mcm优秀论文.docx
- 文档编号:30647595
- 上传时间:2023-08-18
- 格式:DOCX
- 页数:17
- 大小:198.35KB
mcm优秀论文.docx
《mcm优秀论文.docx》由会员分享,可在线阅读,更多相关《mcm优秀论文.docx(17页珍藏版)》请在冰豆网上搜索。
mcm优秀论文
AnEnergyModelfortheStuntPerson
Assumptions:
.Thetotalmass(combinedweight)ofthestuntperson(65kg)andthemotorcycle(135kg)is200kg.
.Themotorcycleandthestuntperson(MASP),areviewedideallyasarectangularsolid,1.8mlong,0.5mwideand1.1mhigh.
.Toguaranteesafety,theeffectof4gisthemaximumlimitthatisallowed.
.Theelephantisviewedideallyasarectangularsolid,6mlong,1mwideand4mhigh.
.Itisreportedthat,toavoidinjury,theverticalvelocityofMASPcan’tbelargerthan6m/swhilereachingtheground[4].
Notations
(thelettersinparenthesesaretheunitsofvariables):
dthewidthofabox
htheheightofabox.
hmaxthealtitudeofthetoppointofMASPjumpingovertheelephant..
lthelengthofabox.
Ltheperimeterofabox(m).
Pthemaximumcompressionstrengthofabox(N).
Pmthesurfacingstrengthofpaper(N/m).
VxthehorizontalvelocityofMASP.
VytheverticalvelocityofMASP.
VsthemaximumverticalvelocityofMASPreachingthegroundsafely(6m/sinourmodel).
Wwork.
TheTrackofMASP
TheIdealModel--IgnoringAirResistance
TheTrackisshownasFigure1:
Figure1.MNisaslopetoacceleratethevelocity
Tojumpovertheelephantsuccessfully,thealtitudeofpointAandBofthetrackmustbenosmallerthanh,thatis:
HA=HB=
where
istheanglebetweenthevelocityandground.Givenan
wecangettheminimumvelocityofv(vmin)andthedistancebetweenpointDandE,whichisLDE=
.
TheAirResistanceModel
--Takingairresistanceintoaccount
TheairresistancehassignificanteffectonthemotionofMASPandcannotbeignoredinthismodelsincetheMASP’svelocityislarge.Basedontheaboveanalysisandconclusions,wecannowtaketheairresistanceintoconsiderationanddothesimilaranalysisconveniently.
ThetrackisshownasFigure2:
Figure2.MNisaslopetoacceleratethevelocity
Sincethegravityismuchlargerthantheverticalairresistance,weonlyfocusonthehorizontalone.Theairresistanceisthequadraticfunctionofvelocity,that’sF=kv2,wherecoefficientkisaconstant[5].Basedonthisconclusion,wegetthefollowingequations:
Forthesameprinciplerelatedbefore,thealtitudeofpointAandpointBmustsatisfy:
HA=HB=
-
Wecanalsoget:
LDE=
Forfurtheranalysis,pleaseseeAppendixA.
TheResearchofBox
TheMaximumCompressionStrengthofaBox
Accordingtoreference[1],themaximumverticalstrengthaboxcansustainisP=10PmLC,whereCisaconstantanddeterminedbythealtitudehofabox.
Unfortunately,wegetlittleinformationabouttherelationshipofCandhoranempiricalformula,whichwehavetodobyourselves.Fromthedatagiveninreference[1],wecandeterminethevalueofCcorrespondingtospecifich(Figure3),whichisrepresentedasvector(h,c
).Thedataareasfollows:
(0.165,5.17583)(0.190,5.00570)(0.230,4.65409)
(0.245,4.60045)(0.304,4.04274)(0.310,4.00022)
(0.335,3.86207)(0.350,3.69107)(0.365,3.62987)
(0.380,3.50039)(0.402,3.30092)(0.450,2.83309)
(0.510,2.34030)(0.545,2.05601)(0.605,1.56072)
Fromobservation,wecanapproximatecwithalinearfunction.UsingtheLeast-SquareMethod,theequationisfittedas
C=C(h)=C1+C2h=(6.459-8.0798h)
10-2,
Theboxesinourmodelareunderverticalandhorizontalpressuresimultaneously,sousingthevectoradditionlaw,thecompressionstrengthismodifiedto:
P=
=20Pm
.
Figure3
CompressionStrengthofaFoldingBox
Therearethreestagesinthecourseofcompressionofabox,i.e.,elasticityphase,plasticityphaseandcollapsephase.Ifthedeformityofaboxisx,thentheelasticityforceFis[2]:
F(x)=axe-bx,
whereaandbareconstantandtobedetermined.
Whenthedeformityoftheboxisx0=
=
Fattainsitsmaximumvalue,whichequalstothemaximumcompressionstrengthP.
ByF’(x0)=0
andMax[F(x)]=F(x0)=P,
weget
TheEnergythatABoxCanAbsorb
Thetotalenergythataboxcanabsorbis:
W=
WcanbedividedintoWvandWh,representingtheverticalandhorizontalabsorbedenergyrespectively:
Wv=
W
Wh=
W
ConstructingtheModel
TheEnergyModel
Inessence,theeffectofcardboardboxestocushionthefallistoeliminatetheenergyofMASP.Toavoidinjury,theverticalvelocityofMASPwhilereachingthegroundcan’tbelargerthanVS=6m/s.Accordingtotheenergyconservationlaw,thedecrementoftheenergyofMASPistheincrementoftheenergyofboxes,ignoringtheairresistance(inthecourseofcushioning)andotherminoringredients.Toguaranteesafety,inthecourseoffalling,theboxescanabsorbatleastasmuchastheformerlyrelateddecrementofenergy.
TheMostDesirableBox
--AnOptimalModel
Themostdesiredboxissuchthattheeffectofcushioncanbeobtainedwithfewerboxesandlowerheight.Themoreenergytheboxcanabsorbperunitvolume,thefewerboxesweneedtocushionthefall,anditismoreconvenienttoavoidbeingseenbycamera.
Thismeansamostdesirableboxisonewithlargest
.
Accordingtotheassumptions,toguaranteesafety,theforcethatthestuntpersoncanendureshouldbenomorethanthemaximumlimit.Hence
F
M(4g+g)=5Mg
Tosatisfythisconstraint,weonlyneedtobesurethat
5Mg,
where
=P
=
.
Pisthemaximumcompressionstrengthofaboxand
showsthenumberofboxescushioningthefallsimultaneously,whichisdeterminedbylanddofabox.Sotheconstraintismodifiedto
5Mg
TosolvethisproblemweproposeanoptimalModel.
Object:
Max=
Subjectto:
5Mg
UsingtheSearchingAlgorithmwithTurboC(seeAppendixB),wegetthesidesofthemostdesirablebox:
L=0.48m,d=0.30m,andh=0.20m.
CushioningtheFall
WedefineWva=
andWha=
wherendenotesthenumberofboxestocushionthefall.
Toguaranteesafety,thefollowingconstraintsmustbesatisfied:
whereSthehorizontaldisplacementofMASPbetweenreachingtheboxandtheground.
Sincethecushionlastsonlyforaveryshorttime,thehorizontalmotioncanbeviewedapproximatelyasuniformspeed-reduction(whoseaccelerationisaconstant).Thissystemofequationscanbemodifiedto:
wherenhdenotesthenumberofboxestocushionthefallhorizontally.
Theresults
Thesizeoftheboxesisthesameasthatofthemostdesiredbox.
Thealtitudeofthetoppointis4.5m(hm=4.5m).UsingTurboC,weknowthat270boxesshouldbeusediftheyarepiledintoa6-storeyedrectangularsolid.
ConsideringthatMASPisgoingdownconstantly,wecanstacktheboxesintheformofastep.Thenumberofboxesisreducedto3
5+3
7+3
9+3
11+3
13+3
15=180.
TheTestofModel
GiventhemassofMASP,wecalculatethesizeofaboxandthenumberneededifthey’restackedintoarectangularsolid.Thenwecalculatethenumberofboxes.
Weadoptdifferentpapers,resultingindifferentPm.TherelationshipbetweenthenumberofneededboxesandPmisshownasFigure4:
Figure4.
Wefixhmax=4.5mandPm=2000N/m,andget:
Figure5.
WefixM=2000KgandPm=2000N/m,andget:
Figure6.
Weanalysetheeffectsofmodificationtotheboxesanddifferentcombinedweightsanddifferentjumpheights.Theresultsareasfollows:
Table
Hmax(m)
M(kg)
Number
V(m3)
Length(m)
Width(m)
Heigth(m)
4.0
150
230
3.723
0.48
0.16
0.21
170
466
5.952
0.48
0.14
0.19
190
334
3.849
0.48
0.12
0.20
210
525
5.998
0.48
0.14
0.17
230
767
6.263
0.48
0.10
0.17
250
1028
8.883
0.48
0.12
0.15
5.5
150
706
11.40
0.48
0.16
0.21
170
1193
15.23
0.48
0.14
0.19
190
1023
11.78
0.48
0.12
0.20
210
1344
15.35
0.48
0.14
0.17
230
1956
16.03
0.48
0.10
0.17
250
2313
19.98
0.48
0.12
0.15
7.0
150
1442
23.26
0.48
0.16
0.21
170
2256
28.81
0.48
0.14
0.19
190
2088
24.05
0.48
0.12
0.20
210
3024
34.55
0.48
0.14
0.17
230
3715
30.31
0.48
0.10
0.17
250
4826
41.70
0.48
0.12
0.15
8.5
150
2438
39.32
0.48
0.16
0.21
170
3151
40.24
0.48
0.14
0.19
190
3529
40.65
0.48
0.12
0.20
210
4726
53.99
0.48
0.14
0.17
230
6018
49.10
0.48
0.10
0.17
250
8253
71.31
0.48
0.12
0.15
10
150
3246
52.35
0.48
0.16
0.21
170
4774
60.95
0.48
0.14
0.19
190
5345
61.58
0.48
0.12
0.20
210
6805
77.74
0.48
0.14
0.17
230
8873
72.41
0.48
0.10
0.17
250
11424
98.70
0.48
0.12
0.15
StrengthsandWeaknesses
Consideringasinglecardboardboxfirst,anoptimalsizeofaboxcanbeworkedout.Thesizeofaboxremainsthesameonlywiththeweightchangeable,sothatthecalculationcanbesimplified.
Thesizeofaboxisobtainedbyapartoptimization,notnecessarilythemostfavorableasawhole,therefore,thepossibleboxesmaybeincreased.
AppendixA
ThetrackisshownasFigure1and2.
Vx=
ax=
Tosolvingthesystemofequations
wecanonlyturnhelptocomputerforsolutionusingnumericalapproximation.ThuswedothefollowinganalysisassumingthattheforceFisthelinearfunctionofvelocity(F=kv).Basedonthisassumption,thehorizontalvelocityandaccelerationare:
Vx=
ax=
TheequationsdescribingthemotionofMASPare
wherexthehorizontaldistancebetweenMASPandpointD.
Fromthese,weget
x(t)=
Sincebothkandtaresmallenough(kisnomorethan0.1
),
x(t)=
y(t)=
Thehorizontalvelocityisvx=
andtheaccelerationisax=
whichshowsthatthemotorcyclereachthetoppointattc=
.
ThetimerequiredtomovefrompointAtopointCandfrompointCtopointB
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- mcm 优秀论文
![提示](https://static.bdocx.com/images/bang_tan.gif)