation”,isthelocationwheretheball'
simpactcausesthesmallestshocktoyourhands.Ifyouhitabaseballclosertothebat'
shandlethantothecenterofpercussion,you'
llfeelaslightforcepushingthehandlebackintothepalmofyourtophand.
3.2ToFindtheCOP
3.2.1AssumptionsabouttheCOPmodel
●Wesimplifythebaseballbattoauniform,smoothroundstick.●Thestickrotatesaboutoneendofitself.
●Theimpactoftheballtothebatishead-oncollision.●Ignorethespinoftheball.
●Weassumethecollisionashard-spherecollision,overlookingdeformation.●Ignorethegravityofthebatatthemomentofcollision3.2.2TheCOPModel
Fig.3.1thecoordinatessystemofbat
Themodelisbaseonthedynamicsequationhowtherigidbeamrevolvesaboutacenteroranaxis.Whatweneedtofigureoutfromtheequationisthe
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