BirgittaLanttoPetterKrusJan-OvePalmbergDivisionofFluidPowerTechnology,DepartmentofMechanicalEngineering,LinkdpingUniversity,S-58183Linkoping,SwedenDynamicPropertiesofLoad-SensingSystemsWithInteractingComplexMechanicalLoadsAload-sensingfluidpowersystemisafeedbacksystemwithseveraltypesofinstabilitymodes.Thispaperdealswithoneofthem,whereloadinteractionthroughamechanicalstructureandthefluidpowersystemmaycauseinstability,e.g.inalorrycrane.Criteriaforinstabilityareevaluated.Thispaperalsoproposessomemethodshowtoavoidthistypeofinstability.IntroductionThehigh-energysavingpotentialofload-sensingsystemsisaspecialadvantage,especiallywhentheconfigurationincludesavariablepump.Controllabilityofsuchsystemsmaybeenhancedbypressure-compensatedcontrolvalvessinceinteractionbetweentheactivatedloadscanbeavoided(seeFig.1).However,thistypeofsystemisafeedbacksystemand,unfortunately,alsoextremelyundamped.Therefore,itisnotunlikelythatinstabilitytakesplaceinsuchsystems.Generally,themostriskyfeedbackisthepumppressureandhighestloadpressurefeedbackscontrollingthepumpregulatorwhichcouldleadtotheso-calledpumpandpump-loadinstabilities,seeKrus(1989).Thereisalsoariskofinstabilityattheloadifafeedbackcomponentsuchasanover-centrevalvecontrolstheactuator.AthirdtypeofinstabilityisdiscussedinLantto(1992).Whenseveralactuatorscontrolthesamemechanicalstructure,e.g.acranearm,theyinteractthroughthestructureandthefluidpowersystem.Then,afeedbackcouldbecreatedasinFig.2whereamovementoftheactuatorwiththelowestloadchangesthehighestloadpressureandthereforetheload-sensingpumppressurewhichchangestheflowtothelowestload.Suchdestabilizinginteractionhasearlierbeenanalyzedbye.g.Rama-chandran(1982)andPannala(1985).Thispapergivesadeeperdiscussionaboutthisthirdtypeofinstabilityrisk,mainlyfoundinload-sensingsystems.ThestabilityanalysisinthispaperisbasedonKrus(1988),Krus(1989),Palmbergetal.(1985),andLantto(1992)andconcernsmainlysystemswithnoncompensatedcontrolvalves.Onlysystemswithanunsaturatedload-sensingpumparediscussedhere.Moreover,onlypositivemassandinertialoadsareconsidered.InstabilityBecauseofLoadInteractionThroughtheMechanicalStructureInthissection,wewillshowhowinstabilitycanbeevaluatedforastructurecontrolledbyanidealload-sensingsystem.ThisContributedbytheDynamicSystemsandControlDivisionforpublicationintheJOURNALOFDYNAMICSYSTEMS,MEASUREMENT,ANDCONTROL.ManuscriptreceivedbytheDynamicSystemsandControlDivisionMarch25,1991；revisedmanuscriptreceivedJuly1992.AssociateTechnicalEditor:A.Akers.sectionwillalsodiscusswhenanormallorrycranereachesinstability.Sincethistypeofinstabilityisdependentonboththefluidpowersystemandthegeometryofthemechanicalstructure,thedesignofthestructureandthefluidpowersystemwillbediscussed.Theanalysiswillshowthatthisinstabilitygenerallymayoccurif1)apositive(ornegative)movementofthelowestloadwillcauseanegative(apositive,respectively)movementofthehighestloadthroughthemechanicalstructureand2)ifafeedbackcomponentcontrolsthesystemsothatanincreaseofthehighestloadpressurewillcauseanincreaseoftheloadflowtothelowestloadthroughthefluidpowersystem.InstabilityinaLoad-SensingSystem.ThesimplestructureinFig.3,controlledbytwocylindersinaload-sensingsystemwithnoncompensatedcontrolvalves,hasthefollowingequa-Fig.1Load-sensingsystemincludingcontrolvalveswithconventionalpressurecompensatorspools/.vHighestloadLoadflowfrLowestload4Loadsensingpressure4Fig.2FeedbackthroughthemechanicalstructureJournalofDynamicSystems,Measurement,andControlSEPTEMBER1993,Vol.115/525Copyright1993byASMEDownloaded22Mar2009to87.RedistributionsubjecttoASMElicenseorcopyright；see/terms/Terms_Use.cfmFig.3Load-sensingsystemwithloadinteractionthroughasimplifiedmechanicalstructuretionsofmotion,describingthesmallmovementsofthetwopistonsinthefrequencydomain:s2/AXLjAXL2,00B2AXi2,ALlJLl-AolUPolAL2APL2-A02AP0%(1)TheyformthetransferfunctionsGmAandGh2ofthemechanicalloadinAppendixwhereM=ml+m2,Bi=bi,M2=m2andB2=b2,whileMfi1=MKi2=m2formsthetransferfunctionsofthemechanicalcouplingasG,=GK2=-AL1AL2m2sALxALlm2s(2)(3)TheseequationsandtheAppendixformtheblockdiagraminFig.4whichdescribesageneralload-sensingsystemwithloadinteractionthroughthemechanicalstructure(blocksGKilandGKi2)andthefluidpowersystem.rf%FKEFig.4Blockdiagramofaload-sensingsystemwithloadinteractionThenextstepistoanalyzethisfeedbacksystembyreducingtheblockdiagramintoclosed-looptransferfunctionswheretheoutputsignalsAXLiandAXL,2arefunctionsofAXViiandAA,seeLantto(1992).Then,thecharacteristicequationisfoundas1H,iiG,ni-+1+G；GmlHmGn+FLSCGmxGvHL2Gmirt+C/j.2j/-pumpGm2G+FLSGn1HLGKj1HL2GK1-+G+G.-Hm-HmGPl+FLsGpGmiGGmiHmH,l+Hsp(Gn+G2+Gp)=0(4)(5)AB,bCdPDLF(s)G(s)H(s)JkKc=area,m2=viscousdampingcoefficientofactuator,Ns/m=volumecapacitance(=V7/3e),m3/Pa=volumetricgradientofpumpdisplacement,mVrev=motor(load)displacement,mVrev=filtertransferfunction=transferfunctionwhereoutputisnormallyflowwhileinputispressure.Theflowisoutputandvalvedisplacementinputwhenthesubscriptisx.=transferfunctionwhereoutputispressurewhileinputisflow.=inertia,kgm=springcoefficient,N/m=flow-pressurecoeff.oforifice(=dq/d(pm-POM),m3/(sPa)KqL,lM,mnPqsVXHe6APe01=flowgainofvalveorifice(=3#/dxvalve),m2/s=length,m=mass,kg=pumpspeed,rev/s=pressure,Pa=flow,m3/s=Laplacetransformoperator,rad/s=volume,m3=displacement,m=effectivebulkmodulus,Pa=dampingratio=smallvariationinalinearizedvariable.=density,kg/m3=angle,rad=frequency,subscriptindicatesabreakfrequency,rad/s.CapitalletterofavariablemayindicateaLaplace-transformedvariable.SubscriptLLSm0PregTVK12J=meter-insideofcylinderload-sensinglinemechanicalloadmeter-outsideofcylinderpumppumpregulatorpump(supply)volumetankcontrolvalveorvalvepackageloadwhichcouplestheactuatorsthroughthestructurethehighestloadthesecondhighestloadshortnoteinsubscript,e.g.,&L,Imeansu526/Vol.115,SEPTEMBER1993TransactionsoftheASMEDownloaded22Mar2009to87.RedistributionsubjecttoASMElicenseorcopyright；see/terms/Terms_Use.cfmAfirstglanceatthisequationusingthefollowingsimplifications:9Aninfinitelyfastpumppressurecontrolwithoutleakagemodelledasaninductance,Gp=l/(LpS)oowhichleadsto#pumP=1/Gp,andconsequentlyAPsAPLii.9Noorificeintheload-sensingline,FLS1.8Constantpressureonthemeter-outsideofthepistons,AP0il=0otAoA=0leadingtoG,AGm1respectivelyAP0i20orAOi20leadingtoG,2=Gm,2.8OnlymassloadswithoutspringforcesandwiththeonlyviscousfrictionappearinginthecylindersareanalyzedwhichleadtoGKAALAALa/(MK%2s)andGK%1=ALAAL1/MKAS).8NodynamicsinthecontrolvalvesleadingtoGAjjG.jsandKCiVy2,respectively.9B2Kc,v,iA2Ltlofthelowestload.givesthefollowingcharacteristicequationwhere=ALAAL,2/(MKACLA)and4,i=ALAALa/(MK:2CL,2).2+-5+12B.Assumethat:+(8)25ilL,KcBx!j-1Then,uAandwfiwillbeachievedfromEqs.(6)and(7)as(9)uA2(10),/WiV1+vw2.,2-.2,wawi2If,/WiVV1-/1W(i(j)iilcoflandcouii2WB.Equations(6)and(7)givethedampingratios8,4andbB.Negativedampingratiosshowoninstability.Here,itispossibletoshowthat5BwillbepositivesincethemassmatrixinEq.(1)mustbepositivesemidefiniteorMKlMK2MlM2(12)motorspeed9L(rad/s)15motorspeed0.(rad/s)ISm45cm3/rovTime(s)10tA/(2ic)*1.52Hz&J(SK)-10.0Hz5.=-0.01-55cm/iw5Time(s)10oiA/(2*).1.70Hzeg/fa)-11.0HzS.-0.005Fig.5Timesimulationsofaload-sensingsystemwithtwointeracting,fixeddisplacementmotorswiththedisplacementsDti1(highestload)andDL2whenDt|1DL2.Thehighestloadstartsat0s,theotherat5sec.KCnM2-MKlALli-40LhWA2iA12UBKCnM,ALl-A72M-2-21!dl2WB(13)TheequationshowsonstabilityonlywhenM212L22ALlAL2BiCLjKc2UA20LXB2CL2Kr.2A2(14)Assumingnoviscousfrictioninthecylinder,B2=0,andnt=0inthesimplifiedcaseinFig.3,whereMKA=M2m2,leadsustothefollowingcrudecriterionsincewAwLAandoBoo:Ifinstabilityshalloccurinthesystem,thepistonareaAL,2ofthelowestloadmustbelargerthanthepistonareaALAofthehighestload.ThismeansthatforthemechanicalstructureinFig.3,instabilitywillalwaysoccursincetheactuatorwiththehighestpressurehasthesmallestpistonarea,seeFig.5.Inreality,thismaynotalwayshappensincethesimplificationsprecedingEq.(6)normallyarenotfulfilledwhichincreasesthedampingandconsequentlythestabilitymarginofthesystem.DesignAspectsofAvoidingInstability.Thisinstabilitytypehasitsoriginineffectivenegativedampingratioofthelowestloadinthetwo-loadsituation.Toincreasethedampingratiooftheactuatorisconsequentlystabilizinge.g.withthemeter-outorifice.Asmentionedearlier,thefeedbacknormallypassesbetweentheactuatorsthroughthemechanicalstructure,butalsothroughtheload-sensinglinetothepumpandpumppressurevolume.Thebestwaytoreducetheinstabilityriskshouldthereforebetodesignthemechanicalstructureproperly.Thisrequiresthatalargemassorinertiaofthedynamiccouplingfromthelowestloadtothehighest,MKUmustbeavoided.Howtocalculatethismassandothersisgivenasanexampleinthenextchapterforalorrycranearm.AflexiblestructureJournalofDynamicSystems,Measurement,andControlSEPTEMBER1993,Vol.115/527Downloaded22Mar2009to87.RedistributionsubjecttoASMElicenseorcopyright；see/terms/Terms_Use.cfmFig.6ModelofalorrycranestructureTotallengthofcylinder1(m)(highesthad)maxTotallengthofcylinder2(m)HighriskFig.7Variousvaluesof2bAlvAAj(mKcv2)asafunctionofthepistondisplacementsofthelorrycrane.Negativevaluesindicatedestabiliza-tionoftheoscillations,whilepositivevaluesindicatestabilization.isalsostabilizingcomparedwithastiffone.Thefeedbacksignalthroughaload-sensingpumpisalsostoppedbyalowpassfilter,thatisanorifice,intheload-sensinglinesinceithastopassthisline.Italsoseemsasifafastpumppressurecontrolintheload-sensingsystemmayeasetheinteractionbetweentheactuatorsthroughthefluidpowersystemandincreasetheriskofinstability.Topressure-compensatethecontrolvalveofthelowestload(s)intheload-sensingsystemreducestheinstabilityrisksinceKC:2isclosetozeroofavalvepackagewithafastandidealpressurecompensatorspoolsuchasinFig.1.AnExample:InstabilityofaLorryCraneArm.Will2hA/wAinEq.(13)bepositiveforthelorrycranestructureinFig.6?Aroughcriterionofacranearmisachievedifthefollowingassumptionsaremade:ThestructureisstiffThemassofthestructureislumpedintotheloadmassm9TheboomisthehighestloadOnlypositiveloadsarediscussed,92-90deg.TheLagrangeequationdescribesthemovementsofthestructure.TisthekineticenergyandMejthetorquefortheangle0-,.d(dfdT,jtw-wMe,=1-2m(kl(eue2)+y2m(eue2)(15)T=-Llcos(dl)+L2cos(62)jLisinO+Zsin)(16)(17)Time(sec)Time(sec)Fig.8Measurementsonthelorrycrane.Thecranepositionintheleftdiagramgaveinstability.M0Mr,ALiAPLi-A0lAPol-BlSAXLlh)AL2APL2-A02AP02-B2sAXL2(18)Foracranestructure,the0,-0,-terms(ij)canbeneglectedcomparedtotheaccelerationterms(i=j).Then,thefollowingdynamicequationsofmotionyieldinthefrequencydomain:Z.,L2cos(0,-02)/,-h0l2iAX,LL2cos(di-d2)LA0,A62iA0lAP0l-BtsAXLlAL2APLl-A02AP02-B2sAXL2ALlAPLliiAX,i-2,/,0-hhA0,A0-(19)(20)Equations(18)and(19)canbereducedtothesameformasinEq.(1)whereM=mLi+Ll+2LlL2cos(ei-d2)M2-mMKX=MK2=mL1L2cos(61-62)+LJ(21)(22)(23)Todescribetheinstabilityrisk,onecontourplotof2bA/uAAi2/(mKCyV2)havebeenmade,implementedonafull-sizelorrycrane,HIAB070fromHIAB-Foco.Thisplot,inFig.7,hasbeendrawnfora