發(fā)動(dòng)機(jī)鏈傳動(dòng)式配氣機(jī)構(gòu)設(shè)計(jì)【說(shuō)明書(shū)+CAD+CATIA+仿真】,說(shuō)明書(shū)+CAD+CATIA+仿真,發(fā)動(dòng)機(jī)鏈傳動(dòng)式配氣機(jī)構(gòu)設(shè)計(jì)【說(shuō)明書(shū)+CAD+CATIA+仿真】,發(fā)動(dòng)機(jī),鏈傳動(dòng),式配氣,機(jī)構(gòu),設(shè)計(jì),說(shuō)明書(shū),仿單,cad,catia,仿真 Anewandgeneralizedmethodologytodesignmulti-stage geardrivesbyintegratingthedimensionalandthe congurationdesignprocess TaeHyongChong * ,InhoBae,Gyung-JinPark DepartmentofMechanicalEngineering,HanyangUniversity,17HaengDang-Dong,SeongDong-Gu, Seoul133-791,SouthKorea Received24July2000;accepted3August2001 Abstract Thispaperproposesanewandgeneralizeddesignmethodologytosupportthedesigneratthepreliminary designphaseofmulti-stagegeardrives.Theproposeddesignmethodologyautomatesthedesignprocessby integratingthedimensionaldesignandthecongurationdesignprocessesinaformalizedalgorithm.The algorithmconsistsoffoursteps.Intherststep,theuserprovisionallysetsthenumberofreductionstages.In thesecondstep,gearratiosofeverystagearechosenbyusingtherandomsearchmethodwithinthespecied ratiorange,andtheratiosareusedasthebasicinputforthedimensionaldesignofgearsinthenextstep.Inthe thirdstep,thevaluesofthebasicdesignparametersofageararechosenbyusingthegenerateandtestmethod. Thenthevaluesofotherdesignparameters,suchaspitchdiameterandouterdiameter,arecalculatedforthe congurationdesigninthenalstep.Inthenalstep,thecongurationdesigniscarriedoutbyusingthe simulatedannealingalgorithm.Thepositionsofgearsandshaftsaredeterminedtominimizethegeometrical volumeofagearboxwhilesatisfyingspatialconstraints.Thesestepsarecarriedoutiterativelyuntilade- sirablesolutionisacquired. C211 2002ElsevierScienceLtd.Allrightsreserved. Keywords:Gear;Multi-stagegeardrives;Dimensionaldesign;Congurationdesign;Simulatedannealing;Generalized newdesignalgorithm 1. Introduction Until now, research on the design of gear drives has focused on the dimensional design of single-stagegeardrives.Inrecentyears,however,theneedfordesigningmulti-stagegeardrives hasbeenincreasedwithmoreapplicationsofthegeardrivesinhighspeedandsmallspace.A MechanismandMachineTheory37(2002)295310 * Correspondingauthor.Fax:+82-2-2296-4799. E-mailaddress:thchonghanyang.ac.kr(T.H.Chong). 0094-114X/02/$-seefrontmatter C211 2002ElsevierScienceLtd.Allrightsreserved. PII:S0094-114X(01)00078-7 numberofcomplicatedproblemsshouldbeconsideredinthedesignofmulti-stagegeardrives, which do not arise in the design of single-stage gear drives. Firstly, the number of reduction (increasing)stagesandthegearratioofeachstageshouldbedeterminedproperlyinconsideration of total gear ratio, available space, and other design requirements. No denite rule has been proposedtodeterminethem.Secondly,theproblemofconguringgeardriveelements(gears, shafts,bearings,.)intoavailablespaceisalsooneofthemajordesignproblems.Pinionsand gearsshouldmeshproperlywitheachotherwhilesatisfyingspatialconstraintsbetweengears,and betweenothermachineelements.Thevolumeorweightofagearboxisalsoconsiderablyaected bythecongurationandthearrangement.Thereareseveralconventionalmethodstoestimate gearsizesrecommendedbygearstandardsorganizationsorresearchers.However,thesemethods donottakethecongurationandthearrangementofthegeardriveelementsintoconsideration, althoughthedimensionsofthemaredirectlyaectedbytheconguration.Thedesignershould havesolvedtheaboveproblemsonlybyusingthetrialanderrormethod,largelydependedonhis intuitionalsense.Thus,thedesignpracticesaretime-consumingevenforexpertdesigners,and oftenendupwithunsatisfactorydesignsolutions. Theobjectiveofthispaperistodevelopanewandgeneralizeddesignmethodologyformulti- stagecylindricalgeardrives(spurandhelicalgeardrives).Theproposeddesignalgorithmsup- portsadesignereectivelyatthepreliminarydesign phaseby integratingandautomatingthe dimensionalandthecongurationdesignprocesses. Thealgorithmconsistsoffoursteps.Intherststep,thedesignerprovisionallysetsthenumber of reduction stages in consideration of total gear ratio and other design requirements. In the secondstep,thegearratiosofeverystagearedeterminedbyusingtherandomsearchmethod,and theratiosareusedasbasicinputforthedimensionaldesignofgearsinthethirdstep.Inthethird step,thethreebasicdesignparametersofmodule,numberofteeth,andfacewidtharedetermined byusingthegenerateandtestmethod.Inthepreliminarydesignphase,itispossibletoconsider onlythethreebasicdesignparameters,whichhavedominanteectsontheoverallsizeofagear, andconsequentlyonthecongurationdesign.Otherdesignparameters,suchaspressureangle, helixangle,addendummodicationcoecient,causerelativelysmallchangeintheoverallsizeof agear,andaregenerallydeterminedlaterinthedetaildesignphaseofageardesignprocess. Strengthanddurabilityofthedesignedgearisguaranteedbybendingstrengthandpittingre- sistanceratingpractices.Inthenalstep,thepositionsofthegearsaredeterminedtominimizethe geometricalvolume(size)ofagearboxbyusingthesimulatedannealingalgorithm,whilemeshing properlybetweenpinionsandgears,andavoidinginterferencesbetweengearsandshafts.The abovefourstepsarecarriedoutiterativelyuntiladesirabledesignsolutionisacquired. Thealgorithmautomatesthepreliminarydesignofmulti-stagegeardrivesbyecientlyinte- grating the dimensional design and the conguration design processes. The availability of the algorithmwillbevalidatedbydesignexamplesoffour-stagegeardrives. 2. The proposed design algorithm Fig.1showstheproposedalgorithmforautomatingthepreliminarydesignphaseofmulti- stagegeardrives.Asmentionedearlier,thealgorithmconsistsoffourdesignsteps,andthesteps arecarriedoutiterativelyuntiladesirablesolutionisacquired. 296 T.H.Chongetal./MechanismandMachineTheory37(2002)295310 InStep1,thedesignerprovisionallysetsthenumberofreductionstagesinconsiderationof totalgearratio,availablespace,andotherdesignspecications.Severalsimpleguideshavebeen proposedtodeterminethenumberofstages.Ingeardesigntexts,itisrecommendedtohandle gearratiosfrom1:1to8:1(or10:1)inasinglereductionforordinaryspurandhelicaldesign practices1.AGMArecommendsaddinganotherstagetothegeartrainifthegearratioofa stageisgreaterthan5:12.Thus,thedesignercanmakeasensiblechoiceforthenumberof reductionstagesfromtherecommendedratiorange.Whenthenaldesignsolutionisnotsat- isfactoryortheiterationexceedsthemaximumnumber,i.e.thedesignisregardedprovisionallyas havingnofeasiblesolution,thedesignercandecideoptionallywhetherornottoproceedwith anothernumberofreductionstages.Itisratherinecienttoautomatethisstepintothealgo- rithm,sincethenumberofstagescanbeselectedinarelativelysmallrange.Moreover,theau- tomationunnecessarilyincreasescomputationtimeinmostcases. In Step 2, the gear ratios of each reduction stage are determined using the random search methodwithinthespeciedratiorange.Nodeniterulehasbeenproposedtodeterminethegear ratios.TheguideproposedbyNiemannetal.3mightbeapracticalone,inwhichgearratiosare determinedbasedontheHertzcontactstressformula.However,thismethodislimitedtothe design of two- and three-stage gear drives, and the designer should previously determine the numberofteethormoduleinordertocalculatethegearratiosofeachstage4,5. We have proposed two types of premises in order to employ the random search method. Firstly,gearratioscanbelimitedtoareasonablerange.Asmentionedearlier,itisreasonable tohandlegearratiosfrom1:1to8:1inasinglereductioninordinaryspurandhelicaldesign Fig. 1.Flowchartforthedesignofmulti-stagegeardrives. T.H.Chongetal./MechanismandMachineTheory37(2002)295310 297 practices. Ratios of even 10:1 are possible 1. Thus, the upper and the lower limits of gear ratios can be set to generally acceptable values according to the above guides, although the denitevaluesofthemarenotknown.Secondly,itisgeneraltochooseagreatervalueforthe gearratiooftherstreductionstagethanthatofthesecondstage.Inthesameway,theratio ofthesecondreductionstageshouldhaveagreatervaluethanthatofthethirdstage,andso forth.Fromthesepremises,arandomvalueisgeneratedforthegearratiooftherstreduction stage u 1 between the lower and upper limits previously specied by the designer. The ratio range from 1:1 to 9:1 has been used for the rst reduction stage of the design examples in Section4(seeTable2).Then,thegearratioofthesecondreductionstage u 2 canbeselectedby settingthegearratiooftherststageasthenewupperlimit.Inotherwords,anotherrandom valueisgeneratedforthegearratioofthesecondstagebetweenthelowerlimitandthegear ratiooftherststagepreviouslydetermined.Thegearratiosforeveryreductionstage u i can be determined by the same way described above. Although the method randomly selects the gear ratios, the gear ratios of every stage shall eventuallyhavepropervalues.Thismaybevalidatedfromthefactthattherearedirectcorre- lationsamonggearratios,thedimensionsandthecongurationofgears,andthevolumeofa gearbox.Thatistosay,gearratiosaectthedimensionsofgears,andthedimensionsofgearsdo thecongurationofthem.Itisobviousthatthecongurationofgearshaveadirecteectonthe volumeofagearbox.ThisfactwillbeclearlyshownbythedesignexamplesinSection4. InStep3,basicdesignparameters(modulem,numberofteethz,andfacewidthb)ofgearsare determinedbyusingthegenerateandtestmethod.Thereareseveralconventionalmethodsto estimategearsizesrecommendedbygearstandardsorganizationsorresearchers.Forexample, AGMA2presentsacompleteguideforthepreliminarydesignprocessofspurandhelicalgears, andDudley6givesageneralwayofestimatinggearsizes.However,thesemethodsdonottake intoconsiderationofthecongurationandthearrangementofthegeardriveelements,although thecongurationofthegearsdirectlyaectsthedimensionsofthem.Onthecontrary,thepro- posedalgorithmintegratesthecongurationandthedimensionaldesignofgearstoconsiderthe relationbetweenthem. Oncethevaluesforthebasicdesignparametersaredetermined,pitchdiameterandouterdi- ameterarecalculatedfrommoduleandnumberofteethforthecongurationdesign.Sincethe purposeofthispaperistoautomatethepreliminaryphaseofthegeardesignprocess,thede- terminationofotherdesignvariables,suchaspressureangle,helixangle,andaddendummodi- cationcoecienthasnotbeenconsidered.Thesedesignvariablescauserelativelysmallchangein the overall size of a gear and are generally determined in the detail design phase. Thus, the variableshavexedvaluesinthedesignprocess.Thisisoneofthekeypointsofenablingtheuse ofthegenerateandtestmethodforthedimensionaldesign,althoughtheeciencyofthemethod isnotgoodinmostcases.Anotherkeypointisthatthesearchtimeofthemethodcanbereduced considerablybylimitingthesearchspaceofthedesignvariables.Firstly,itisrecommendedby standardsorganizations7tousestandardvaluesformodule,andthusitmaybetreatedasa discretevariable.Moreover,theupperandthelowerlimitsofitcanbegivenaccordingtothe applicationofthegeardrive.Secondly,thenumberofteethisobviouslyanintegervariable.The minimumvalueofthenumberofteethinpinioncanbespeciedaccordingtopressureangle,and themaximumvalueofitcanbelimitedtoaconventionalvalue.Finally,supposingthattheface widthisspeciedbyanintegermultipleofmodule(facewidthfactor),asisincommonpractice,it 298 T.H.Chongetal./MechanismandMachineTheory37(2002)295310 canbetreatedasadiscretevariable.Itisalsopossibletospecifytheupperandthelowerlimitsof ittoconventionalvaluesinaccordancewiththeapplication. Oncethedesignvariablesaredetermined,thenstrengthratingpracticeiscarriedoutusingthe AGMAratingformulas8forbendingstrengthandpittingresistanceratingtotestthevalidityof thedimensionaldesignsolution.Ifthegeardoesnotsatisfytheratingpractice,thedesignrestarts withincreasingvaluesofthebasicdesignparametersfromStep2.Thus,thecongurationdesign inStep4iscarriedoutonlyforthegearssatisfyingstrengthanddurabilitycriteria. InStep4,thecongurationdesigniscarriedouttominimizethevolumeofagearboxbyusing thesimulatedannealingalgorithm.Sincetheouterdiameterandthefacewidthofagearhave beendeterminedfromthepreviousdesignsteps,althoughthevaluesareprovisional,thecon- guration design might be considered as a problem of packing gearsof xed size in three-di- mensionalspace.Therehavebeenseveralresearchestosolvethree-dimensionalpackingproblems using optimization techniques 911. In particular, Szykman and Cagan 9,10 have reported signicantlygoodresultsfortheoptimalpackingproblemsofthree-dimensionalelementsofxed Fig. 2.Flowchartofsimulatedannealingalgorithmforthecongurationdesignofageardrive. T.H.Chongetal./MechanismandMachineTheory37(2002)295310 299 sizeusingasimulatedannealingalgorithm.Theproblemofpackinggearsinathree-dimensional spaceisproblematictoconventionalgradient-basedoptimizationmethodsduetodiscontinuities andseverenonlinearitiesinitsobjectivefunctionspace.Simulatedannealingiswellsuitedtothe problem,becauseitiszero-orderalgorithmrequiringnoderivativeinformation,andthusdis- continuitiescanbeeasilydealtwith12,13. Fig. 2 shows the owchart of the simulated annealing algorithm used in this paper for the congurationdesignofageardrive.Startingfromaninitialrandompoint,thealgorithmtakesa stepandthefunctionisevaluated.Whenminimizingafunction,anydownhillstepisacceptedand theprocessrepeatsfromthisnewpoint.Anuphillstepmaybeaccepted.Thus,itcanescapefrom thelocaloptima.Thisuphilldecisionismadeby theMetropoliscriteria. Astheoptimization process proceeds, the length of the step declines and the algorithm closes in on the global optimum. 3. Objective function formulation for conguration design TheobjectivefunctionFforthecongurationdesignisformulatedsimplyasthelinearsum- mationofthevolumeofavirtualgearbox,i.e.aboxcompletelyboundingthegears,andthe spatialconstraints,asshowninEq.(1) F W box P box V box X nc i1 W i P i C i jj; 1 where W box , P box arethe weighting factor and thenormalizing factor for the volume V box of a virtualgearbox,respectively. W i and P i aretheweightingfactorsandthenormalizingfactorsfor ithconstraint,C i .Thetotalnumberofconstraintsisnc.ThevaluesofnormalizingfactorsP box and P i areonedividedbythemaximumvaluesofV box andC i atthecurrentposition,respectively.The spatial constraints C i should be satised to congure the gear drive elements properly. The constraintsconsistoffourtypesofspatialconstraints;thecenterdistanceconstraintsforproper meshingofpinionandgear,thefacedistanceconstraintsformatingofco-axisgears,thegear interferenceconstraintstoavoidtheinterferencebetweengears,andtheshaftinterferencecon- straintstoavoidtheinterferencebetweengearandshaft.AstheobjectivefunctionFminimizes, thevaluesoftheconstraintsapproachzero. InordertoconrmthevalidityofthecongurationdesignalgorithmusingEq.(1),thecon- gurationdesignofsixcylindershasbeencarriedout.Thecylindersconsistofthreecylinderswith thesamediameterof10mm,andthreecylindersof20mm.Theheightofeverycylinderis10mm. Thiscongurationisfortheanalogyofgearmeshingofathree-stagereductiongeardrive. Fig.3showstheoptimalcongurationsofthesixcylinders.Theglobaloptimalcongurationis in Fig. 3(a) with its bounding box having a volume of 24000mm 3 . Fig. 3(b) shows another possibleconguration,i.e.alocaloptimum,havingavolumeof24500mm 3 .Thisconguration alsomightberegardedasagooddesign,thoughitisnotaglobaloptimum.Theconstraintsto locateproperpositionsofcylindersareshowninEqs.(2)(13). C 1 d 1 d 2 =2C0 x 1 C0 x 2 2 y 1 C0 y 2 2 q 0; 2 300 T.H.Chongetal./MechanismandMachineTheory37(2002)295310 C 2 z 1 C0 z 2 0; 3 C 3 d 3 d 4 =2C0 x 3 C0 x 4 2 y 3 C0 y 4 2 q 0; 4 C 4 z 3 C0 z 4 0; 5 C 5 d 5 d 6 =2C0 x 5 C0 x 6 2 y 5 C0 y 6 2 q 0; 6 C 6 z 5 C0 z 6 0; 7 C 7 z 2 j C0 z 3 jC0 b 2 b 3 =20; 8 C 8 x 2 C0 x 3 0; 9 C 9 y 2 C0 y 3 0; 10 C 10 z 4 j C0 z 5 jC0 b 4 b 5 =20; 11 C 11 x 4 C0 x 5 0; 12 C 12 y 4 C0 y 5 0; 13 wherecoordinatesx;y;zrepresentthecenterpositionofacylinder.Thediameterandtheheight (facewidth)ofacylinderaredandb,respectively.Thesubscriptsinthevariablesrepresentthe numberofthecylindersinFig.3. Theaboveconstraintsmaybeclassiedintothecenterdistanceconstraintsandthefacedis- tanceconstraintsaccordingtothespatialrelationbetweencylinders.Theconstraints C 1 and C 2 representthecenterdistanceconstraintsforpropermeshingofcylindersintherststage.This might be regarded as an analogy of meshing of pinion and gear in a gear drive (see Fig. 4). Similarly,C 3 C 4 andC 5 C 6 representthecenterdistanceconstraintsforthesecondandthethird stage,respectively.Fromtheseequations,thegeneralrepresentationofthecenterdistancecon- straintscanbederivedasshowninEq.(14). Fig. 3.Optimalcongurationsofsixcylinders:(a)globaloptimalconguration;(b)alocaloptimalconguration. T.H.Chongetal./MechanismandMachineTheory37(2002)295310 301 d 2iC01 C0 d 2i C1 =2C0 x 2iC01 C0 x 2i C0C1 2 y 2iC01 C0 y 2i C0C1 2 q 0; z 2iC01 C0 z 2i 0 i 1;2; .;n; 14 wherenisthetotalnumberofstages. Theconstraints C 7 C 9 representthefacedistanceconstraintsforcylinders2and3inthexy planeandinthezdirection.Thismightberegardedasananalogyofmatingrelationofco-axis gearsinageardrive(seeFig.5).Similarly, C 10 C 12 representthefacedistanceconstraintsfor cylinders4,5andEq.(15)showsthegeneralrepresentationofthefacedistanceconstraints. z 2i C12 C12 C0 z 2i1 C12 C12 C0 b 2i C0 b 2i1 C1 =20; x 2i C0 x 2i1 0; y 2i C0 y 2i1 0 i 1;2; .;n C01: 15 Fig.6showsoneofthecongurationdesignresultsusingtheproposedformulation.Inthis case,thevaluesfortheweightingfactorswerechosentounitytoevaluatetheeciencyofthe algorithm. Final volume of the bounding box was 24962mm 3 , and the number of function Fig. 5.Schematicoffacedistanceconstraints. Fig. 4.Schematicofcenterdistanceconstraints. 302 T.H.Chongetal./MechanismandMachineTheory37(2002)295310 evaluationwas61201.Thecongurationresultwasconsiderablygoodcomparedtotheoptimal congurationsinFig.3.Thisindicatesthatthesimulatedannealingalgorithmwiththeobjective functionformulationofEq.(1)canbeusedtoeectivelyconguregears. Meanwhile,theconstraintsforthecongurationdesignofmulti-stagegeardrivesalsoinclude theinterferenceconstraintsbetweengearsandshafts.Forexample,theinterferenceconstraints forathree-stagegeardriveareshowninEqs.(16)(27).TheinequalitysignsofC 13 C 24 meanthat iftheinterferenceconstrainthasavaluesmallerthanzero,thenthereisnointerferenceandthe constraintisnotincludedintheobjectivefunction. C 13 d o1 d o5 =2C0 x 1 C0 x 5 2 y 1 C0 y 5 2 q 0; 16 C 14 d o1 d o6 =2C0 x 1 C0 x 6 2 y 1 C0 y 6 2 q 0; 17 C 15 d o2 d o5 =2C0 x 2 C0 x 5 2 y 2 C0 y 5 2 q 0; 18 C 16 d o2 d o6 =2C0 x 2 C0 x 6 2 y 2 C0 y 6 2 q 0; 19 C 17 d s1 d o4 =2C0 x s1 C0 x 4 2 y s1 C0 y 4 2 q 0; 20 C 18 d s1 d o6 =2C0 x s1 C0 x 6 2 y s1 C0 y 6 2 q 0; 21 C 19 d s2 d o6 =2C0 x s2 C0 x 6 2 y s2 C0 y 6 2 q 0; 22 C 20 d s3 d o1 =2C0 x s3 C0 x 1 2 y s3 C0 y 1 2 q 0; 23 C 21 d s3 d o2 =2C0 x s3 C0 x 2 2 y s3 C0 y 2 2 q 0; 24 C 22 d s4 d o1 =2C0 x s4 C0 x 1 2 y s4 C0 y 1 2 q 0; 25 C 23 d s4 d o2 =2C0 x s4 C0 x 2 2 y s4 C0 y 2 2 q 0; 26 Fig. 6.Congurationresultofthesixcylinders:(a)two-dimensionalrepresentation;(b)three-dimensionalrepresen- tation. T.H.Chongetal./MechanismandMachineTheory37(2002)295310 303 C 24 d s4 d o4 =2C0 x s4 C0 x 4 2 y s4 C0 y 4 2 q 0; 27 wherecoordinates x;y;z and x s ;y s ;z s representthecenterpositionsofagearandashaft,re- spectively.Thediametersd o andd s representtheouterdiameterofagearandashaft,respectively. Theconstraints C 13 C 16 areincludedtoavoidinterferencesbetweenthegears.Forexample, C 14 meansthatthedistancebetweengear2andgear5mustbelargeenoughtoavoidtheinterference betweentheouterdiametersofthem.ThereisnointerferenceinthecongurationofFig.7,butin somecase,gear5canbeplacedonthebottomofgear4inthezdirection.Inthiscase,thereisa possibleinterferencebetweengear2andgear5(seeFig.8).Inaddition,theinterferencebetween gear1andgear5possiblyoccurs,ifgear1islocatedontheright-handsideofgear2inthex directionC 13 .Thegearinterferenceconstraintscanberepresentedinthegeneralformulationas showninEq.(28). d o2iC01 C0 d o2jC01 C1 =2C0 x 2iC01 C0 x 2jC01 C0C1 2 y 2iC01 C0 y 2jC01 C0C1 2 q 0; d o2iC01 C0 d o2j C1 =2C0 x 2iC01 C0 x 2j C0C1 2 y 2iC01 C0 y 2j C0C1 2 q 0; d o2i C0 d o2jC01 C1 =2C0 x 2i C0 x 2jC01 C0C1 2 y 2i C0 y 2jC01 C0C1 2 q 0; d o2i C0 d o2j C1 =2C0 x 2i C0 x 2j C0C1 2 y 2i C0 y 2j C0C1 2 q 0 i 1;2; .;n C01; j i 2;i 3; .;n: 28 Fig. 7.Acongurationofathree-stagegeardrive. Fig. 8.Interferencebetweengear2andgear5. 304 T.H.Chongetal./MechanismandMachineTheory37(2002)295310 Similarly, C 17 C 24 representtheinterferenceconstraintsbetweenthegearsandtheshafts,and the meaning of the constraints are identical to those of C 13 C 16 . Eq. (29) shows the general representationoftheshaftinterferenceconstraints. d si C0 d o2j C1 =2C0 x si C0 x 2j C0C1 2 y i C0 y 2j C0C1 2 q 0 i 1;2; .;ns C02; j i 1;i 2; .;