03年美国数学建模A题.docx
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03年美国数学建模A题.docx
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03年美国数学建模A题
CardboardStackingTypeWhenitComestoCrashing
Summary
Thisarticleisaboutastuntpersononamotorcyclewilljumpoveranelephantandlandinapileofcardboardboxestocushiontheirfall.Firstlyweneedtoprotectthestuntperson,secondly,weneeduserelativelyfewcardboardboxes.
Inquestion1,weneedfigureoutthesize,theamountandthestackedmethod.Firstweanalyzetheshocktosinglecardboardbox,anddrawthemotioncurveofthestuntperson&motorcycle.NextweapplyNewton'slawsofmotionandlawofenergyconservationtobuildthemotionmodel.Finallywefigureoutthesizeofthecardboardboxis
thenumberofthecardboardboxesis138,thecostingis$690.
Inquestion2,onthemotioncurveofthestuntperson&motorcycle,wefindthatwecanstacktheboxestosteps.Thiswaynotonlycanmakethestuntpersonsafe,butalsoreducethecosting.Itisgreatlyoptimizingthemodel.Sowebuildthestackedstepmodel,andwealsocanfigureouttheminimumnumberofthecardboardboxesis:
andtheminimumcostingofthecardboardboxesis:
itcansave$220.
Atlast,wealsotalkaboutsomeotherstackedway,wecomparetheconventionalstackedtotheintercrossstacked.Firstweuseeightcardboardboxestoanalyzetheforce,andthenpopularizetoncardboardboxes,finallywefindthattheshockforcetointercrossstackedcanbearis
timestoconventionalstacked.
Inquestion3,wegeneralizetodifferentcombinedweights(stuntperson&motorcycle)anddifferentjumpheights,duetoformula
(1)and
(2),wefigureoutthenumberandcostingtodifferentcombinedweightsandjumpheights.Atlast,wegetaconclusion:
whenthecombinedweighisinvariant,theamountisincreasingalongwiththejumpheight;whenthejumpheightisinvariant,theamountisincreasingalongwiththecombinedweight.
Contents
CardboardStackingTypeWhenitComestoCrashing1
Summary1
1.Introduction3
2.RestatementoftheProblem3
3.AnalysisoftheProblem4
4.Assumptions5
5.theSymbols5
6.ModelDesignandSolve6
Model1theMotionModelBasedonLawofEnergyConservation6
Model2theStackedStepModel8
Model3theSizeandAmounttoDifferentCombinations11
7.SensitivityAnalysis12
8.FurtherDevelopment12
9.StrengthsandWeaknessesoftheModels13
1.Strengths13
2.Weaknesses14
References15
1.Introduction
Anexcitingactionsceneinamovieisgoingtobefilmed,andyouarethestuntcoordinator!
Astuntpersononamotorcyclewilljumpoveranelephantandlandinapileofcardboardboxestocushiontheirfall.ThisprocesslikesFigure1.
Airbagrigsarecommonlyrentedforhighfallstunts.However,airbagsaredesignedonlytocatchhumans.Catchingthemotorcycleintheairbagwouldriskdamagingit,andreplacementcostsmightwellexceedthebudgetofourunnameddirector.Thealternativeistouseacardboardboxrig-astackofboxesthatwillcrushandabsorbtheimpactofthemotorcycleandstuntman.
Figure1:
Astuntpersononamotorcyclefliesswiftlyandskipsanelephant
Theanswertokeepingtheelephantsafeissimple:
theelephantjuststandsinplaceanddoesnothing.Thesafetyofthestuntmanisamoredifficultproblem.
Themotorcycle’smovementisaccordingtotheNewton'slawsofmotion.Theformulais:
.
Theprocessofthemotorcyclelandinapileofcardboardboxestocushiontheirfallisaccordingtolawofenergyconservation.Thedefinitionis:
theenergyisnotemergingwithoutfoundation,andnormissingwithoutfoundation.Itisonlyfromoneformchangingtoanotherform,orfromoneobjecttoanotherobject.Thetotalenergywillnotchange.
2.RestatementoftheProblem
Hardlyamoviemadetodayiswithoutsomekindofamazingstuntwork.Weallareheldbreathlesswhenapersonfallsoutofa20-storybuildingorwhenaheart-stoppingcarchaseendsinaspectacularcrash.
Toreducethechancesofdamageorinjury,stuntdesignersusedevicesthatstretchoutthetimeittaketostopabody'smomentum.Thelongertheperiodoftimeusedinchangingthemomentum,thelessforcewillbereleaseduponimpact.Thecardboardboxesareoneofthedevicesused.
Ourtaskisto:
(1)Determinewhatsizeboxestouse
(2)Determinehowmanyboxestouse
(3)Determinehowtheboxeswillbestacked
(4)Determineifanymodificationstotheboxeswouldhelp
(5)Generalizetodifferentcombinedweights(stuntperson&motorcycle)anddifferentjumpheights
3.AnalysisoftheProblem
Firstwemustknowthatwearenowfacedwiththefollowing:
(1)Theprimaryobjectiveistosafelycatchthestuntmanandmotorcycle.
(2)Asasecondaryobjective,wewishtominimizethecostandsizeoftheboxrig.
Afteranalyzing,wehavesomeapproaches:
(1)Firstweinvestigatetherelationshipbetweenthesize/shape/materialofaboxandtheworkrequiredtocrushit.Wecallthisquantitythecrushenergyofabox.
(2)Wereviewthemostcommonlyavailablecardboardboxtypes.Werestrictourconsiderationtoonlythesetypesbecausecustomboxesaremuchmoreexpensivethanstandardones.
(3)Weanalyzethetrajectoryofthemotorcycleandtheuncertaintyinitslandinglocation.Thisdeterminestheproperplacementoftheboxrigandhowlargeanareaitmustcovertosafelycatchthemotorcycleandstuntman.
(4)Usingthecrushenergyformula,weestimatethenumberofboxesthatmustbecrushedinordertoarrestthemotionofthemotorcycleandstuntman.
(5)Basedontheresultsfromoursimulation,costandconstructionconsiderations,andsafetyrequirements,wemakeasetofstrongrecommendationsregardingplacement,size,construction,andstackingtypeoftheboxrig.
Theprocessofastuntpersononamotorcyclejumpoveranelephantandlandinapileofcardboardboxescancomposeoftwoprocesses.Thefirstisthestuntpersononamotorcyclejumpoveranelephant,wecancallthisprocessisflatthrowing.WecanuseNewton'slawsofmotiontoanalyzingthisproblem.Secondly,wewillanalyzetheprocessofthemotorcyclecrashedtothecardboardboxes.Throughanalyzetheforcetothecardboardboxes.Wecangettheseconclusions:
(1)Theman&motorcyclemovestowardstheuncompressedbox.
(2)Whentheman&motorcyclecontactstheboxandbegintocompressit,turnitfromuncompressedintopartlycompressed,theboxabsorbsadefiniteamountofthekineticenergyoftheman&motorcycle.
(3)Theman&motorcyclecontinuestocompressthebox,theboxisinthepartlycompressedstate,andexertsnoforceontheman&motorcycle.
(4)Thevolumeoftheboxdecreasestothelimitandturnsintothecompressedstate(arigidbody)andcollideswiththeman&motorcycle.
(5)Thecompressedboxisforcedintomovementbythecollision,whiletheman&motorcycleisdeceleratedbythebox.
Atlastwecanuselawofenergyconservationtoworkoutthesizeandnumberofthecardboardboxes,andthenwecandiscusstheamountofthecardboardboxesandcostingtodifferentcombinedweights(stuntperson&motorcycle)anddifferentjumpheights.
4.Assumptions
(1)Duringtheentireprocess,thestuntperson&motorcycleareconsideredasawholeobject.
(2)Thecardboardboxesareapproximatelyconsideredascuboids.
(3)Thecardboardboxesaremadeupofthesamematerial.
(4)Thecostingofthecardboardboxesisdirectratiotothesurfacearea.
(5)Theareawherethecardboardboxesarestackedisflatandwithoutanyobstacles.
(6)Comparedtotheresistanceoftheboxes,theairisveryslightandcanbeneglected.
5.theSymbols
:
thecomponentofvelocityalong
axis
:
thecomponentofvelocityalong
axis
:
theenergytobreakdownabox
:
theheightofthedivingplatform
:
thecombinedweightofstuntperson&motorcycle
:
thenumberoftheboxesontheupright
:
thelengthofthecardboardboxes
:
theheightofthecardboardboxes
:
thelengthofasideofthebox
:
thepoundsperinchofedgecompressiveforceparalleltotheflute
:
thevelocityintheverticaldirectionwhenthestuntmanhasgonethroughacarton
:
thevelocityintheverticaldirectionwhenthestuntmanhasgonethroughtwocartons
:
thevelocityintheverticaldirectionwhenthestuntmanhasgonethroughthreecartons
:
thevelocityintheleveldirectionwhenthestuntmanhasgonethrough23/3cartons
:
thevelocityintheleveldirectionwhenthestuntmanhasgonethrough46/3cartons
:
thevelocityintheleveldirectionwhenthestuntmanhasgonethrough23cartons.
6.ModelDesignandSolve
Model1theMotionModelBasedonLawofEnergyConservation
Tofigureoutthesizeandnumberofthecardboardboxestosafethestuntperson.Firstweneedanalyzetheindividualcardboardbox.
Themovementoftheman&motorcycleisinfluencedbygravityandcollisionwithboxes;themovementofacompressedboxisinfluencedbygravity,collisionwithman&motorcycle,collisionwithotherboxesandgroundsupport&frictionifitcontactstheground,whileanuncompressedboxcanbeonlyinfluencedbygravityandsupportfromboxesbeloworfromthegroundbeforeitturnsintocompressedstate.
Figure2.theshocktoonebox
Tocalculatetheabilityofacardboardboxtoabsorbtheimpactofthestuntpersonandmotorcycleweestimatetheenergyrequiredtocrushthebox.Thisestimationisbasedonacombinationofphysicalconsiderationsandexperimentalboxescrushing.
Therearethreepartsofthecardboardboxcanbeartheshock,theyaretheuppersurface,thesidesurfaceandverticalface.Sotheenergyoftheshockis
.
Butthesufferedshockoftheuppersurfaceandthesidesurfaceisfarinferiortotheverticalface.Sothat
.
CommercialcardboardisratedbyEdgeCrushTest(ECT),which
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