MCMTeam8008paper.docx
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MCMTeam8008paper
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Summary
Thechosenofball-pointhavedirectlyinfluenceontheoutputvelocityofballsandthereactionforcesonthehandofbatters.Agoodball-pointistheplaceonthebatbarrelwherethecontactbetweenbatandballresultsinthebesthit-theballleavesthebatwiththegreatestspeedandtheplayer'shandsfeelverylittlevibrationfromtheimpact.Inthispaper,myprojectusesaPhysicsknowledge,MATLABaswellasSimulationTechnologysoftwaretodeterminewhereabaseballbathittingabaseballwouldresultintheleastvibration,asingle“sweetspot”oralarger“sweetzone”.
Asforthefirstproblem,inordertodevelopamodeltoexplainwhythesweetspotisn’tattheendofthebat,weutilizethreeequations:
conservationoflinearmomentum,conservationofangularmomentum,andthedefinitionofcoefficientofrestitution,andconsideraspecialcircumstancethatthereisnoenergylossduringthecollisionbetweenthebatandball.Thenbysimplification,wecanderivethemodelaboutthelocationofsweetspot
andwewillverifytherationalityofthismodelbyMATLABSimulationTechnology.
Thenforthesecondone,byamelioratingthemodeldevelopedabove,itrequiresustojudgewhether“corking”abatcanenhancethe“sweetspot”effect.Onthebaseofthemodelabove,wetakerecoilfactorofthebat,COR,BPFandrelationsbetweenthemintoconsideration.Thenweobtainanewmodelwhichreflectstheaffecttothe“sweetspot”by“corking”abat.
.Wewillconfirmthatthemodelisreasonablebyqualitativeanalysis,andwewillcometoconclusionthat“corking”abatcanenhancethe“sweetspot”effect.
Astothethirdproblem,ourtaskistoanswerthequestionwhetherthematerialoutofwhichthebatisconstructedmatterthesweetzone.Throughthedeeplyexplainfortheimprovedmodel,wetendtothinkthatthemetalbatsoutperformwoodones.
Finally,becauseofthestrongpracticalsignificanceoftheissue,wecollectsomecertainexistedmodelforustoinfertoandindirectlyvalidatethemodel.
Keywords:
sweetspot,coefficientofrestitution,simulation,thelossofenergy,thebaseballbat,recoilfactor
Content
Introduction……………………………………………….3
background……………………..…………………………3
Statementoftheproblems…………………………….3
Assumptionsofthemodels……………………..……..4
Variabledefinition……………………..……………….4
Steptosolvetheproblem……………………..……………4
Stepone…………………………………………………4
Steptwo………………………………………………...6
Stepthree……………………………………………….6
Stepfour………………………………………………..7
Stepfive………………………………………………...8
Thepromotiontothemodel……………………..…………9
Theevaluationtothemodel……………………..…………9
StrengthsandWeaknesses………………..……………….13
TheReferences……………………………………………14
Appendix……………………..……………………………15
ExploretheSweetspotoftheBaseballbat
Introduction
1.1Background
Doyouknowthesweetspotofbaseballbat?
Duringabaseballgame,everybatterwantstomaketheballtotravelasfaraspossible.Asweknow,thechosenofball-pointhavedirectlyinfluenceontheoutputvelocityofballsandthereactionforcesonthehandofbatters.Soitisnecessaryforustofindagoodball-point.Therearemanydefinitionsforwhatconstitutesthe"SweetSpot"onabaseballbat.Fromaplayer'sviewpoint,thesweetspotistheplaceonthebatbarrelwherethecontactbetweenbatandballresultsinthebesthit-theballleavesthebatwiththegreatestspeedandtheplayer'shandsfeelverylittlevibrationfromtheimpact.Mostofteninthepublishedresearchthesweetspotisusuallyreferedtoaseitherthecenterofpercussion(COP)orthenodeofthefirstbendingmode.
Whilealmosteverybatonthemarket(especiallyneweraluminumbats)isclaimedtohavea"wider"sweetspot.Theamountofcontroversyoverthemetalversuswoodbatissue,therehavebeensurprisinglyfewscientificstudiescomparingtheperformanceofwoodandmetalbaseballbats.Inthispaper,wedevelopamodeltodescribethatthemetalbathasa"wider"sweetspotcomparedtowoodone.
1.2Statementoftheproblems
Thenextquestionsshouldbesolved:
1.Developamodeltovalidatethefindingthatthesweetspotisn’tattheendofthebat.
2.Makeyourchosentoaffirmornegativetheversionthat“corking”abatcanenhancethe“sweetspot”effectandgiveyourreasonbyamelioratingthemodeldevelopedabove.
3.Forfurtherdetails,judgeifthebats’materialmattersthelocationofthesweetspot.
4.Discusswhethermetalbatshavea“wider”sweetzone.
5.AndthendepicttherationalityofwhyMajorLeagueBaseballprohibitsmetalbats.
1.3Assumptionsofthemodel
1.3.1Supposethatthebaseballbatishomogeneousandthesurfaceofitissmooth.
1.3.2Supposethatthebaseballbatissincere.
1.33Ignoretheimpactofairresistancewhentheballtravelsintheair.
1.4Variabledefinitions
:
Thebatmomentofinertia(abouttheCM).
:
Thequalityofbaseballbat.
:
Thequalityoftheball.
:
Theball’sinitialvelocity.
:
Thefinalvelocityofball.
:
Theinitialvelocityofbat.
:
Thefinalvelocityofbat.
:
Theinitialangularvelocityofbat.
:
Thefinalangularvelocityofbat.
:
Thedistancefromthecentroidtosweetspot.
:
Thedistancefromfulcrumtocentroid.
:
Restitutioncoefficient.
:
Thedistancefromfulcrumtothetopofthebat.
:
Theball-wallCOR.
:
Thebatrecoilfactor.
Steptosolvetheproblems
Step1
DesignIssuesfortheSweetspotoftheBaseballbat
Fig.1.Stuctureofthebaseballbat
Fig.2.Thesystemofthebatandball
(Vfamongthesystemrepresents
Vballamongthesystemrepresents
Vbatamongthesystemrepresents
)
Intheball-batsystem,astheinteractiontimeofcollisionisveryshortwhilethesignificantchangesinmotionstateaftercollision,sointernaleffectandtheothereffectcanbeignored.ThenweconsiderthatthesystemisConservationofMomentum.
.
Wepurposethebatandballtobeasystem.ThenthissystemwillsatisfyConservationofAngularMomentumtotheCM,
(1)
Wecanknowfromthedefinitionoftherestitutioncoefficientthat,
.
(2)
Iftherestitutioncoefficientequalstoone,knownfromConservationofenergytheorem,itsuggestthatit’snotcausethelosstothetotalofthemechanicalenergybeforeandaftercollision.Thencombiningformula
(1),
(2)withe=1(3)(Iftherewasnolossofenergyduringthebouncethenewouldbe1),wecanreceivethenextformula:
(4)
Asallweknown,itwillproduceenormousenergywhentheballandthebatviolentcollision.Energycan’tnotbedestroyed,onlycanbetransmittedfromonetoanother.Alltheenergycannotbepassedtotheballactuallyiswasted.Asthebatwillhaverecoilforceandrotationaboutanaxisperpendiculartotheplaneoftheswing,whiletheybothcanreducetheenergywhatshouldbepassedtotheball,andthendecreasetheoutputvelocityofball.Thuswhentheimpactspotofballisthespotwheretheenergytransmittedfromthebattoballutmost,thebathasnofinalrotationalenergy.Thentheangularmomentummustbezero,thatis
.Sotheformula(4)willbelistsystematicallytobe
Thenweknowthat,
while
initcanbeobtaininthelab.
Step2
Amelioratingthemodeldevelopedabove
The“corking”ofabatresultintheReboundcoefficientincrease.
So
Basedonphysicalknowledgewecanseethatwhentheballhitthesweetspot,reboundcoefficientequaltoone.
So
Step3
Theanswertothequestion3
Fromtheformulaabove,wecanknowthatwhenthereboundcoefficientincreases,thenitwillresultinthedecreaseofthemechanicalenergy.Atthesametime,wecanreceivethatthefinalvelocityoftheballdecreasesandmakethesweetzonenarrowatlast.Thenindirectlyitwilllowertheaudiences’enthusiasmtoappreciateabaseballgamebecauseitwillbetoodifficultforthemtowatchawinningball.Inaword,“corking”abatcanenhancethe“sweetspot”effect,andwhattobeworseisthatitcanresultinthedecreaseofthequalityofappreciatingabaseballgame.
Step4
Doesthematerialoutofwhichthebatisconstructedmatter?
Wecanobtainthelocationofthesweetspot
frommodel1.ItisclearthatthispointisnottheCOPlocation,becausethepositionisafunctionoftheballandbatvelocityinadditiontothepropertiesofthebat.Thevalueof
and
areeasytoobtaininthelab,butthevaluesof
and
aredeterminedbythehitterswingingthebatandthepitcherthrowingtheball.
Thevalueof
forarangeoftheseparameterscanbeobtained,butitisinstructivetofirstseehow
varieswitheachofthesevariable.If
Islessthanone,thesquarerootcanbeexpandedandtheresultis
Where
and
hasbeenreplacedby
.
Throughtheformulaabove,weknowthat
thedistancefromtheCMofthebattotheimpactpointoftheball,increasesasthemassofthebatincreases,andincreasesasthemomentofinertiaofthebatincreases,andincreasesthemore”wristy”istheswing.Asdifferentmaterialhasdifferentmomentofinertia,whilethroughtheformulaaboveweknowthatthemomentofinertiacanaffectthedistancefromtheCMofthebattotheimpactpointoftheball,therefore,thematerialoutofwhichthebatisconstructedmatters.Atthesametime,themassofabatcanalsoinfluenthedistancefromtheCMofthebattotheimpactpointoftheball.Thatis,themodelcanpredictthedifferent