冲压技术英文资料doc.docx

冲压技术英文资料doc.docx

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冲压技术英文资料doc

冲压技术英文资料

StampingDieStripOptimizationforPairedPartsV.Vamanu,T.j.NyeMechanicalEngineeringDepartmentMcMasterUniversityHamilton,OntarioOctober30,2000AbstractInstamping,operatingcostaredominatedbyrawmaterialcosts,whichcantypicallyreach75oftotalcostsinastampingfacility.Inthispaper,anewalgorithmisdescribedthatdeterminesstampingstriplayoutsforpairsofpartssuchthatthelayoutoptimizesmaterialutilizationefficiency.Thisalgorithmpredictsthejointly-optimalblankorientationonthestrip,relativepositionsofthepairedblanksandtheoptimumwidthforthestrip.Examplesaregivenforpairingthesamepartstogetherwithonerotated180º,andforpairsofdifferentpartsnestedtogether.ThisalgorithmisideallysuitedforincorporationintodiedesignCAEsystems.KeywordsStamping,DieDesign,Optimization,MaterialUtilization,MinkowskiSum,DesignToolsIntroductionInstamping,sheetmetalpartsofvariouslevelsofcomplexityareproducedrapidly,ofteninveryhighvolumes,usinghardtooling.Theproductionprocessoperatesefficiently,andmaterialcostscantypicallyrepresent75oftotaloperatingcostsinastampingfacility[1].Notallofthismaterialisusedintheparts,however,duetotheneedtotrimscrapmaterialfromaroundirregularly-shapedparts.Theamountofscrapproducedisdirectlyrelatedtotheefficiencyofthestampingstriplayout.Clearly,usingoptimalstriplayoutsiscrucialtoastampingfirm’scompetitiveness.Thedegreeofthistrimlossisdeterminedatthetoolingdesignstagewhenthestriplayoutiscreated.Asapartorpartsarelaidoutonthestrip,thedesignerchoosestheorientationoftheparts,widthofthestrip,and,inthecaseofmultiplepartsblankedtogether,theirrelativepositions.Ideally,thematerialutilizationismaximized.Thevalueofeventinyimprovementsinmaterialutilizationcanbegreat;forexample,inastampingoperationrunningat200strokesperminute,asavingsofjust10gramsofmaterialperpartwillaccumulateintoasavingsofmorethanatonneofrawmaterialpereight-hourshift.Thematerialutilizationissetduringthetoolingdesignstage,andremainsfixedfortheusuallylonglifeofthetool.Thus,thereissignificantvalueindeterminingtheoptimalstriplayoutbeforetoolingisbuilt.Thistaskiscomplicated,however,sincechangingeachvariableinthelayoutcanchangeboththepitchdistancealongthestripbetweenadjacentpartsandstripwidthsimultaneously.Evaluatinglayoutefficiencymanuallyisextremelychallenging,andwhileexactoptimalalgorithmshavebeendescribedforthelayoutofasinglepartonastrip,sofaronlyapproximatealgorithmshavebeenavailableforthelayoutofpairsofpartstogether.Nestingsolutionsforpairsofpartsisanimportantproblemsinceitisempiricallyknownthatnestingpairsofpartscanoftenimprovematerialutilizationcomparedtonestingeachpartonaseparatestrip.Thispaperaddressesthecommoncasesinwhichagivenpartisnestedwithasecondcopyofitselfrotatedat180º,andwhentwodifferentpartsarenestedtogether.Inthispaperwedescribeanewalgorithmthatprovidestheoptimalstriplayoutforthesetwocases.PreviousWorkOriginally,striplayoutproblemsweresolvedmanually,forexample,bycuttingblanksfromcardboardandmanipulatingthemtoobtainagoodlayout.Theintroductionofcomputersintothedesignprocessledtoalgorithmicapproaches.Perhapsthefirstwastofitblanksintorectangles,thenfittherectanglesalongthestrip[2].Variationsofthisapproachhaveinvolvedfittingblanksintonon-overlappingcompositesofrectangles[3],convexpolygons[4,5]andknowninterlockingshapes[6].Afundamentallimitationexistswiththisapproach,however,inthattheenclosingshapeaddsmaterialtotheblankthatcannotberemovedlaterduringthelayoutprocess.Thisaddedmaterialmaypreventoptimallayoutsfrombeingfound.Apopularapproachtoperformingstriplayoutistheincrementalrotationalgorithm[6-10,16].Init,theblank,orblanks,arerotatedbyafixedamount,suchas2º[7],thepitchandwidthofthelayoutdeterminedandthematerialutilizationcalculated.Afterrepeatingthesestepsthroughatotalrotationof180ºduetosymmetry,theorientationgivingthebestutilizationisselected.Thedisadvantageofthismethodisthat,ingeneral,theoptimalblankorientationwillfallbetweentherotationincrements,andwillnotbefound.Althoughsmall,thisinefficiencyperpartcanaccumulateintosignificantmateriallossesinvolumeproduction.Meta-heuristicoptimizationmethodshavealsobeenappliedtothestriplayoutproblem,bothsimulatedannealing[11,12]andgeneticprogramming[13].Whilecapableofsolvinglayoutproblemsofgreatcomplexityi.e.manydifferentpartsnestedtogether,general2-Dnestingofsheets,theyarenotguaranteedtoreachoptimalsolutions,andmaytakesignificantcomputationalefforttoconvergetoagoodsolution.Exactoptimizationalgorithmshavebeendevelopedforfittingasinglepartonastripwherethestripwidthispredetermined[14]andwhereitisdeterminedduringthelayoutprocess[15].Thesealgorithmsarebasedonageometricconstructioninwhichoneshapeis‘grown’byanothershape.Similarversionsofthisconstructionarefoundunderthenames‘no-fitpolygon’,‘obstaclespace’and‘Minkowskisum’.Fundamentally,theysimplifytheprocessofdeterminingrelativepositionsofshapessuchthattheshapestouchbutdonotoverlap.Throughtheuseofthisconstructioninthispaper,theparticularversionusedistheMinkowskisum,efficientalgorithmscanbecreatedthatfindthegloballyoptimalstriplayout.Fortheparticularproblemofstriplayoutforpairsofparts,resultshavebeenreportedusingtheincrementalrotationalgorithm[7,16]andsimulatedannealing[11],butsofarnoexactalgorithmhasbeenavailable.Inwhatfollows,theMinkowskisumanditsapplicationtostriplayoutisbrieflyintroduced,anditsextensiontonestingpairsofpartsisdescribed.TheMinkowskiSumTheshapeofblankstobenestedisapproximatedasapolygonwithnvertices,numberedconsecutivelyintheCCWdirection.Asthenumberofverticesincreases,curvededgesontheblankcanbeapproximatedtoanydesiredaccuracy.Giventwopolygons,AandB,theMinkowskisumisdefinedasthesummationofeachpointinAwitheachpointinB,1Intuitively,onecanthinkofthisprocessas‘growing’shapeAbyshapeB,orbyslidingshape–Bi.e.,Brotated180ºaroundAandfollowingthetraceofsomereferencepointonB.Forexample,Fig.1showsanexampleblankA.Ifareferencevertexischosenat0,0,andacopyoftheblankrotated180ºi.e.,–AisslidaroundA,thereferencevertexon–AwilltraceoutthepathshownastheheavylineinFig.2.ThispathistheMinkowskisum.MethodsforcalculatingtheMinkowskisumcanbefoundincomputationalgeometrytextssuchas[17,18].SamplePartAtobeNested.MinkowskiSumheavylineofsamplePartlightline.Thesignificanceofthisisthatifthereferencevertexon–Aisontheperimeterof,Aand–Awilltouchbutnotoverlap.Thetwoblanksareascloseastheycanbe.Thus,foralayoutofapairofblankswithonerotated180ºrelativetotheother,definesallfeasiblerelativepositionsbetweenthepairofblanks.AcorollaryofthispropertyisthatiftheMinkowskisumofasinglepartiscalculated.Withitsnegative,i.e.,.AcompleteexplanationofthesepropertiesoftheMinkowskisumisgivenin[15].Theseobservationswerethebasisforthealgorithmforoptimallynestingasinglepartonastrip.Thesituationwhennestingpairsofpartsismorecomplex,sincenotonlydotheoptimalorientationsoftheblanksandthestripwidthneedtobedetermined,buttheoptimalrelativepositionofthetwoblanksneedstobedeterminedaswell.Tosolvethisproblem,aniterativealgorithmissuggestedGivenBlanksAandBwhereB–Awhenablankispairedwithitselfat180º1.SelecttherelativepositionofBwithrespecttoA.TheMinkowskisumdefinesthesetoffeasiblerelativepositionsFig.2.2.‘Join’AandBatthisrelativeposition.CallthecombinedblankC.3.NestthecombinedblankConastripusingtheMinkowskisumwiththealgorithmgivenin[14]or[15].4.Repeatsteps1-3tospanafullrangeofpotentialrelativepositionsofAandB.Ateachpotentialposition,evaluateifalocaloptimamaybepresent.Ifso,numericallyoptimizetherelativepositionstomaximizematerialutilization.LayoutOptimizationofOnePartPairedwithItselfThefirststepintheaboveprocedureistoselectafeasiblepositionofblankBrelativetoA.Thispositionisdefinedbytranslationvectortfromtheorigintoapointon,asshowninFig.3.Duringtheoptimizationprocess,thistranslationvectortraversestheperimeterof.RelativePartTranslationNodeson,showingTranslationVectort.Initially,adiscretenumberofnodesareplacedoneachedgeof.Thetwopartsaretemporarily‘joined’atarelativepositiondescribedbyeachofthetranslationnodes,thenthecombinedblankisevaluatedforoptimalorientationandstripwidthusingasingle-partlayoutproceduree.g.,asin[14]or[15].Inthisexample,consistsof12edges,eachcontaining10nodes,foratotalof120translationnodes.Thepositionofeachnodeisfoundvialinearinterpolationalongeachedge,whereisvertexIontheMinkowskisumwithacoordinateof,.Definingapositionparameterssuchthats0atands1at,coordinatesofeachtranslationnodecanbefoundas23Ifmnodesareplacedoneachedge,,thepositionparametervaluesforthenode,,arefoundas4Calculatingtheutilizationateachofthe1