2010MCM特等ProblemA:TheSweetExplainthe“sweetspot”onabaseballbat.Everyhitterknowsthatthereisaspotonthefatpartofabaseballbatwhere umpoweristransferredtotheballwhenhit.Whyisn’tthisspotatofthebat?Asimpleexplanationbasedontorquemightseemtoidentifyofthebatasthesweetspot,butthisisknowntobeempiricallyincorrect.Developamodelthathelpsexplainthisempiricalfinding.Someplayersbelievethat“corking”abat(hollowingoutacylinderintheheadofthebatandfillingitwithcorkorrubber,thenreplacingawoodcap)enhancesthe“sweetspot”effect.Augmentyourmodeltoconfirmordenythiseffect.DoesthisexplainwhyMajorLeagueBaseballprohibitsDoesthematerialoutofwhichthebatisconstructedmatter?Thatis,doesthismodelpredictdifferentbehaviorforwood(usuallyash)ormetal(usuallyaluminum)bats?IsthiswhyMajorLeagueBaseballprohibitsmetalbats?PeterDiaoMouPrinceton,NJWedeterminethesweetspotonabaseballbat.Wecapturetheessentialphysicalsoftheball-batimpactbytakingtheballtobealossyspringandthebattobeanEuler-Bernoullibeam.Toimpartsomeintuitionaboutthemodel,webeginbypresentingarigid-bodymodel.Next,weuseourfullmodeltoreconcilevariouscorrectandincorrectclaimsthesweetspotfoundintheliterature. Finally,wediscussthesweetspotandtheperformancesofcorkedandaluminumbats,withaparticularemphasisonhoopmodes.確定了棒球棒的最佳擊球點。把棒球視作彈性體，把棒視為歐拉—伯努利梁，從而抓住了棒—球作用過程的物理本質(zhì)。為了把對該問題的直觀判斷引入模型當中，用一個剛體模型作為初始模型。然后，提出了的波模型，并對一些文獻中正確的和不正確的判斷做了整合。最后，并對塞有軟木的棒球棒和鋁質(zhì)棒球棒的最佳擊球點進行了。Althoughahittermightexpectamodelofthebat-baseballcollisiontoyieldinsightintohowthebatbreaks,howthebatimpartsspinontheball,howbesttoswingthebat,andsoon,wemodelonlythesweetspot.轉(zhuǎn)，樣最好揮動等等，僅對最擊球點行建。Thereareatleasttwonotionsofwherethesweetspotshouldbe—animpactlocationonthebatthateither（對最佳擊球點至少有兩種定義） forttothehands （對手部的痛感最小izestheoutgoingvelocityoftheball（球速最大Wefocusexclusivelyontheseconddefinition.（只專注于第二個定義Thevelocityoftheballleavingthebatisdeterminedby（決定球速的因素theinitialvelocityandrotationoftheball（球的初速度和角速度theinitialvelocityandrotationofthebat（棒的初速度和角速度therelativepositionandorientationofthebatandball,and（棒的相對位置和方向theforceovertimethatthehitter’shandsapplyonthehandle.（手作用于握柄的時間）Weassumethattheballisnotrotatingandthatitsvelocityatimpactisperpendiculartothelengthofthebat.Weassumethateverythingoccursinasingleplaneandwearguethehands’interactionisnegligible.Intheframeofreferenceofthecenterofmassofthebat,theinitialconditionsarecompleyspecifiedbytheangularvelocityofthebat（棒的角速度thevelocityoftheballand（球的速度thepositionoftheimpactalongthebat（作用點的位置決定Thelocationofthesweetspotdependsnotonjustthebatalonebutalsoonthepitchandontheswing.最佳擊球點的位置不僅取決于棒，還取決于揮動軸和揮動的力Thesimplestmodelforthephysicsinvolvedhasthesweetspotatthecenterofpercussion[Brody1986],theimpactlocationthatminimizes forttothehand.Themodelassumesthebat1tobearigidbodyforwhichthereareconjugatepoints:Animpactatonewillexactlybalancetheangularrecoilandlinearrecoilattheother.Bygripatoneandimpactingattheother(thecenterofpercussion),thehandsexperienceminimalshockandtheballexitwithhighvelocity.Thecenterofpercussiondependsheavilyonthemomentofinertiaandthelocationofthe1hands.Wecannotacceptthismodelbecauseitbotherroneouslyequatesthetwodefinitionsofsweetspotandfurthermoreassumesincorrectlythatthebatisarigidbody.一種最簡單的模型將撞心視為最佳擊球點[Brody1986]，球擊打該處時對手產(chǎn)生的痛感最小。這個模型把棒球棒假設為剛體，棒上存在一對點：在撞心處的擊打力會平衡握柄端的后坐力。握住一端擊打撞心會使得手的痛感最小而棒球以高速飛離棒球棒。撞心的位置主要取決于轉(zhuǎn)動慣量和手的位置。由于該模型不僅錯誤地Anothermodelpredictsthesweetspottobebetweennodesofthetwolowestnaturalfrequenciesofthebat[Nathan2000].Givenafreebatallowedtooscillate,itsoscillationscanbe posedintofundamentalmodesofvariousfrequencies.Differentgeometriesandmaterialshavedifferentnaturalfrequenciesofoscillation.Theresultingwaveshssuggesthowtoexcitethosemodes(e.g.pluckingastringatthenodeofavibrationmodewillnotexcitethatmode).Itisambiguouswhichdefinitionofsweetspotthismodeluses.Usingthedefinition,itwouldfocusonthe fortableexcitationsofvibrationalmodes:Choosingtheimpactlocationtobenearnodesofimportantfrequencies,aminimumof fortablevibrationswillresult.Usingtheseconddefinition,theworryisthatenergysentintovibrationsofthebatwillbelost.Thismodelassumesthatthemostimportantenergiestomodelarethoselosttovibration.另一種模型認為最佳擊球點位于棒的前兩階固有頻率的節(jié)點之間[Nathan2000]。給定一個振動的球棒，它的振動可以分解為許多固有頻率不同的基準模態(tài)。不同的幾何Thismodelraisesmanyquestions.Whichfrequenciesgetexcitedandwhy?TheFouriertransformofanimpulseingeneralcontainsinfiniymanymodes.Furthermore,woodisaviscoelasticmaterialthatquicklydissipatesi ergy.Isthenotionofanoscillatingbatevenrelevanttomodelingabat?Howvalidistheconditionthatthebatisfree?Oughtthesystemtobecoupledwithhandsonthehandle,orthearm’sbonestructure,orpossiblyeventheball?Whattypesofoscillationsarerelevant?Acylindricalstructurecansupportnumerousdifferenttypesofmodesbeyondthetransversemodesusuallyassumedbythismodel[Graff1975].棒是的這個條件有多強？應不應該考慮握柄端對棒的作用，手的骨骼結(jié)構(gòu)，還有很多種振動模態(tài)[Gaff17]。Followingthecenter-of-percussionlineofreasoning,howdowemodeltherecoilofthebat?Followingthevibration-nodeslineofreasoning,howdowemodelthevibrationsofthebat?Interaltheoryofimpactmechanics[Goldsmith1960],thesetwoeffectsarethemainones(assumingthatthebatdoesnotbreakordeformpermanently).Brody[1986]ignoresvibrations,Cross[1999]ignoresbatrotationbutstudiesthepropagationoftheimpulsecoupledwithball,andNathan[2000]emphasizesvibrationmodes.Ourapproachreconcilesthetensionamongtheseapproacheswhileemphasizingthecrucialroleplayedbythetime-scaleofthecollision.從前面對撞心的分析，該如何對棒的反應進行建模？從對振動模態(tài)的分析來看，種效應起主導作用（假設棒不會斷裂也不會發(fā)生變形Brody[1986]忽略了振動，Cross[1999]忽略了棒的轉(zhuǎn)動，轉(zhuǎn)而研究在與球的耦合作用下沖擊在棒中的，Nathan[2000]則專注于振動模態(tài)的研究。的方法在綜合這三種方法的同時，還主Ourmaingoalistounderstandthesweetspot.Asecondarygoalistounderstanddifferencesbetweenthesweetspotsofdifferentbattypes.Althoughmarketersofoftenemphasizethesweetspots,thereareotherrelevantfactors:easeofswing,tendencyofthebattobreak,psychologicaleffects,andsoon.Wewillarguethatitdoesn’tmattertothecollisionwhetherthebatter’shandsaregripthehandlefirmlyorifthebatterfollowsthroughontheswing;thesecircumstanceshavenobearingonthetechniquerequiredtoswingthebatorhowthebat’spropertiesaffectit.的首要目標就是理解最佳擊球點的含義。次要目標就是理解不同類型的棒球棒的最佳擊球點的區(qū)別。盡管棒球棒廠商專注于最佳擊球點，但是其他因素比如揮舞的輕松程度，棒是否容易斷裂，棒球手的心理因素等等也有關(guān)系。會證明棒球手是否Ourprisanizedasfollows. ,resenttheBrodyrigid-bodymodel,illuminatingtherecoileffectsofimpact.NextresentafullcomputationalmodelbasedonwavepropagationinanEuler-Bernoullibeamcoupledwiththeballmodeledasalossyspring.Wecomparethismodelwithothersandexplorethelocalnatureofimpact,theinteractionofrecoilandvibrations,androbustnesstoparameterchanges.Weadjusttheparametersofthemodeltocommentonthesweetspotsofcorkedbatsandaluminumbats.Finally,weinvestigatetheeffectofhoopfrequenciesonaluminumbat.了Brdy然后建立了一個純計算模型，型以歐拉—伯努利梁理論為基礎，當作彈性體并考慮了與球的耦合作用。此模型與其他模型做了比較，并研撞擊的本質(zhì)，反沖與振動之間的相互作用驗證了參數(shù)的魯棒性。通過改數(shù)對塞的hoop（這個“模態(tài)”不好翻譯）AsimpleWebeginbyconsideringonlytherigidrecoileffectsofthebat—ballcollision,muchasinBrody[1986].Forsimplicity,weassumethatthebatisperfectlyrigid.Becausethecollisionhappensonsuchashorttime-scale(around1ms),wetreatthebatasafreebody.Thatistosay,we’renotconcernedwiththebatter’shandsexertingforceonthebatthatmaybetransferredtotheball.棒是剛體。由于撞擊過程的持續(xù)時間很短（大約1毫秒，進一步把棒視為剛體。也就是說，不關(guān)心棒球手會對球產(chǎn)生力的作用。ThebathasmassMandmomentofinertiaIaboutitscenterofmass.Fromreferenceframeofthecenterofmassofthebatjustbeforethecollision,theballhasinitialvelocityviinthepositivex-directionwhilethebathasinitialangularvelocityi.Inoursetup,viandihaveoppositesignswhenthebatterisswingingattheballasinFigure1,inwhicharrowspointinthepositivedirectionsforcorrespondingparameters.棒球棒的質(zhì)量為M，繞質(zhì)心的轉(zhuǎn)動慣量為I。以撞擊之前棒的質(zhì)心標系為參考坐標系，球的初速度為vi

Initial

Distancefrom棒的初始角速度為i。假設正方Figure1中所示。撞擊點位于距離棒的質(zhì)心l處 假設碰撞為正碰撞

碰撞之后，球速為vf，棒球棒質(zhì)心

Initial的速度為V，角速度為（這 物理量均表示大小，不帶符號）做恢復系數(shù)，通常用e表示，e0示完全非彈性碰撞，e1表示完全彈性碰撞。在本模型中，做出如下

Figure1Thee對于棒上任何一點都相等eisconstantalongthelengthofthee對于任何vieisconstantforallvi給出模型的基本條件之后，有如下方程 動量守恒(Conservationoflinear MVm(v 角動量守恒(Conservationofangular 恢復系數(shù)(Definitionofcoefficientof e(vl)vV 由上述式子解出vifvf

mM1M這里 2 2 I是棒的有效質(zhì)量。(TheeffectivemassoftheForcalibrationpurposes,weusethefollowingdata,whicharetypicalofaregulationbatconnectingwithafastballinMajorLeagueBaseball.TheresultsareplottedinFigure2.為了最佳擊球點的位置有一個初步計，使用以下數(shù)據(jù)，這些數(shù)據(jù)全美棒球的典型棒球棒的數(shù)據(jù)。結(jié)果如ure2所示。mM0.145m kg67m/60rad/eFigure2Finalvelocityvf(solidarcattop),swingspeedil(dottedrisingline),andeffectivemass(dashedfallingcurve)asafunctionofdistancel(inmeters)fromcenterofmass. umexitvelocityis27m/s,andthesweetspotis13cmfromthecenterofmass.Missingthesweetspotbyupto5cmresultsinatmost1m/sdifferencefromumvelocity,implyingarlativlywideswet5cm時最大速度減小1m/s,可見最佳擊球點的范圍較大。Fromthisexample,weseethatthesweetspotisdeterminedbyamultitudeoffactors,includingthelength,mass,andshofthebaseballbat;themassofthebaseball;andthecoefficientofrestitutionbetweenbatandball.Furthermore,thesweetspotisnotuniquelydeterminedbythebatandball:Italsodependsonthe ingbaseballspeedandthebatter’sswingspeed.Figure2alsoshowsintuitivelywhythesweetspotislocatedsomewherebetweenthecenterofmassand ofthebarrel.Asthepointofcollisionmovesout2wardalongthebat,theeffectivemassofthebatgoesdown2,sothatagreaterfractionoftheinitialkineticenergyisputintothebat’srotation.Atthesametime,therotationinthebatmeansthatthebatismovingfasterthanthecenterofmass(orhandle).Thesetwoeffectsworkinoppositedirectionstogiveauniquesweetspotthat’snotateitherendpoint.外移，棒的有效質(zhì)量減小2，因此的初始動能轉(zhuǎn)化為棒的轉(zhuǎn)動動能。另外由于棒的However,thismodellsonlypartofthestory. Indeed,someofourstartingassumptionscontradicteachother:然而，這個模型還只是揭示了問題的一部分。實際上最初的假設是相互的Wetreatedthebatasafreebodybecausethecollisiontimewassoshort.Inessence,duringthe1msofthecollision,theball“sees”onlythelocalgeometryofthebat,notthebatter’shandsonthehandle.Ontheotherhand,weassumedthatthebatwasperfectlyrigid—butthatmeansthattheball“sees”theentirebat.2”goesup”,”goes由于碰撞時間極短，把棒視為剛體。實際上，在1毫秒的接觸時間內(nèi)，球只對棒的局部起作用，不會影響到握柄端。而另一方面，假設棒球棒是剛體，Wealsoassumedthatisconstantalongthelengthofthebatandfordifferentcollisionvelocities.Experimentalevidence[Adair1994]suggeststhatneitherissuecanbeignoredforanaccuratepredictionofthelocationofthesweetspot.還假設恢復系數(shù)對于棒長和任意初速度是常數(shù)，實驗表明對于精確分析而言，Weneedamoresophisticatedmodeltoaddress 需要建立一個更完善的模型來彌補這些缺O(jiān)urWedrawfromBrody’srigid-bodymodelbutmoresofromCross[1999].OnecoulddescribeourworkasanadaptionofCross’sworktoactualbaseballbats.Nathan[2000]attemptedsucptionbutwasmisledbyincorrectintuitionabouttheroleofvibrations.WedescribehisapproachanderrorasawaytoexplainCross’sworkandtomotivateourwork.從Brody的剛體模型出發(fā)，但是地采用了Cross的模型。的工作可以看是Crss的成果應用于實際棒球棒的變型。atan所誤導，對振動所起的作用估計不足。描述了他的方法和錯誤，以此來闡釋PreviousBrody’srigid-bodymodelcorrectlypredictstheexistenceofasweetspotnotattheendofthebat.Thatmodelsuffersfromthefactthatthebatisnotarigidbodyandexperiencevibrations.Onewaytoaccountforvibrationsistomodelthebatasaflexibleobject.Beamtheories(ofvaryingdegreesofaccuracyandcomplication)canmodelaflexiblebat.VanZandt[1992]wasthetocarryoutsuchanysis,modelingthebeamasaTimoshenkobeam,afourth-ordertheorythattakesintoaccountbothshearforcesandtensilestresses.Theequationsarecomplicatedandwewillnotneedthem.VanZandt’smodelassumestheballtobeuncoupledfromthebeamandsimplytakestheimpulseoftheballasagiven.Theresultingvibrationsofthebatareusedtopredictthevelocityofbeamattheimpactpoint(bysummingtheBrodyvelocitywiththevelocityofthedisplacementattheimpactpointduetovibrations)andhencetheexitvelocityoftheballfromtheequationsofthecoefficientofrestitution[VanZandt1992].理論（各種梁理論的精度和復雜度不同）適用于這一彈性體。VanZandt第一個將梁理了剪應力和軸向應力。該理論的方程形式很復雜，在這里不列出來。VanZandt的速度除了包括Brody模型中的速度外，還要加上振動引起的速度，再由恢復系數(shù)的表達式即可求出球的最終速度[VanZandt1992]。Cross[1999]modeledtheinteractionoftheimpactofaballwithuminumbeam,usingtheless-elaborateEuler-Bernoulliequationstomodelthepropagationofwaves.Inaddition,heprovidedequationstomodelthedynamicalcouplingoftheballtothebeamduringtheimpact.Afterdiscretizingthebeamspatially,heassumedthattheballactsasalossyspringcoupledtothesinglecomponentoftheregionoftheimpact.Cross使用了較不精確的歐拉-伯努利梁理論來分析鋁質(zhì)棒球棒受到棒球沖擊時的振波的。除此之外，他還提供了棒的耦合作用的方程。在將棒球棒之后Cross’sworkwasmotivatedbybothtennisracketsandbaseballbats,whichdifferimportantlyinthetime-scaleofimpact.Thebaseballbat’scollisionlastsonlyabout1ms,duringwhichthepropagationspeedofthewaveisveryimportant.Inthislocalviewoftheimpact,theimportanceofthebaseball’scouplingwiththebatisincreased.Crs為1毫秒左右，這段時間內(nèi)振動波的速度是很重要的。在撞擊的局Crossarguesthattheactualvibrationalmodesandnodepointsarelargelyirrelevantbecausetheinteractionislocalizedonthebat.Theboundaryconditionsmatteronlyifvibrationsreflectofftheboundaries;animpactnotcloseenoughtothebarrelendofthebatwillnotbeaffectedbytheboundarythere.Inparticular,apulsereflectedfromafreeboundaryreturnswiththesamesign(deflectedawayfromtheball,decreasingtheforceontheball,decreasingtheexitvelocity),butapulsereflectedfromafixedboundaryreturnswiththeoppositesign(deflectedtowardstheball,pushingitback,increasingtheexitvelocity).Awayfromtheboundary,weexpecttheexitvelocitytobeuniformalongnon-rotatingbat.Cross’smodelpredictsalloftheseeffects,andheexperimentallyverifiedthem.Inourmodel,weexpectsimilarphenomena,plusthenarrowingofthebarrelnearthehandletoactsomewhatlikeaboundary.件只在振動波到邊界的時候才起作用：沖擊點離棒的末端足夠近時，邊界條件才轉(zhuǎn)動的棒球棒來說，球速是一致的。Cross的模型考慮了上述所有因素，他還用實驗證明了他的考慮。在的模型中，也認為這樣的現(xiàn)象會發(fā)生，同時還考慮了握柄Nathan’smodelalsoattemptedtocombinethebestfeaturesofVanZandtandCross[Nathan2000].HistheoryusedthefullTimoshenkotheoryforthebeamandtheCrossmodelfortheball.Heevenacknowledgedthelocalnatureofimpact.Sowheredowedivergefromhim?Hiserrorstemsfromanoveremphasisontryingtoseparateouttheball’sinteractionwitheachseparatevibrationalmode.理論和Cross的棒球模型。他還承認了沖擊的局部效應。那么和他的區(qū)別在哪里呢？Thesignofinconsistencycomeswhenheusesthe“orthogonalityofeigenstates”todeterminehowmuchagivenimpulseexciteseaode.Theeigenstatesarenotorthogonal.Manytheoriesyieldsymmetricmatricesthatneedtobediagonalized,yieldingtheeigenstates;butTimoshenko’stheorydoesnot,duetothepresenceofodd-orderderivativesinitsequationsNathan’sstoryplaysoutbeautifullyifonlytheeigenstateswereactuallyorthogonal;butwehavenumericallycalculatedtheeigenstates,andtheyarenotevenapproximayorthogonal.Heusestheorthogonalitytodrawimportantconclusions:但是鐵摩辛柯梁理論的方程中含有奇次導數(shù)項，因此得不到對稱矩陣。Nathan的理論在模態(tài)具備正交性時很完美，但是計算了這些模態(tài)之后，發(fā)現(xiàn)它們根本不正交。Thelocationofthenodesofthevibrationalmodesare振動模態(tài)的節(jié)點位置很High-frequencyeffectscanbecompley高頻率效應可以完全忽Wedisagreewithbothofthese.（這兩點都不同意（這里突然拿出運動方程顯得有些突兀，下面給出推導過程連續(xù)系統(tǒng)的振動方程如下

MyKyF離散為n度系統(tǒng)之后，這里M是nn質(zhì)量矩陣（對稱矩陣，K是nn剛度矩陣（正定矩陣y(t是撓度向F(t是外力向量。本方程的解可由假設法或傅里葉變換得到，這里為了得到y(tǒng)sin(t該式對于任何t均成立，則有：

(K2M)0M1K2由此可解出n個模態(tài)向量n個相應的固有頻率。yM1KyM1令FM1[1

n即可得到文中這部分的相關(guān)方程，可見 為對稱矩陣，而非文中所說的稱矩陣還需的是，文中的下標均使用了愛因斯坦求和約定（Einstein Thecorrectderivationstartswiththefollowingequationofmotion,wherekisthepositionofimpact, yiisthedisplacementandFiistheexternalforceontheithsegmentofthebat,and isasymmetric3matrix:y(t) y(t)F Wewritethesolutionsasy(t) an(t),wherethecolumns4 areeigenmodes eigenvaluesn2.Explicitly,Hjkknn2 and indicatesthekthcomponent thentheigenmode.Thenwewritetheequationof a(t)2a(t)F kn a(t)2a(t) Inthelaststepweusedthefactthattheeigenmodesformacompletebasis.（這里運用了模態(tài)矩陣的正交性）FNathan’sp rusesontheright-handsidesimply knFkscaledbyanormalizationconstant.Atglance,thisseemslikeaminortechnicaldetail,butthephysicshereisFimportant.Wecalculatethat

termsstayfairlylargeforevenhighvalueofncorrespondingtothehighfrequencymode(kisjustthepositionofimpact).Thismeansthattherearesignificanthighfrequencycomponents,atleastat.Infact,thehighfrequencymodesarenecessaryfortheimpulsetopropagateslowlyasawavepacket.InNathan’smodel,onlytheloweststandingmodesareexcited;sotheentirebatstartsvibratingassoonastheballhits.Thiscontradictshisearlierbeliefinlocalizedcollision(whichweagreewith)thatthecollisionisoversoquicklythattheball“sees”onlypartofthebat.Nathsoclaimsthatthesweetspotisrelatedtothenodesofthelowestmode,whichcontradictslocality:Thelocationofthelowestordernodesdependsonthegeometryoftheentirebat,includingtheboundaryconditionsatthehandle.Nathan在他的里對右端項使用了正交化常數(shù)。乍一看這只是一個小的數(shù)學處理 但是這里的物理意義是很重要的計算后發(fā)現(xiàn)，即使在n 很大的時候，右端項 對應的數(shù)值依然很大，這和高頻模態(tài)有關(guān)（k是沖擊點位置。實際上，高頻模態(tài)對沖擊的慢速是必要的。在Nathan的模型里，只有最低的幾階模態(tài)受到激發(fā)，因此整是同意的）相，撞擊很快就結(jié)束了以至于球與棒的作用僅限于局部區(qū)域。Nathan還聲稱最佳擊球點的位置和最低階模態(tài)的節(jié)點位置有關(guān)，這也和局部撞擊相矛 WhiletheinconsistencyintheNathanmodelmaycancelout,webuildourmodelonamorerigorousfooting.Forsimplicity,weusetheEuler-BernoulliequationsratherthefullTimoshenkoequations.Thedifferenceisthattheformerignoreshearforces.Thisshouldbeacceptable;Nathanpointsoutthathismodelislargelyinsensitivetotheshearmodulus.Wesolvethedifferentialequationsdirectlyafterdiscretizinginspacerather intomodes.Intheseways,wearefollowingtheworkofCrossNathan的模型的缺點可能抵消它的長處，因此在更加嚴謹?shù)幕A上建立的模型。為簡化起見，采用歐拉—伯努利方程而不是完整的鐵摩辛柯方程。前者忽略了剪切力。這一點是可以接受的；Nathan他的模型對剪切模量很不敏感。我們直接對該偏微分方程離散求解，而沒有采用模態(tài)分解。從這個意義上來說，沿用了Cross的方法。Ontheotherhand,ourmodelextendsCross’sworkinseveralkey另一方面，的模型在以下幾個方面擴展了Cross的工作Weexamineparametersmuchclosertothoserelevanttobaseball.Cross’modelfocusedontennis.Featuringuminumbeamofwidth0.6cmbeinghitwithaballof42gataround1m/s.Forbaseball,wehaveuminumorwoodbatofradiuswidth6cmbeinghitwithaballof145gtravellingat40m/s(whichinvolves5000timesasmuchimpactenergy).采用的是棒球的參數(shù)。Cross的模型則專注于網(wǎng)球，以一根直徑為0.6cm的鋁棒球棒，擊打一個重145g、速40m/s的棒球（動能為前者的5000倍。Weallowforavaryingcrosssection,animportantfeatureofareal考慮了截面沿棒長的變化，這是實際棒球棒的一個重要特Weallowthebattohavesomeinitialangularvelocity.Thiswillletusscrutinizetherigid-bodymodelpredictionthathigherangularvelocitiesleadtothe umpowerpointmovingfartherupthebarrel.Toreiterate,themainfeaturesofourmodel再次重申，模型的主要特點是anemphasisontheballcouplingwiththe重點考慮球與棒的耦合Finitespeedofwavepropagationinashorttime-scale,振動波的速度有Adaptiontorealistic更接近真實的棒Thesearenaturaloutgrowthsoftheapproachesinthe這些是文獻中所述方法的自然延伸MathematicsofOur取梁的微元段dx列力平衡方程和力矩平衡方程y方向力平衡：x 2 xdx Fs(Fs sdx)F 化簡得：2yFsl 力矩平衡：

x MFdx(M dx) 化簡得：

F 由材料力學：2 EI 綜上得：

2y

2(EI2y)F(x,t)l OurequationsareadiscretizedversionoftheEuler-Bernoulli 2y(z, 2y(z, F(z, t z z （ （

2 2(YI2y)F(z,t)

l isthemassdensity,（質(zhì)量密度（應該為線密度l

isthedisplacement,（撓度istheexternalforceinourcase,appliedbytheball),（外力YistheYoung’smodulusofthematerialaconstantand（楊氏模量IisthesecondmomentofareaR4/ forasoliddisc).（截面慣性矩（A是棒的截面積，z是棒的自然坐標，對z離散，步長為，采用中心差分）WediscretizezinstepsofTheonlyforceisfromtheballinthenegativedirectiontothekthsegment.Ourdiscretizedequationis:d2yiF(t)Y ( 2y y 2Ii(yi12yiyi1)Ii1(yi2yi1這里把第 個單元（即棒球作用的單元）上的分布力等效為集中力（乘以單元步長Ourdynamicvariablesarey1throughyN.Forafixedleftend,retendy1y00.Forafreeleftend,retendy1y0y0y1y1Theconditionsontherightendareogous.ThesearethesameconditionsthatCrossFinally,wehaveanadditionalvariablefortheball’sposition(relativetosomezerow(t).Initially,w(t)ispositiveand w(t)isnegative,sotheballismovingfromthepositivedirectiontowardsthenegative.Letu(t)w(t)yk(t).Thisvariablerepresentsthecompressionoftheball,andwereplace F(t) withF(u(t),u(t)).Initially,u(t)0andu(t)vball.TheforcebetweentheballandthebattakestheformofhysteresiscurvessuchastheonesshowninFigure3.因此棒球向負方向運動。令u(t)w(t)yk(t，u(t)表示球的壓縮量，作用力是時間的函數(shù)，因此也是u(t和u(t的函數(shù)。初始時刻u(t0u(tvball。球與棒的作用力與球的壓縮量的關(guān)系如圖3所示。Figure3Ahysteresiscurveusedinourmodeling, umcompression1.5Thehighercurveistakenwhenu(t)0(compression)andthelowercurvewhenu(t)0(expansion).Whenu(t)0,theforceiszero.Theequationofmotionfortheballisw(t)u(t)y(t) Wehaveeliminatedthevariable圖中上方的曲線為壓縮過程，u(t)0，下方的曲線為擴張過程，u(t)0作用力為零。球的運動方程如下：kw(t)u(t)y(t)k

這 就消去了wWehaveyettospecifythefunctionF(u(t),u(t)).Ascanbeseenin s[BaseballResearchCentern.d.],theballcompresssignificantly(oftenmorethan1cm)inacollision.Thecompressionand pressionislossy.Wecouldmodelthislossbysubtractingafractionoftheball’senergyafterthecollision;thatapproachisgoodenoughformanypurposes,butweinsteadfollowNathananduseanonlinearspringwithhysteresis.但是還沒有確定函數(shù)F(u(t),u(t))。正如在里看到的，棒球在撞擊過程中顯著SinceWFdx,thetotalenergylostistheareabetweenthetwocurvesinFblemwithcreatinghysteresiscurvesisthatonedoesnotknowthe compression(i.e.wheretostartdrawingthebottomcurve)untilaftersolvingtheequationsofmotion.Inpractice,wesolvetheequationintwosteps.3，用兩步來解方程。Themainassumptionsofourmodelderivefromthemainassumptionsofeach模型的主要假設就是對這兩個方程（方程（1）和（2））的假設Theistheexactformofthehysteresiscurveoftheball.Cross[1999]arguesthattheexactformofthecurveisnotveryimportantaslongasthedurationofimpact,magnitudeofimpulse, umcompressionoftheball,andenergylossareroughly續(xù)時間，沖擊的大球的最大壓縮量和能量損失大概正確。BoththeTimoshenkoandEuler—Bernoullitheoriesignoreazimuthalandlongitudinalwaves.Thisisafundamentalassumptionbuiltintoalloftheapproachesintheliterature.Assumingthattheimpactoftheballistransverseandtheballdoesnotrotate,theassumptionisjustified.鐵摩辛柯和歐拉伯—努利梁理論都忽略了方位波和縱波。這是文獻中所有方法本假設。假設棒球的沖擊是橫向的，棒球不發(fā)生旋轉(zhuǎn)，這樣假設是合理的Theassumptionsofourmodelarethesameasthoseintheliterature,sotheyarebytheliterature’s（）上法”求解，這里給出全部求解過程對方程（1）散后的棒的振動方程為：d2yi F(t) ( 2y y l l 2Ii(yi12yiyi1)Ii1(yi2yi1對方程

散后的球的運動程為

i , d2u

)F(u,du)

[

2I

l

l

k

k

k

k yk1(k1)21（）聯(lián)立方程（）和（）求解。由圖，假設作用力與壓縮量的關(guān)系曲線的形式，假設壓縮段為直線，松弛段為：1.52將方程（3）和（4）聯(lián)寫成矩陣形式從而轉(zhuǎn)化為常微分方程組的初值問題，可利用的相關(guān)函數(shù)求d2 dtd2 dt

y2 y1d2

dt

d2 Y dtY d2 k

(N6)(N k dt yk1 yk2 yN2 dt

d

N2Fdt

需注意的是，上式左端未知量為加速度項，右端未知量為位移項，在球棒作用期間，位移的變化遠遠快于加速度的變化，因此這是一個剛性問題（F()表示成u()的函數(shù)）其中 T

k

k

k

k

3k 3 11Ik11Ik(2Ik12Ik (Ik1+4Ik+Ik1(2Ik2IkIk+l寫成便于用ode求解的形式是：其中

PTP[y2, y2,u,y,y y T I 0

Q[y2,

2(N6)2(NyN2,u,y2, 設球壓縮到最大對應的時間為t1，從時刻t1到球離開棒的時間為則 F(u) 初始條件為

y2y1y0y1 yN2u其中w為棒的角速度

i 這里由于t1和t2的值未知，而t1t21.4ms，只好對t1和t2的值進行SimulationOurmodel’stwomainfeaturesarewavepropagationinthebatandpessonofheba.heaersustaedbyheasmetyofhepotnigurea.hspotasoeveasheie-scaeofhecoson:hebaleaeshebat.4saferipac.Dungandafercoson,shockwaespopaaehouhheba.模型的兩個主要特點就是同時振動波在棒中的和球的壓縮擴張程。后a.4毫秒離開棒球棒。碰撞期間和碰撞后，振動波在棒球棒中。Inthisexample,thebatstruck60cmfromthehandle.Whatdoesthecollisionlooklikeat10cmfromthehandle?Figure4bshowstheanswer:Theotherendofthebatdoesnotfeelanythinguntilabout0.4msanddoesnotfeelsignificantforcesuntilabout1.0ms.Bythetimethatportionofthebatswingsback(almost2.0ms),theballhasalreadyleftcontactwiththebat.Thisilluminatesanimportantpoint:Weareconcernedonlywithforcesontheballth twithinthe1.4mstime-frameofthecollision.Anywavestakinglongertoreturntotheimpactlocationdonotaffectexitvelocity.答案：該處知道撞擊發(fā)生后大約0.4毫秒才發(fā)生振動，直到大約1.0毫秒后才達到最位移。到位移重新達到最小值時（撞擊發(fā)生2.0毫秒之后球已經(jīng)飛離了棒。這也說Havingdemonstratedthebasicfeaturesofourmodel,wenowreplicatesomeofresultsbutwithbaseball-likeparameters.InFigure5a,weshowthattheeffectsoffixed-free-boundaryconditionsareinagreementwithCross’s球的參數(shù)。在圖5a中，表明在固端邊界和邊界條件下的結(jié)果都和Cross的模Asweexpected,fixedboundariesenhancetheexitvelocityandfreeboundariesreducethem.Fromthisresult,weseetheeffectofthesh ofthebat.Thehandledoesindeedactlikeafreeboundary.Thedistancebetweentheboundariesistoosmalltogetaflatzoneintheexitvelocityvs.positioncurve.Ifweextendthebarrelby26cm.aflatzonedevelops(Figure5b;noticethechangeinaxes).Intuitively,thisflatzoneexistsbecausetheball“sees”onlythelocalgeometryofthebatandtheboundariesaretoofarawaytohavesubstantial正如預料的，固端邊界增大了球速，邊界減小了球速。從這一結(jié)果可以看出棒的形狀的影響。握柄端的作用相當于一個邊界。兩個邊界之間的距離較短時，最佳擊球區(qū)較窄，如果增大棒長26cm，最佳擊球區(qū)會變寬（圖5b。直觀的解釋是，在兩端邊界距離較遠時，棒球的作用僅限于沖擊的局部區(qū)域，最佳擊球區(qū)就更加平緩Fromnowon,weusean84-cmbatfreeonbothends,wherepositionzerodenotesthehandleend.Inthisbasecase,thesweetspotisat70cm.Weinvestigatethedependenceoftheexitspeedontheinitialangularvelocity.Accordingtorigid-bodymodels,thesweetspotisexactlyatthecenterofmassifthebathasnoangularvelocity.InFigure6, theresultsofchangingtheangularvelocity.Ourresultscontrastgreatlywiththesimpleexamplepresentedearlier.Whiletheangular-rotationeffectisstillthere,theeffectivemassplaysonlyanegligibleroleindeterminingtheexitspeed.Inotherwords,thebatisnotarigidbodybecausetheentirebatdoesnotreactinstantly.Thedominatingeffectisfromtheboundaries:ofthebarrelandwherethebarreltrsoff.Thesefreeendscauseasignificantdropinexitvelocity.Increasingtheangularvelocityofthebatincreasestheexitvelocity,inpartjustbecausetheimpactvelocityisgreater(byafactorofdistancefromthecenterofmassofthebat).

witimes位置為70cm。研究了球速對初始角速度的依賴性。根據(jù)剛體模型，在棒球棒沒有初始角速度時，最佳擊球點就在棒的質(zhì)心位置。在圖6中，給出了改變角速度后棒球棒不是剛體。起主要作用的是邊界條件：棒的末端和棒的細端。這些邊界使速度等于角速度wi乘以作用點距離質(zhì)心的距離lInFigure7a,weshowthatnearthesweetspot(at0.7m),inincreasingangularvelocityactuallydecreasestheexcessexitvelocity(relativetotheimpactvelocity).Weshouldexpectthis,sinceathigherimpactvelocity,moreenergyislosttotheball’scompressionandpression.Toconfirmthisresult,wealsorecreatetheplotinFigure7bbutwithoutthehysteresiscurve—inwhichcasethiseffectdisappears.Thisexampleisoneofthefewplaceswherethehysteresiscurvemakesadifference,confi