pneumatic drives system design modelling and control

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PneumaticDrives PeterBeaterPneumaticDrivesSystemDesign,ModellingandControlWith244Figuresand14Tables123 Prof.Dr.-Ing.PeterBeaterFachhochschuleSüdwestfalenFachbereichMaschinenbau-AutomatisierungstechnikLübeckerRing259494SoestGermanyLibraryofCongressControlNumber:2006939785ISBN-103-540-69470-6SpringerBerlinHeidelbergNewYorkISBN-13978-3-540-69470-0SpringerBerlinHeidelbergNewYorkThisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broad-casting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.DuplicationofthisofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.ViolationsareliableforprosecutionundertheGermanCopyrightLaw.SpringerisapartofSpringerScience+BusinessMedia©Springer-VerlagBerlinHeidelberg2007springer.comTheuseofgeneraldescriptivenames,trademarks,etc.inthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.Coverdesign:eStudioCalamar,SteinenTypesettingbyauthorandSPiPrintedonacid-freepaperSPIN:1174555662/3100/SPi543210 PrefaceTheideatouseairfortransmittingpowerisveryold.Ctesibiusinan-cientGreecedescribedacatapultusingpneumaticcylinderstofirststoreenergyandthenrapidlyaccelerateanarrow.HeronofAlexandriadevel-opedautomatictempledoorswhichopenedandclosedbymeansofhotair.AndfromtheGreekwordforbreathhecoinedthetermthatwasusedastitleforhisbookandtodaydescribesawholeindustry:ʌȞİȣµĮτȚțȩȢ-pneumatics.Pneumaticcomponentsandsystemshavebecomeanimportanttopicfortextbooks.Mosthavetheirfocusonthedescriptionofthesteady-statebe-haviour,practicalproblemsliketroubleshootingorBooleanalgebratohelpdesigningcontrolalgorithms.Onlyafewtextbookscoveringthetheoreti-calanalysisanddesignofpneumaticsystemshavebeenpublished(Zal-manzonetal.1965;Andersen1967;Anderssonetal.1975).Buttheywerewrittenatatimewhendigitalcomputerswerenoteasilyavailabletoengi-neersandthereforecontainfewmaterialaboutmodellingandsimulation.Thisbooktriestobridgethegapbetweenscientificdisciplines(fluidmechanics,thermodynamics,mathematics,control,etc.),theconventionalapproachtodescribepneumaticcomponentsandsystemsbytheirsteady-statebehaviour,thewishofadesignengineertotesthisdesignbeforeac-tuallybuildinghardwareandtheresultingneedformathematicalmodelsinordertousetoday’spowerfuldigitalcomputers.Thebookcoversfirstthebasiclawsofnatureandthenthedesignandmodesofoperationofpneumaticcomponents,includingequationstomodeltheirstaticanddynamicresponse.Inthethirdpartofthebooksys-temsaredescribed:binarymodecylinderdrives,positioncontrolleddrivesandcomputeraidedanalysisofcomplexsystems.Wheneverapplicable,thisbookcontainsequationsthatcandirectbeusedfortheanalysisanddesignofdrives.Butinanumberofcasestheap-proachesofdifferentmanufacturersvaryconsiderablysuchthatnouniquemathematicalmodelcanbegiven.Anexampleisthecalculationofthepermissiblestrokelengthofacylinderunderaxialcompressiveload.Thisbookwillbeusefultoengineersandscientistswhowanttounder-standthedynamiceffectsoccurringinapneumaticcircuitinordertode- VIPrefacesignanoptimumsystem.Thetoolofchoiceisadigitalcomputerwithapackagefortime-domainsimulationusingthemodelspresented.Thisbookcontainsanumberofmeasurementstoillustrateandvalidatethetheory.MostofthemwerecarriedoutinthecontrollabinSoestwithgreatcare.However,theirqualityshouldnotbecomparedwiththosefromspecialisedphysicslabswhoseequipmentisqualifiedforscientificre-searchwhileintheSoestlabweusedevicestypicallyfoundinproductionmachines.Thisbookisnottheworkofasingleperson,butmanypeoplehelpedme.Firstofall,mystudentswiththeresultsoftheirdiplomaormasterthe-sisesortheirworkaslaboratoryassistant.Theybuilttest-rigs,madethe3Ddrawingsortooktime-consumingmeasurementsandarelistedinal-phabeticalorder:RalfBergmann,EvaBrückner,YannDécaillet,DanielDiers,ThomasGrosserüschkamp,RolandHenze,KaiHansmeier,IlonaHuppert,PeterIles,OliverJürgens,JoachimLütticke,NormaAliciaMon-tealegreAgramont,MichaelOtte,AnsgarPäschke,CorneliusSchaffranek,DavidSchlüter,ThomasSchulze-Rudolphi,TorstenVolmerandMichaelVoß.IamparticularlythankfultoIlonaHuppertforherhelpinproof-readingthewholemanuscript.IamindebtedtoseveralpeopleattheCampusSoestoftheFach-hochschuleSüdwestfalenfortheirsupport.Firstofall,tomylaboratoryengineerHans-JoachimRatajczak,whosolvedeveryelectricalormeas-urementproblem,andtoAndreasHülsbeck,whodidthesameontheme-chanicalside.IamgratefultoDr.M.Göttert,FestoAGEsslingen,andProf.G.Bel-forte,PolitecnicoTorino,fortheirfeedbackonanearlydraftofthisbook.Last,butcertainlynotleast,Iwanttothankmywifeforherpatienceandencouragementduringthelongtimethisbooktookmetofinish.November2006PeterBeater Contents1Introduction.............................................................................................12PropertiesofCompressedAir...............................................................52.1MathematicalModelofAir..............................................................62.2AtmosphericAir...............................................................................82.3DefinitionsRelatedtoCompressedAir............................................93ThermodynamicProcesses...................................................................113.1ConstantVolumeProcesses............................................................113.2ConstantPressureProcesses...........................................................133.3ConstantTemperatureProcesses....................................................183.4ReversibleProcesseswithoutHeatTransfer..................................183.5PolytropicProcesses.......................................................................203.6GeneralProcesses...........................................................................223.7SonicVelocity................................................................................234SomeResultsfromFluidMechanics...................................................254.1Viscosity.........................................................................................264.2ContinuityEquation........................................................................274.3FreeDischargefromNozzles.........................................................284.4OrificeFlow....................................................................................324.4.1IncompressibleFlow...............................................................324.3.2CompressibleFlow..................................................................344.5FrictionalFlow...............................................................................365EngineeringFlowRateCalculations...................................................415.1MathematicalFlowRateModel.....................................................415.2FlowRateCharacteristicsofRestrictions.......................................485.3SimplifiedFlowCalculations.........................................................495.4FlowCapacitySpecificationsinDataSheets.................................50 VIIIContents6ModellingofLongLines......................................................................556.1Steady-StateLossesofLongLines.................................................556.1.1FluidMechanicsModel...........................................................576.1.2EmpiricalModels....................................................................586.1.3TestResults.............................................................................596.2Steady-StateLossesofFittings.......................................................626.3TimeDomainModels.....................................................................656.3.1DerivationofTimeDomainModel.........................................656.3.2TestResultsintheTimeDomain............................................696.4FrequencyDomainModels............................................................767Electro-MechanicalConverters...........................................................817.1Solenoids........................................................................................817.1.1SwitchingSolenoids................................................................837.1.2ProportionalSolenoids............................................................857.1.3Pulse-WidthModulation.........................................................867.2VoiceCoilandPlungerTypeSystems...........................................937.3PiezoelectricActuators...................................................................947.3.1.StackTranslators...................................................................947.3.2.Benders...................................................................................947.3.3PiezoelectricElementsinPneumaticValves...........................958Cylinders................................................................................................998.1StrokeCushioning........................................................................1028.2MathematicalModel.....................................................................1128.3CylinderParameters.....................................................................1168.3.1SealFriction...........................................................................1168.3.2CylinderLeakage...................................................................1228.3.3CoefficientofHeatTransfer..................................................1239Non-StandardLinearActuators........................................................1279.1Multi-PositionandTandemCylinders..........................................1279.2RodlessCylinders.........................................................................1309.2.1Split-SealorSlotType..........................................................1309.2.2CableType.............................................................................1329.2.3MagneticType.......................................................................1329.3Bellows.........................................................................................1339.4Rolling-DiaphragmCylinders......................................................1379.6BrakeChambers...........................................................................1399.5MuscleActuators..........................................................................1409.6ImpactandKnockingCylinders...................................................142 ContentsIX10Semi-RotaryActuators.....................................................................14510.1CylinderBasedActuators...........................................................14510.2VaneTypeActuators..................................................................14811AirMotorsandAirTurbines..........................................................15111.1VaneMotors...............................................................................15311.1.1PrincipleofOperationofVaneMotors...............................15411.1.2MathematicalModel............................................................15611.1.3SpeedControl......................................................................16411.2AirTurbines................................................................................16812DirectionalControlValves...............................................................17112.1DesignofDirectionalControlValves........................................17312.2OperationofDirectionalControlValves....................................17512.3SimulationModelofDirectionalControlValves.......................18113Shut-OffValves.................................................................................18513.1Non-ReturnValves.....................................................................18513.2Non-ReturnValveswithOverride..............................................18813.3ShuttleValves.............................................................................18913.4TwinPressureValves.................................................................19013.5QuickExhaustValves................................................................19114PressureControlValves...................................................................19314.1SpringControlledPressureRegulators.......................................19314.1.1DesignofDirectActingValves...........................................19614.1.2SimulationModelofaPressureReducingValves..............19914.1.3Linearmodel........................................................................20214.1.4Non-LinearEffects..............................................................20314.1.5DesignofPilotOperatedValves.........................................20514.2ElectricallyOperatedPressureRegulators.................................20714.3PressureRegulatorswithClosed-LoopControl.........................20914.3.1ReportsaboutCommercialValves......................................21214.4PressureReliefValves................................................................21214.5Soft-StartValves.........................................................................21315FlowControlValves.........................................................................21515.1ThrottlingValve.........................................................................21515.2One-WayFlowControlValve....................................................21615.3DelayValve................................................................................21715.4AutomaticShut-OffValves........................................................218 XContents16ProportionalDirectionalControlValves........................................22116.1DesignofProportionalDirectionalControlValves...................22216.2OperationofProportionalDirectionalControlValves...............22416.3SimulationModelofProportionalControlValves.....................23016.4ReportsaboutExperimentalandCommercialValves................23217Stroke-TimeControl........................................................................23517.1CircuitsusingQuickExhaustValves.........................................23717.2Meter-OutControl......................................................................23917.3Meter-InControl.........................................................................24117.4CircuitsusingTwoPressures......................................................24217.5OilCushioning............................................................................24418PositionControlofPneumaticSystems..........................................24718.1MathematicalModelforControlSystemDesign.......................24918.2ModelofControlValves............................................................25018.3PressureDynamics.....................................................................25318.4EquationofMotion.....................................................................25618.5ControlLaws..............................................................................25818.5.1SingleLoopControllers.......................................................25918.5.2AdditionalLoops.................................................................26018.5.3StateFeedbackControl........................................................26018.5.4ReconstructionoftheVelocityandAccelerationSignal.....26318.5.5Non-LinearControlLaws....................................................26318.6PerformanceofaCommercialSystem.......................................26519ControlofActuatorsforProcessValves.........................................26919.1CharacteristicsofProcessControlSystems...............................27119.2Positioners..................................................................................27319.2.1PneumaticPositioners.........................................................27519.2.2AnalogueElectro-PneumaticPositioners............................27619.2.3DigitalPositioners...............................................................277 ContentsXI20DigitalSimulation.............................................................................28120.1ModellingApproaches...............................................................28220.2PrinciplesofObject-OrientedModelling...................................28620.3TheObject-OrientedModellingLanguageModelica.................28820.4FluidPowerLibrariesinModelica.............................................28920.4.1ExamplesofLibraryModels...............................................29020.4.2ComplexComponentModelofthePneumaticLibrary......29220.5LibrarySolutionforExample.....................................................29320.6Multi-DomainModels................................................................294References...............................................................................................297Index.......................................................................................................319 NomenclatureUnlessotherwisespecifiedinthetext,thefollowingsymbolsareusedcon-sistentlythroughoutthisbook:2Across-sectionalarea,mAsystemmatrixbpressureratiobinputvectorcspeedofsound,m/s3Csonicconductance,m/(s⋅Pa)Cddischargecoefficientfororificecpspecificheatcapacityatconstantpressure,J/(kg⋅K)cvspecificheatcapacityatconstantvolume,J/(kg⋅K)d,D(internal)diameter,mFforce,N2ggravitationalacceleration,m/s2hcoefficientofheattransfer,J/(m⋅K)Htotalenthalpy,Jielectriccurrent,ALlength,mmgasmass,kg•mmassflowrate,kg/sMtranslationallymovingmass,kgnpolytropicindexofexpansionorcompressionnrotationspeed,rpmppressure,absolutepressurePa,gaugepressurebarPpower,kWQheat,J3qvvolumeflowrate,m/sRgasconstant,J/(kg⋅K)ReReynoldsnumbersdisplacement,distance,mttime,sTtemperature,°CorK XIVNomenclatureuinputsignaltocontrolblockUinternalenergy,Jvvelocity,m/s3vspecificvolume,m/kg3Vtotalvolume,mWtotalwork,Jwflowvelocity,m/sxposition,mxstatevectorGreeklettersγratioofspecificheatcapacitesorindexofisentropicexpansionorcompression∆incrementζdampingcoefficientofasecondordersystemηefficiencyλfrictionfactorµdynamicviscosity,kg/(m⋅s)2νkinematicviscosity,m/s3ρdensity,kg/mτtimeconstant,sψflowfunctionωangularvelocity,rad/sSuperscripts.(overdot)quantityperunittimeLogarithmsarewrittenasLnaturallogarithm,lnx=Lmeaninge=xLcommonlogarithm,log10x=Lmeaning10=x 1IntroductionPneumaticsisthedisciplinethatdealswithmechanicalpropertiesofgasessuchaspressureanddensity,andappliestheprinciplestousecompressedgasasasourceofpowertosolveengineeringproblems.Themostwidelyusedcompressedgasisair,andthusitsusehasbecomesynonymouswiththetermpneumatics.Hydraulicsisthedisciplinethatdealswiththeme-chanicalpropertiesofliquids,andappliestheprinciplestosolveengi-neeringproblems.Gasesandliquidsarebothfluidsasopposedtosolids.Pneumaticsandhydraulicsaresimilarinmanyrespectsandoftende-scribedbythegenerictermfluidpower.Theuseofairasanenergytransfermediumcanbetracedbackmorethan2000years,andthevaryingareasofapplicationreflectthechangesintechnologysincethen.Theindustrialuseonalargerscalebegan1888whena1,500kWcentralcompressorstationwasinstalledinParistosup-plythecitywithcompressedair(Neermann1989).Withtheevolutionofelectricpower,thisformofenergytransferbecameobsolete,butthecom-petitionbetweenfluidpowersystemsandelectricsystemsisstillgoingon.Around1900themostoftenusedpneumaticcomponentswerepneu-matichammers,e.g.inshipyards.Astechnologyevolved,rivetinghasbeenreplacedbyweldingandpneumatichammersarenowusedmostlyonconstructionsites.Industrialapplicationofpneumaticsforautomationstartedaround1950whenthedemandforautomationinindustrialproduc-tionlinesincreasedandsuitableelastomericmaterialsforvalveandpistonsealsbecameavailable.Manymachinesrequiresomecontrollogicforsafeoperation.Inthe1960sfluidicelementsweredevelopedwhichusetheCoandaeffecttogiveBooleanANDorORfunctions.Theywereworkingatabout0.3barpressureandconsideredtobethepneumaticequivalentofelectroniccon-trol(e.g.KirshnerandKatz1975;Espositio2000).Manypresentationsatfluidpowerconferencesdiscussedtherelativemeritsoffluidicorvalvelogic.Pneumaticsensorsforanumberofquantitiesandevencompleteflightcontrolsystemforjetaircraftweredeveloped(RaymondandChe-noweth1993:117–125).Today,fluidicscanonlybeseeninmuseumsandvalvelogicismostlyrestrictedtosimplemachinesthatoperateinexplo-siveenvironments.Typically,controlisdonebyadigitalcomputerinform 21Introductionofaprogrammablelogiccontroller,PLC,withmagneticsensorsforpistonpositionandelectricallyoperatedvalves.Today,themostimportantpropertyofthemediumairisthesimpleconversionofpressuretoforceandtranslationaldisplacementusingapis-toninacircularbore.Theseactuatorsareofsimpledesign,canbeveryfastanddonotoverheatevenifstalledindefinitely.Thegeneratedforcescanbeeasilycontrolledbyapressureregulator.Theabsenceofheatgen-erationallowsforverycompactdesignsandanexcellentratioofpowertoweight:aturbineinahand-heldgrinderweighsonly185gandhasanout-putpowerof2kW.Pneumaticcylindersarethereforewidelyusedwhenmassesofupto20kghavetobetransportedoverrangesofupto1minaminimumoftime.Oneoftheadvantagesoffluidpowerdrivesistheeasywayinwhichthedeliveredpowercanbecontrolled.Asimplevariablerestrictionissuffi-cienttoreducethepowercontinuouslyfromthenominalvaluetozero.ThisisthereasonthathydraulicdriveswereverypopularinmachinetoolsbecauseittakesverysophisticatedsignalprocessingandpowerelectronicstoachievethesamebehaviourwithelectricACmachineswhichofferoth-erwiseanumberofadvantages.Airdoesnotgeneratesparks,posesnohealthhazardandcaneasilybestored.Pneumaticactuatorscanthereforebeusedintheexplosiveenvi-ronmentsofchemicalplants.Ifnolubricationisused,aircanbeventedfromthecomponentdirectintotheatmosphere;aseparatereturnlineisnotnecessary,butsomeformofsilencingisusuallyapplied.Leaksdonotcausecontaminationorelectricshocksandarethereforeoftenignoredleadingtoavoidablelossesandhighoperatingcosts.Asimplebottleisenoughtostorepneumaticenergyforalongtimeandevenunderseveretemperatureconditions.Forintermittentuseastoragetankcanbeusedandasmallcompressorsufficesforfilling.Atmosphericairisfreeandthishasledtostatementsthatcompressedairisacheapformofenergy.However,mostoftheelectricinputenergytothecompressorisconvertedtoheat,dependingonthesystembetween60and90%.Pneumaticenergyisthereforemuchmoreexpensivethanthealreadyexpensiveelectricity.Aruleofthumbisthatacompressortakingin11l/soffreeairatatmosphericpressurewillrequireanelectricinputpowerofabout4kWtoproduceanoutputpressureof7bar(Falkmann1975b:476).However,inamodernfactorytheheatfromthecompressorscanberecoveredtoreducethecostofcompressedairupto80%(Ruppelt2003:489).Itisthereforeimpossibletomakegeneralstatementsaboutthecostofcompressedairortheenergycostofpneumatictools.Partofthecostdisadvantageisoffsetbythefactthatpneumaticcomponentsarein-expensiveandcanoperateatahighnumberofcyclesperworkday. 1Introduction3Thehighcompressibilityofairmakescontrolofactuatorvelocityverydifficult.Duetothelowviscosityaircanusuallynotbeusedtolubricatethemachineryitactuates.Theadvancesinelectronicshelpedtodevelopcontrolsystemsforelec-tricdrivesthatmadethemsuperiortoformerlyusedfluidpoweractuators.Thistechnologycanalsoenhancetheperformanceofpneumaticdrives.Examplesareopenorclosed-loopcontrolledcylindersformanufacturingorassemblytasks,pressurecontrolledchambersinlorrybrakingcircuitsorpositioncontrolledactuatorsforprocessvalves.Thisbookisorganisedinthreeparts.Thefirstpartconsistsofchapters2–7andgivesthetheoreticalbackground.Chapter2describespropertiesofcompressedair,followedbyashortreviewofthermodynamicprocessesandsomeresultsfromfluidmechanics.InChap.5modelsaregiventode-scribetheflowratecharacteristicsofrestrictions,inChap.6severalmod-elsoflonglinesarederived.Chapter7describestheconversionofelectriccurrenttomechanicalquantities,likeforceordisplacement.Thesecondpartofthisbookgivesadescriptionofpneumaticcompo-nents.ThestandardcylinderwhichisthemostimportantconverterfrompressuretoforceismodelledinChap.8.Thefollowingchaptersdescribenon-standardlinearactuatorslikebellowsandchambersforbrakingsys-temsinlorries,bussesortrainsandsuspensionsinpassengercars,andsemi-rotaryactuators.MotorsandturbinesaremodelledinChap.11.Therestofthispartisdevotedtovalves.Chapter12introducesdirectionalcontrolvalveswhichareneededtodirectsupplypressuretotheappropri-ateactuatorports.Chapter13describesshut-offvalves.Theimportantclassofpressurecontrolvalvesfollows.FlowcontrolvalvesarepresentedinChap.15andinthefollowingchapterproportionaldirectionalcontrolvalvesaredescribed.Thethirdpartofthisbookdescribessystems,i.e.thecombinationofcomponentstofulfilacertaintask.InChap.17methodstocontrolthestroke-timeofadrivearepresentedfollowedbyanalysisanddesignofpo-sition-controllers.InChap.19theuseofpneumaticdrivesintheprocessindustryisdescribedandthelastchaptergivesanintroductiontodigitalsimulationofpneumaticsystems.Therearemanyinterestingtopicsthatcouldnotbecoveredinthebook.Foremost,thegeneration,preparationandtransportationofcompressedairandsomeareasofapplicationlikepneumaticconveyingorvacuumtech-nology.Butalsoformanyaspectsofcircuitdesignlikesafetyrequire-mentsandregulations,minimisationofswitchingfunctionsoradvancedcontroltopicslikecontinuouspathcontrolthereaderisreferredtotheap-propriatetexts. 2PropertiesofCompressedAirPneumaticdrivesusecompressedair1tostoreandtransmitpowerorsig-nals.Itspropertiesarethereforesignificantforthebehaviourofthedrivesandagoodmathematicalmodelisneededforreliablenumericalanalysisandsimulation.Clean,dryairisamechanicalmixtureofapproximately78%byvol-umenitrogenand21%oxygen.Theremaining1%consistsofminorquantitiesofsomefourteenothergases.Thecompositionofairremainssubstantiallythesamebetweensealevelandanaltitudeofabout20km,butitsdensityvarieswithpressureandtemperature.Atstandardtechnical5referenceconditions,withapressureof10Pa,atemperatureof20°Candarelativehumidityof65%,thedensityofairis1.185kg/m³.Atthispres-sureandtemperature,1kgofairhasathereforeavolumeof0.844m³.Atstandardtemperatureandpressure,themeanvelocityofgasmole-culesisoftheorderof500m/s,withameanfreepathbetweenintermo--7-8lecularcollisionoftheorderof10to10m.Therateofcollisionisre-sponsibleforthepressureexertedbyair.Theeffectofachangeintemperatureistomodifythevalueofthemeanvelocity.Theresultantpressurethereforevarieswiththetemperature.Similarly,anychangeinvolumeormasseffectsthepressure.Thuspres-sure,temperatureandvolumeareinterrelated.Forallfollowingderivationstheassumptionismadethatthecom-pressedaircanbetreatedasacontinuum.Allquantitiesofinterestsuchasdensity,velocityandpressureareassumedtobedefinedeverywhereinspaceandtovarycontinuouslyfrompointtopointwithinaflow.Whenthecomponentslikevalvesorcylindersaremanufacturedwithtypicalme-chanicalengineeringtools,thisassumptionisusuallyvalid.However,ifmicrotechniqueslikephotolithographyorsilicon-basedmicromachiningtechniquesareusedtomanufactureverysmalldevicestheabovestatedas-sumptionmaybeviolatedanddifferentmodellingapproachesrequired.Foranintroductiontomicrofluidicssee(NguyenandWereley2002).1Thereareonlyfewreportsabouttheuseofothergases;amongthem(FernandezandWoods2000). 62PropertiesofCompressedAirHotGas(Missiles)10100MobileApplications(TruckBrakes)Drives110ProcessControlVacuumAbsolutePressue(MPa)0.10GaugePressure(bar)Fig.2.1.PressurerangesofpneumaticsystemsTypicalpressurerangesofpneumaticsystemsaregiveninFig.2.1.Vacuumsystemsoperatewithpressuresbelowatmosphericpressureandrequiredifferentmodelsthantheothersystemsandwillthereforenotbedescribedinthisbook.Processcontrol,drivesandmanysystemsinlorriesortrainsoperatewithpressuresofupto20barandthemathematicalmodelinChap.2.1canbeapplied.Highpressuregasfromhotgasgen-eratorsasusedinmissileshastobedescribedbydifferentlawsandwillalsonotbeconsideredinthistext.2.1MathematicalModelofAirThereareseveralwaystomodeltherelationshipbetweengasmass,pres-sure,temperatureandvolume.Onewayistomeasuretherelevantsetsofparametersandusetablesandinterpolation.Thisgivesthemostaccurateresultsbutistimeconsuming;bothwhencarriedoutwithpenandpaperorduringasimulation.Anotherwayistouseagaslaw.Thesimplestgaslawisbasedontheassumptionsthatthemoleculesareperfectlyelastic,arenegligibleinsizecomparedwiththelengthoftheirmeanfreepathandex-ertnoforceoneachother.Thisgasiscalledidealgasandtherelationbetweenmass,pressure,temperatureandvolumeisgivenby 2.1MathematicalModelofAir7p⋅V=m⋅R⋅T,(2.1)wherepabsolutepressureinPa,Vvolumeinm³,mgasmassinkg,RgasconstantinJ/(kg.K),TabsolutetemperatureinK.Asimplecheckwhetherthismodelisadequateforthemodellingofpneumaticdrivesisacomparisonbetweenmeasuredandcalculatedvaluesforrelevantparametersets.Tabulateddataisgivenin(ChadwickandBrady1957;BaehrandSchwier1961;Brower1990).Figure2.2showstherelativeerrorerelinpercentbetweentabulatedandcomputedvaluesofthedensityρ=m/V,ρtable−ρidealgas(2.2)erel=⋅100.ρtableForrelevantcombinationsofpressureandtemperatureforpneumaticdrivestheerrorislessthan5%,i.e.intheregionof200K0.528(subsonicflow)°γ−1…∆p¸∆p¸»pψ=°…¬«1¹«1¹»¼1(4.21)®°1⊬2·γ−1γp2°∆¸for≤0.528(chokedflow)°∆«γ+1¸¹γ+1p¯1Whenthementionedassumptions-wellroundednozzle,negligibleap-proachvelocity-aremet,themassflowratecanbedescribedbyEqs.(4.17)and(4.21)withanerroroflessthan1percent(Perry1949).Themodelisevenaccurateenoughtobeusedformeasurementpurposes.Intheabovederivationtheapproachvelocitywasassumedtobenegli-giblesmallandsettozero.Ifthisassumptionisnotmade,theexitvelocityweisgivenforsubsonicflowby(Schmidt1963:275)γ−1(4.22)p1γ»peºγ2we=2…1−»+w1.ρ1γ−1¬p1¼Bialas(1973:15)studiestheeffectofanon-zeroupstreamvelocityw1whenthenozzleisinstalledinapipe.Theerrorduetoneglectingtheinitialvelocityisless 324SomeResultsfromFluidMechanicsthan3.5%ifthenozzlediameteris70%orlessofthepipediameter.Theup-streamstaticpressurewilldifferfromthetotalpressurebylessthan1%ifthera-tiooftheupstreamdiametertonozzlediameterismorethan2.212(Andersen1967:25).Mo(1989)analysesaspoolvalvewheretheinitialairvelocityreaches35%ofthesonicvelocityandhederivesamodeltoincludethiseffect.Equations(4.17)and(4.21)arederivedforsteady-stateflow.Manyexperimentsshowthatitcanalsobeusedfordynamicflow(Kastneretal.1964;BurrowsandPeckam1977).ThompsonandArena(1975)giveapredictionofcriticalmassflowforrealgasesincontrasttotheidealgasusedabove.4.4OrificeFlow4.4.1IncompressibleFlowEquations(4.17)and(4.21)arevalidforwellroundednozzleswithoutflowcontractionandaninitialvelocityofalmostzero.Theyhavetobeextendedtodescribeflowthroughthin,sharpedgedorificeswhereflowcontractionandlossesoccur.Forincompressibleflow,e.g.inoil-hydraulicsystems,theequation2⋅(p−p)(4.23)q=A⋅C⋅12vdρ3whereqvvolumeflowrateinm/s,2Aflowareainm,Cddischargecoefficient,3ρfluiddensityinkg/m,p1upstreampressureinPa,p2downstreampressureinPa,isoftenusedtodescribeflowthroughanorifice.Equation(4.23)isanex-cellentmodelofturbulentflowifthedischargecoefficientCdisknown.Itisintroducedasacorrectionfactormainlytotakeaccountofthejetcon-traction.Alsoreductioninmassflowduetofriction,heatlossesandve-locityprofileeffectsareincludedinthedischargecoefficient.Oftencon-stantvaluesaround0.8areusedforoil-hydraulicorificesthoughCdactuallydependsonanumberofparameters.Inadditiontothegeometricshapeandtheconditionoftheedge-sharporslightlyrounded-theRey-noldsnumberisveryimportant.Therearemanymathematicalmodelsfordifferenttypesoforifices(Beater1999).Asanexample,themodelfromLichtarowiczetal.(1965)isgiveninEq.(4.24). 4.4OrificeFlow33LDdD12pp12Fig.4.4.Schematicviewofashorttubeorifice»Lº0.0051…120⊬L·D»=1−β4…+∆1+2.25¸−h»(4.24)C…CRe∆D¸1+7.5(log0.00015Re)2»d…Dmaxh«h¹10h»¬¼42∆pDH1−βwhereReh==Re,ρνCdLCdmax=0.827−0.0085,DhDhhydraulicdiameter,Dh=dforcircularholesinm,β=D1/D2ratioofdiameters,modelisvalidfor10b(subsonicflow)•10T1-bp°°∆¸m=®11∆¸(5.1)°«¹°°Tpp⋅C⋅ρ⋅0for2≤b(chokedflow)°1oTp°¯11wherem&massflowrateinkg/s,p1upstreampressureinPa,Csonicconductanceinm³/(s.Pa),ρ0densityofairatreferenceconditionsinkg/m³,T0temperatureofairatreferenceconditionsinK,T1upstreamtemperatureofairinK,p2downstreampressureinPa,bcriticalpressureratio. 425EngineeringFlowRateCalculationsThismethodtomodeltheflowratethroughapneumaticcomponentwillbeusedintheremainderofthisbookanditsparameterscanbecharacter-isedasfollows:ThesonicconductanceCofacomponentistheratiobetweenthe•massflowratemthroughthecomponentandtheproductofup-streampressurep1andthemassdensityatstandardconditionswhentheflowischoked.Thecriticalpressureratiobofacomponentistheratiobetweenthedownstreampressurep2andtheupstreampressurep1atwhichtheairvelocityachievessonicspeed.Chokedflowoccurswhentheratiobetweenthedownstreampres-surep2andtheupstreampressurep1islessthanthecriticalpressureratiobofthatcomponent.Afurtherreductionofthedownstreampressurep2doesnotincreasethemassflow.Equation(5.1)canapproximatethetheoreticalfluidmechanicsequa-tionsoftheflowratethroughanidealnozzlewithanerrorofonly0.33%.InthestandardISO6358thereferenceconditionsaredefinedas:-temperatureT0=293.15K,-pressurep0=100kPa,-gasconstantR0=288J/(kg.K),-relativehumidity65%,-densityρ0=1.185kg/m³.Sanvilleproposedhismodelin1971.Itbecameaprovisionalrecom-mendationofaCETOPstandardin1973andanISOstandardin1989.•Figure5.1givesameasuredrelationshipbetweenmassflowratemandthepressureratiop2/p1foroneflowpathofadirectionalcontrolvalveandthecalculatedvaluesaccordingtoEq.(5.1).Thecriticalpressureratiobhastypicallyavaluebelow0.5,forvalvesusuallyabove0.2,butcanalsobeassmallas0,e.g.forsilencersorproportionaldirectionalcontrolvalves(Ballard1974;Wikander1988).ForanorificewithaCdasgiveninEq.(4.28),bcanbearound0.125andthedifferencebetweenbothmodelsisverysmall. 5.1MathematicalFlowRateModel430.6chokedsubsonic0.40.2ModelMassFlowRateMeasurement00.050Error(g/s)-0.0500.20.40.60.81p/p21Fig.5.1.Massflowrateasafunctionofpressureratio(top),andmodellingerror(bottom);measurementset-upaccordingtoFig.5.2,modelaccordingtoEq.(5.1).Wiedmann(1979)reportsforaLavalnozzleeventhehighvalueofb=0.9.ThevalueusedinFig.5.1isb=0.225,markedwithadotdashedline.Both,themeasuredsignalandthecomputedmassflowrateareshowninFig.5.1andbothlinesliedirectlyaboveoneanother.Thelowerpartofthefigureshowsthereforetheerrorbetweenmeasurementandmodel.However,notalwayscanthemodelbefittedtoameasuredsignalwithsuchasmallerror.TodeterminethecoefficientsbandC,theISOstandardgivestwomeasurementset-ups,dependingwhetherthecomponenthastwoports,Fig.5.2,oroneinletportandanoutletdirecttotheatmosphere,Fig.5.3.Thestandardalsogivesdetailedrecommendationsaboutthedimensionsofthemeasurementset-upandacceptableerrorsofthesensors.TheISOmodelhasbeenusedforseveraltypesofpneumaticcompo-nentsandgenerallyfoundtobeadequate.Pott(1995)modelledverysmallorificesandnozzleswithadiameterof25µm.Nguyen(1987)usedittodescribetheflowthroughaproportionalcontrolvalve.TherehasbeencriticismaboutthewaythecoefficientsbandCaretobedeterminedaccordingtotheISOstandard.Itstipulatesthatonlyfourpointsontheellipsehavetobemeasuredandusedforthecomputationofb.Thissmallnumberofmeasurementscanleadtoapoorfitofthemodel(Hanetal.2001).Anotherobjectionisthatalargeamountofairisneeded 445EngineeringFlowRateCalculationstotesthighcapacityvalvesinthisway.Lebig(1971)discussesthedis-chargeofairfromavesseltocalculatetheparametersofthenozzlemodelfromthemeasuredpressuretrajectory.Thisapproachhasfoundmanysup-porters(Ohligschläger1988;Haack1991;Hanetal.2001;delasHeras2001).p1∆pGFig.5.2.Measurementset-upaccordingtoISO6358forcomponentswithtwoports.ThedeviceundertestisG.Themassflowrateisreducedbyclosingtheflowcontrolvalve.TheresultingcurveslooklikeFig.5.1(top).p1GFig.5.3.Measurementset-upaccordingtoISO6358forcomponentswithoneportanddirectoutlettotheatmosphere.ThedeviceundertestisG.Themassflowrateisreducedbyreducingtheupstreampressure.TheresultingcurveslooklikeFig.5.4.0.60.40.2MassFlowRate(g/s)000.10.20.30.40.50.6p(MPa)1Fig.5.4.Computedmassflowrateasafunctionofinletpressurep1foraset-upasinFig.5.3.Thedownstreampressurep2istheconstantatmosphericpressure. 5.1MathematicalFlowRateModel45Fromamodellingpointofview,thereisanothershortcomingoftheISOmodel.Eq.(5.1)describeschokedflow(p2/p1≤b)andsubsonicflow(p2/p1>b).However,ifthepressureratiobisalmostunity,theflowmodeislaminarandEq.(5.1)modelsthisflowmodepoorly1.Thissituationcanarisewhenthepoppetinapressureregulatorisbarelyopenedtoreplenishasmallleakageflow.TheReynoldsnumbercanbeassmallas15(Istóketal.2003).TheflowcanthenbeconsideredasincompressibleandEq.(4.37)showsthatforthislaminarflowmodetheflowrateisproportionaltothepressuredifferential∆p=p2-p1whileitisasquarerootfunctioninEq.(5.1).Thisisnotonlyapoorphysicalmodelbutleadstoseverenu-mericalproblemsforanintegrationalgorithmwithvariablestepsizebe-causethegain,i.e.thechangeofthemassflowratewithrespecttothechangeofthepressuredifferential,goestoinfinityifthepressuredifferen-tial∆p=p2-p1goestozero.Thegainforsubsonicflowisgivenby⊬2·∆⊬p·¸∆2¸•∆-b¸dmd∆T0∆p2+∆p¸¸=∆C⋅ρ0⋅()p2+∆p⋅1-∆¸¸d∆pd∆pT∆1-b¸∆1¸∆∆∆¸¸¸∆«¹¸«¹2(5.2)⊬p·T⊬p·∆2-b¸p⋅C⋅ρ0∆2−b¸T∆p+∆p¸20T∆p+∆p¸0«2¹1«2¹=C⋅ρ⋅1-+.0T(1-b)221⊬p·∆2−b¸∆p+∆p¸2()«2¹(1-b)p+∆p1-2(1-b)2Takingthelimitfor∆pgoingtozerogivestheinfinitegain:•dmlim=∞.∆p→0d∆p(5.3)Thiseffectisalreadyknownfromthemodellingofincompressibleflowthroughsharpedgedorifices,seeEq.(4.25).ForsimulationsinthisbookathirdflowmodeisaddedtotheISOmodel,1Kaasaetal.(2004)presentanothermethodtoimprovetheISOequations.Theirapproachisespeciallysuitedforcontrolanalysisanddesign. 465EngineeringFlowRateCalculations°°⊬p·Tp°k⋅p⋅∆1-2¸⋅02≥0.999(laminar)°l1∆p¸Tp°«1¹11°°2°⊬p·°∆2-b¸•°T∆p¸pm=p⋅C⋅ρ⋅0⋅1-1for0.999>2>b(subsonic)®∆¸1oT∆1-b¸p(5.4)°11°∆¸°«¹°°°Tp°p⋅C⋅ρ⋅02≤b(choked)°1oTp°11°¯withthelineargainkl.2(5.5)⊬0.999-b·kl=1000⋅C⋅ρ01−∆¸.«1-b¹Thechoiceofthelimitp2/p1>0.999forthelaminarregionisarbitrarilydone.AmorephysicalorientedwaywouldbetobasethelimitontheReynoldsnumberoftheflowthroughthecomponent.However,thiswouldrequireareferencediameterthatcannotbeeasilydeterminedforcompli-catedcomponentslikevalves,especiallyifnoadditionalinformationlikeadrawingisavailable.Testsalsoshowedthatthelaminarregionistypicallyonlyactiveinthesteadystateofasimulationandanotherchoiceofthelimitdoesnotgivedifferentresults.Figure5.5showsasimpleexample.Initially,volume1ischargedwithapressureof9barwhilevolume2hasatmosphericpressure.Whenstartingthesimulation,airflowsthroughthenozzleintovolume2wherethepres-surerisesuntilthereisequalpressureinbothvolumes.Figure5.6showsthecomputedpressuretrajectoryintheupperpart.Fromtime0tot1theflowmodeischoked,fromt1tot2subsonicandafterthatlaminar.InthelowerpartofFig.5.6thecomputingtimesformodelEq.(5.1)andEq.(5.4)areshown,scaledtothecomputingtimeforthestandardISOequations.TheISOmodelgivesthesametrajectoriesbutneedsmorethan50timeslongerthanthemodelaccordingtoEq.(5.4);100%insteadof2%.Andmostofthetimeisneededforthecomputationofthesteadystatewhenthereisnosignificantflowthroughthenozzle. 5.1MathematicalFlowRateModel47Volume1Volume2NozzleFig.5.5.SimpletestsystemtoshowtheeffectsoftheinfinitegainoftheISOmodelforsmallpressuredifferentialp18p26412p,p(bar)20t1t2100ISO50modifiedCPUtime(%)000.010.020.030.04Time(s)Fig.5.6.TrajectoriesofthesimpletestsystemandrequiredCPUtimeforsystemaccordingtoEq.(5.4)(modified)andEq.(5.1)(ISO),respectively(integrationmethodLSODAR2,tolerance1e-8)2HindmarshAC(1983)ODEPACK,asystematizedcollectionofODEsolvers.ScientificComputing.Edt.R.S.Steplemanet.al.North-Holland,Amsterdam,asimplementedintheusedsimulationsystemDymola 485EngineeringFlowRateCalculationsEquations(4.17)and(4.21)or(5.1)areverytediousforhandcalcula-tions.Inthepast,theflowfunctionΨwasthereforetabulatedandap-proximationsofthemodelequationswereused.Forsubsonicflow,acriti-calpressureratioof0.250.9p1thefollowingequationcanbeusedwithanerrorbelow2%(Haack1991:43):2⋅p⋅(p−p)(5.6)q=C212v1−b5.2FlowRateCharacteristicsofRestrictionsThevaluesofthecriticalpressureratiobandthesonicconductanceCde-pendontheparticulardesignofacomponent.Typically,theyaredeter-minedfrommeasurementsor-inanincreasingnumberofcases-giveninthemanufacturer’sdatasheet.ForgeometricallysimplecomponentsformulashavebeenpublishedtocomputebandC.ForlonglinesEqs.(6.8)to(6.11)canbeused.Anap-proximationforrestrictions-definedtohavearatiooflengthLtodiame-terdoflessthan10-wasgivenbyGidlund(1977):d(5.7)b=0.41+0.272D2C=0.128⋅d(5.8)whereDinnerdiameteroflineinmm,dinnerdiameterofrestrictioninmm,Llengthofrestrictioninmm,3Csonicconductanceindm/(s.bar).LdDFig.5.7.Schematicviewofarestrictioninalongline 5.3SimplifiedFlowCalculations49Belforteetal.(1995a,1995b)carriedoutalargenumberofexperimen-talflowratetestsonsharp-edgedorificesinthediameterrangefrom4to10mmandratiosoflengthtodiameterfrom0.33to10andfoundthatC8dm3(5.9)≈.22dmin⋅bar⋅mmToensureastableflowrate,theratioL/dshouldbegreaterthan1.AgelandCodina(1996)studytherelationbetweensoundpressurelevelandflowrateparametersofpneumaticsilencersandgiveagraphofCversussilencerdiameter.5.3SimplifiedFlowCalculationsForcomponentsthatareconnectedinseriesthecriticalpressureratiobandthesonicconductanceCoftheoverallsystemcaneasilybecalculatedbyaniterativetechniquewithanerroroftypically3–4%,onlyinextremecasesupto10%(Gidlund1977,Eckersten1975a:187–189).Forthefirsttwocomponents,thevalueofanauxiliaryvariableαisgivenbyC(5.10)α=1b⋅C12wheretheindex1referstothefirst(upstream)componentandtheindex2tothesecond(downstream)component.Ifα<1,theflowbecomeschokedfirstincomponent1whenthepressuredropacrossbothcomponentsisin-creased.Ifα=1,theflowbecomeschokedinbothcomponentsatthesametime.Ifα>1,theflowbecomeschokedincomponent2.ThesonicconductanceC1,2oftheseriesconnectionisgivenby°°°Cforα≤1°1°°°C=®1,22°⊬1−b·(5.11)°α⋅b+()1−b⋅α2+∆1¸−1°11∆b¸«1¹°α⋅Cforα≥1°22⊬1−b·°α2+∆1¸°∆b¸¯«1¹ 505EngineeringFlowRateCalculationsAndthecriticalpressureratioisgivenby2⊬1−b11−b2·(5.12)b1,2=1−C1,2∆∆2+2¸¸.«C1C2¹Forathirdcomponentthesecalculationsarerepeatedwherethealreadycomputedparametersb12andC12becomeb1andC1andthecoefficientsofthethirdcomponentbecomeb2andC2,respectively.FortheseriesconnectionofnidenticalnozzleswithparametersbandCWiedmann(1979)foundb(5.13)bseries=3nandC(5.14)Cseries=.3nThisresultexplainsthatpneumaticvalves,whichcanbethoughtofaconcatenationofseveralrestrictions,canhavevaluesofbthataremuchlowerthan0.53,thetheoreticalvalueforanozzle.ForaparallelconnectionofncomponentsthepressureratiobandsonicconductanceCcanbecalculatedby(Eckersten1975a:189–190):n(5.15)Cparallel=ƒCi,i=1CnC(5.16)paralleli=ƒ.1−bparalleli=11−bi5.4FlowCapacitySpecificationsinDataSheetsSomemanufacturerspublishnumericalvaluesofthecriticalpressureratiobandthesonicconductanceCoftheirvalves.Sometimescharacteristiccurvesaregiventhatshowthemassflowrateasafunctionofsupplypres-sureandworkingpressure,seeFig.5.8.Foragivensupplypressure,e.g.6bar,thecurvestartsonthey-axisforaworkingpressureofalso6barandthereforeaflowrateofzero.Ifthepressureattheworkingportisreduced,airflowsthroughthevalve.Thecorrespondingmassflowrateisgivenon 5.4FlowCapacitySpecificationsinDataSheets51109criticalpressureratio876543PortPressure(bar)210010203040MassFlowRate(l/min(ANR))Fig.5.8.Flowcharacteristicsofavalve.Characteristiccurveofflowrateasafunctionofsupplyandworkingpressure:forasupplypressureof6barandaworkingportpressureof4barthemassflowrateis20l/min(ANR).thex-axis.Forasupplypressureof6barandapressureof4barattheworkingportthemassflowrateis20l/min(ANR)inFig.5.8.FromthesecurvestherequiredparametersforbandCcanbeobtained.InFig.5.8thetransitionbetweensubsonicandchokedflowismarkedwithadotdashedlinethatalsoshowsthevalueofthecriticalpressureratiob.ThestandardISO6358alsodefinesanalternativesetofcoefficientsandequations,respectively.Thecoefficientofcompressibilityeffectsisgivenby1(5.17)s=.1−bTheeffectiveareaAisgivenbyA=C⋅ρ0s⋅R⋅T0.(5.18)Forchokedflow,i.e.when∆p/p1≥1/s,themassflowrateisgivenby•p(5.19)m=A1.s⋅R⋅T1 525EngineeringFlowRateCalculationsForsubsonicflow,i.e.∆p/p1<1/s,themassflowrateisgivenby•p1∆p⊬∆p·(5.20)m=A2∆1−s¸.∆¸R⋅T1p1«2p1¹TheJapanesestandardJISB8390(anon.2000f)alsocharacterisestheresistanceofavalvebythesizeoftherestriction3.Inthiscaseitistheef-fectivecross-sectionalareaS.Thestandarddefinesthedischargeofavol-umeasmeasurementmethod4.Theinitialgaugepressureinthevesselis5barandthevalveisclosedifthepressureinthevesselreaches2bar.Thesizeofthevesselhastobechosensuchthatthedischargetimetisintheorderofseveralseconds.Theeffectivecross-sectionalareaisthengivenbyV⊬p0+0.1·293(5.21)S=12.1⋅⋅log∆¸⋅t10∆p+0.1¸T«¹2whereSeffectivecross-sectionalareainmm,Vinnervolumeofvesselinl,p0initialgaugepressureofvesselinMPa,presidualgaugepressureofvesselinMPa,tdischargingtimeofairins,TroomtemperatureinK.AcomparisonoftheISOmethodwiththeJISmethodshowsonlya5%differenceoftheeffectiveareas(Itohetal.1989).SometimesmanufacturersgivethenominaldiameterND.Itisthedi-ameterofaholewiththesameareaasthatofthesmallestflowareaofthecomponent.ThestandardISO6358definescoefficientsandequationstoaccuratelymodeltheflowratethroughapneumaticvalve.However,whenselectingavalveduringtheinitialdesignphasesuchadetailedmodelisnotneededandsometimesnotevenpracticalbecauselongercalculationsarerequired.AsimplerflowcapacityspecificationisthenominalflowrateQnthatgivestheflowrateforanupstreampressurep1=6barandadownstreampressurep2=5bar.Aruleofthumbstatesthatthepressuredropacrossthemeteringorificeofadirectionalcontrolvalveofacylinderdriveshould3ThisstandardcorrespondingtoISO6358hasbeissuedinJapan2000andmostmanufacturersstoppedusingSandgivebandCinstead(Zhangetal.2005).4ThereareseveralproposalstousethedischargeofavolumealsofortheISOstandardtoreducetherequiredamountofenergy(Benchabaneetal.1994;Ka-washima2004). 5.4FlowCapacitySpecificationsinDataSheets53notexceed1barwhenusingasupplypressureof6bar.Aftercalculatingtheflowratefortherequiredpistonspeed,anappropriatevalvecanbese-lected.Anothermethodtodescribetheflowcapacityofavalveistheflowco-efficientkv(anon.1962;Haack1991).ItwasfirstusedintheUStochar-acteriseprocessvalvesandusedimperialunits.ThemeasurementsaremadewithwaterasfluidandinEuropekvgivesthevolumeflowrateinl/minforapressuredropof1baracrossthevalve.Assumingavalvewithacriticalpressureratioofb=0.528andsubcriticalflow,theflowcoeffi-cientkvcanbeusedtocalculatetheflowofgasthroughavalveatrefer-encetemperature:q=28.16⋅k⋅p⋅(p−p)(5.22)vv112whereqvvolumeflowrateinl/min(ANR),kvflowcoefficientinl/min/bar(ANR),p1absoluteupstreampressureinbar,p2absolutedownstreampressureinbar.However,asshowninChap.5.3thefirstassumptionmaynotbeappli-cabletopneumaticvalveswherethecriticalpressureratiobcanbemuchlower.AnotherproblemwiththismethodisthefactthattheresistanceofavalvemaysignificantlydependontheReynoldsnumberwhichismuchhigherifairisusedinsteadofwater.Areliabledeterminationofthemassflowrateforairisthereforenotpossible;especiallyifthepressureratioismuchlessthanunity.Therearetwovariationsofthismethod.Oneusestheflowcoefficient3Kvthatgivestheflowrateinm/hforapressuredropof1bar.TheotherusesUSunitstodefinetheflowcoefficientCvinUSgal/minforapressuredropof1psiacrossthevalve.In1990thismethodhasbeenpublishedasstandardANSI/(NFPA)T3.21.3-1990andmanyUSmanufacturersofpneumaticvalvespublishCvvaluesoftheirproducts.Whilethisapproachcanleadtoproblemswhenaccuratemodelsofre-strictionsareneededfordigitalsimulation,Berninger(2002)canbefol-lowedwhenhestatesthat“theCvisawidelyrecognizedmethodofclassi-fyingvalvesbyflowcapacityandcanbeeffectivelycontinued”.Inparticular,whenCundiff’s(2001:391)ruleofthumbisappliedthat“gooddesignpracticerequiresthatthepressuredropacrossavalvehastobearound10%ofline(upstream)pressure”andthereforesubsonicflowandapressureratioofalmostunityispresent.ThemostdetaileddescriptionofarestrictionisgivenbytheISOmodel.Aconversionfromonedescriptiontoanothercanonlybedoneapproxi- 545EngineeringFlowRateCalculationsmately:eventhecalculationofCvfrombandCleadstovaluesthatareslightlyhigherthanthoseobtainedfrommeasurementsaccordingtotheANSIstandard(Berninger2002).AnumberofconversionformulasisgiveninTable5.1.Table5.1.Formulastoapproximatelyconvertflowratespecifications(anon.2000d:15.53,Murrenhoff1999:62)TaskFormulaCalculatethenominalflowrateinl/min(ANR),Qn=216⋅Cforb=0givensonicconductanceCinl/s/barQn=247⋅Cforb=0.25Qn=294⋅Cforb=0.5Calculatethenominalflowrateinl/min(ANR),Qn=66⋅kvgivenCv,Kv,kvQn=1100⋅KvQn=984⋅CvCalculatethenominalflowrateinl/min(ANR),2Qn=37.6⋅dgivenholewithdiameterdinmmCalculatethesonicconductanceCinl/s/bar,2C=0.128⋅dgivenholewithdiameterdinmm 6ModellingofLongLinesThecomponentsofapneumaticcircuitareconnectedthroughlines.Todaytheyaretypicallymadeofplastictubesfromnylonorpolyurethane.Thematerialischosentomeettherequiredpermissiblepressure,bendingra-dius,temperatureetc.(seealsoMünzerandMüller-Lohmeier2004).Cop-pertubesareusedforheavydutyapplicationslikeactuationofprocessvalvesinchemicalplants.Forplastictubespush-infittingscanbeusedwhichhavealowresistanceandtheadditionaladvantagethatnotoolsarerequiredtoinstallorremovethem.Whendesigningoranalysingacircuit,differentaspectsmayhavetobestudied:thesteady-stateresistance,thedynamiceffectsinatimedomainsimulationorthefrequencyresponse.Thefollowingsubchaptersgivemodelsforallthreecases.6.1Steady-StateLossesofLongLinesMajordesigncriteriaforlonglinesistheresistance.TheISOstandard4414givesatablethatbasestheflowratesupona10%pressuredropin30mofISO65grademediumwroughtironpipeat20°C(anon.1998).Typicalairvelocityisbetween10and40m/s(WillandStröhl1990:335;Stoll1999:54),typicalReynoldsnumbersarebetween(Hennig1982)4⋅103≤Re≤4⋅106,andthemaximumpressuredropshouldbebelow0.1bar(Falkman1975b:473–477).Fig.6.1.Schematicviewofaplastictube 566ModellingofLongLinesTable6.1.Maximumrecommendedflowratestobeusedinpneumaticsystempipinginl/s(ANR)(anon.1998)WorkingInsidediameterofpiping(mm)pressurebar6913160.200.180.410.911.70.400.280.621.42.60.630.380.851.93.50.800.441.02.24.11.000.521.22.64.91.250.621.43.15.81.600.751.73.87.02.000.912.04.58.42.501.12.55.5103.151.33.06.7124.001.63.78.3155.002.04.610196.302.55.613238.003.17.0162910.03.98.7193612.54.810244516.06.11331576bar42206-50m4+tubeg/s20128°C24inout200200040006000Time(s)Fig.6.2.Pressuredrop,massflowrateandtemperatureasafunctionoftime;andmeasurementset-up,linelengthL=50m,innerdiameterD=5.7mm 6.1Steady-StateLossesofLongLines57Unlessthelinesare“long”theyaretypicallynotconsideredduringthedesignoranalysisofapneumaticcircuit.Ifnecessary,theirresistance,i.e.thepressuredropalongtheline,canbeincludedinthenumericalanalysis.Asecondstepistheinclusionofthedynamicbehaviourwhichiscausedbythecompressibilityofair.Atypicalexampleoflonglinesarerailwaybrakesystemswherethelengthcanbeintheorderof400mandmore(Kageetal.1985;MurtazaandGarg1989).Inthiscaseboththefrictionandthedynamicsmustbetakenintoaccount.6.1.1FluidMechanicsModelThefirstmodelofalinewithlengthLandinnerdiameterDfollowsaderivationbyFalkman(1975a).Itisbasedonthegeneralenergyequationindifferentialform,Eq.(3.31):w⋅dw+g⋅dz+v⋅dp+dw=0(3.31)TwithfrictionworkdwF,accordingtodLw2(6.1)dw=dw=λFTD2whereλisthefrictionfactordefinedinEq.(4.40)andgiveninEqs.(4.42)to(4.45)asafunctionoftheReynoldsnumber.SubstitutingdwFinEq.(3.31)andassumingz1=z2gives:dLw2(6.2)w⋅dw+v⋅dp+λ=0.D2InFig.6.2ameasurementset-upisshown.Theinputpressuretoalonglinewithaninnerdiameterof5.7mmandalengthof50misincreasedfrom0to6.25barandthenreducedto0bar.Themassflowratefollowswithamaximumof4.3g/s.Theoutputtemperaturedropsonlyslightly,themaximumdifferencetotheinputtemperatureis0.8°C1.AnimportantobservationfromFig.6.2isthatthetemperaturesoftheenteringandtheleavingairarealmostidentical.Hennig(1977)statesthatasignificanttemperaturedroptakesplaceonlyforvelocitiesabove100m/s(seealsoWroten1969).Foranidealgasthismeans1Forathoroughanalysisofthetemperatureofairventingintotheatmospheresee(Barth2003). 586ModellingofLongLinesp⋅v=p1⋅v1.(6.3)Usingthecontinuityequationwv(6.4)=w1v12anddividingbywgivesdwv1⋅p⋅dpλ⋅dL(6.5)++=0.wp⋅w22⋅D11Thisequationcanbeintegratedtoyield⊬w·v22λ⋅L(6.6)ln∆2¸+1⋅()p−p+=0.∆w¸2⋅p⋅w2212⋅D«1¹11UsingEqs.(6.3)and(6.4),thenon-linearequation(6.6)canbemanipu-latedtocontainonlythesquareoftheentrancevelocityw1andnotw2,butitcannotbesolvedanalyticallyforw1withoutmakingadditionalassump-tions.Typicallythechangeinvelocityissmall,leadingtow1=w2andtherefore222p−pλ⋅L⋅ρ⋅w21+11=0(6.7)2⋅p2⋅D1Equation(6.7)willbecalledFluidMechanicsmodelor“FM”inthefol-lowingtextandlegends.AcomparisonwithChap.4.5showstheeffectofthecompressibilityofthefluidonthefriction.6.1.2EmpiricalModelsAnempiricallyderivedmodelgivesthetwoparametersfortheISOnozzlemodelfromChap.5.1.ItwasfirstpublishedbyEckersten(1975a:185)withoutderivation;asimilarmodelisgivenbyGidlund(1977).ThevaluesforthepressureratiobandsonicconductanceCaregivenby474C(6.8)b=,D2 6.1Steady-StateLossesofLongLines590.029D2(6.9)C=L+510D1.25whereDinnerdiameterinm,Llengthinm,3Csonicconductanceinm/(s⋅Pa).InFigs.6.5to6.10thecomputedcurvesfromthismodelarelabelled“Atlas”.Thisapproach-modellingthelineasasinglerestriction-hasbeenconfirmedbymeasurementstakenbyBideauxandScavarda(2000).Eschmann(1994:62)givesasimilarmodelforlineswithalengthof0.25m>1,-normaliseddensityvariations∆ρ/ρ<<1. 6.4FrequencyDomainModels77ForalineterminatedbyanimpedanceZ,theratiooffrequencytrans-formsofthepressureamplitudesp1(ω)andp2(ω)orm&1(ω)andm&2(ω),re-spectively,aregivenby2p(ω)1(6.36)2=p(ω)A1A+1211Zor•(6.37)m2(ω)1=.•A+A⋅Zm1(ω)2221Ifthelineisblockedattheend(index2)theimpedanceZgoestoinfin-ity.Hence,p(ω)11(6.38)2==;p(ω)Acosh()Γ⋅L111theinputresistanceisthengivenbyA⋅Z+A(6.39)W=1112.iA+A⋅Z2221Forcircularhomogenouslinesandameanmassflowofzero,theparame-tersaregivenby(Kohl1973),(6.40)⊬ω·1+()γ−1⋅J∆¸∆ω¸jω«T¹Γ(ω)=c⊬ω·1−J∆¸∆ω¸«ν¹whereΓpropagationoperator,cspeedofsound,Aarea,A=πD²/4,with8⋅πν(6.41)ω=TAPr2Fortheimpedanceofanorificesee(Kohl1972:42) 786ModellingofLongLineswherePrPrandtlnumber,forairPr=0.708,8⋅π⋅ν(6.42)ω=,νAand⊬·(6.43)∆3ω¸2⋅J8j⊬·1∆∆ω¸¸∆ω¸«T,ν¹J=∆ω¸⊬·«T,ν¹∆3ω¸3ωJ8j⋅8j0∆∆ω¸¸ω«T,ν¹T,νwhereJi(...)Besselfunctionoffirstkindandorderi.ThecharacteristicimpedanceZ0isgivenby−12(6.44)c»⊬ω·⊬⊬ω··ºZ=…1−J∆¸⋅∆1+(γ−1)⋅J∆¸¸».0A…∆ω¸∆∆ω¸¸»¬«ν¹««T¹¹¼Theinputvariablesforthemodelarethefrequencyω,themeanpres-sureandthetemperatureTwhichdirectaffectthedynamicviscosityν.Theinputamplitudeisconsideredtobeverysmall.Ingeneral,thehigherthemeanpressurethehigherarethepressurepeaksatresonance.Figure6.24showsthecomputedpressureratio|p2/p1|foralinewithalengthof2mandaninnerdiameterof2.5mmandfourdifferentvaluesofthemeanpressure.Themodelwasderivedundertheassumptionthatthemeanmassflowiszero,e.g.ablockedlineoralineterminatedbyavolume.Kohl(1972)pre-sentsgraphsthatshowthatasuperimposedflowreducesthedampingoftheline.Forexampleasuperimposedflowwithavelocityof34m/sandanormalisedfrequencyoff/fν=10reducesthedynamicpressurelossbyalmost10%(Kohl1973:19).Forlineswithrectangularcross-sectionorothergeometricshapetheseequationshavetobemodified(Schaedel1968;Kohl1973;Kanaietal.1985). 6.4FrequencyDomainModels798bar94bar2bar1bar7521|p/p|31050100150Frequency(Hz)Fig.6.24.ComputedfrequencyresponsesofalineL=2m,D=2.5mmfordif-ferentmeanpressuresForthedesignofcontrolsystemstheapproximationbyafiniteordermodelissometimeshelpfultoknowtheeigenvaluesormodalcharacteris-tics(Nichols1962;KaramandFranke1967).Woodsetal.(1985)presentanalgorithmtocomputestatevariablemodelsforanarbitrarilychosennumberofmodes.Hougenetal.(1963)givefirstandsecondorderap-proximations.Sometimesitmayevenbesufficienttocalculatethefrequenciesfiofthefirstimodesofalosslessline.Dependingontheboundaryconditionstheyarec(6.45)fi=()2i−1,withi=1,2,...4Lforablockedlineandcf=i,withi=1,2,...(6.46)i2Lforanopenline. 7Electro-MechanicalConvertersToday,agreatmajorityofpneumaticdrivesiscontrolledbycomputers,bothPLCsforon-offvalves,embeddedcontrollersorIPCsforcontinuousoperation.Thesesystemsrequiretheconversionoftheelectricsignalout-putbythecontrollertoapneumaticsignal,usuallyapressure.Typicallythisisperformedbyasolenoidthatconvertstheelectriccurrenttoaforcewhichopensorclosesaflowpathinavalve.Forlowpowerapplicationsvoicecoilsareused.Forexplosiveenvironmentswhereultralowpoweroperationisimportantpiezoelectricdevicesareavailable.7.1SolenoidsSolenoidsusethephysicaleffectthatcurrentflowingthroughanelectricconductor,e.g.anenamelledcopperwire,generatesamagneticfield.Thismagneticfieldcanbeamplifiedbyarrangingthewireintheformofacoilwithagreatnumberofwindings.Anironfluxcagearoundthesolenoidcoilhasasubstantialimpactontheresultantmagneticforces.Themag-neticfieldgeneratedbythesolenoidcoilcausesaforceofattractiononthemagneticcorelocatedatthecentrewhichisalsofrequentlyreferredtoasthearmatureorina2/2-wayvalveasplungerwhichopensorclosesaflowpath.Thisprincipleisusedinbothon-offcontrolvalvesandpropor-tionalcontrolvalves.However,thedesignofthesolenoiddiffersconsid-erably.Inonecase,abinarymodeofoperationisrequired,i.e.aneithercompletelyopenorcompletelyclosedon-offvalve.TheswitchinghastobefastandtherequiredelectriccurrentlowtofacilitateasolidstateswitchasinaPLC.Intheothercase,theforceactingontheplungershouldde-pendproportionallyonthecurrentandbeindependentofthestroke.TheforceoftheconventionalDCsolenoidishighlydependentonthepositionofthearmaturebecausetheairgapwhichcanbeseenasaresis-tanceforthefluxbetweenfluxcageandplungerchangeswiththeposition,seeFig.7.2. 827Electro-MechanicalConvertersFig.7.1.Schematicviewofadirectcurrentsolenoidwithcoil,coreandspringandmagneticfieldlinesWhenusingalternatingcurrentACforasolenoid,thedesigninFig.7.1needsanadditionalshadingcoiltomaintainthemagneticfieldastheACvoltagefluctuates.Otherwisetheholdingforcewoulddroppracticallytozerowitheachzerocrossoverofthesinusoidalmainsvoltageandthecorewouldbeliftedandattractedagaintwiceperperiod.Theresultingveryloudoscillationwouldtaketheformofaveryunpleasanthum.Thisshadingcoil,sometimescalledashadingring,consistsofasingleturnwindingofheavycopperwireandismountedinthefaceofthelami-natedironcoremagnetassembly.Intheenergisedstateavoltageisin-ducedintheshadingcoilbytheprimaryflux,andthisvoltagegeneratesacurrentintheshadingcoilwhichisphase-offset.Thiscurrentthengener-atesasecondaryfluxwhichisphase-offsetwithrespecttotheprimaryflux.Theadditionalsecondaryfluxproducesaquitedifferentresultantforcewhoseminimumvaluenolongerdropstozero.Fig.7.2.ForceasafunctionofplungerdisplacementforseveralvaluesofDCcur-rent 7.1Solenoids837.1.1SwitchingSolenoidsForon-offvalvestherearecoilsforeitherdirectcurrentDCoralternatingcurrentAC.TypicalvaluesaregiveninTable7.1.Therearelowpowerandhighpowercoils,e.g.foravoltageof24Vthepowercanbe1.15Wor4W.ACsolenoidsaregivenapowerratingwithtwovalues,e.g.4/2.5VAwhere4VAistheinrushpowerwhichlastsforafewmillisecondswhilethearmaturepullsinand2.5VAisthecontinuingholdingpower.Thenominalvaluesofforceandstrokeoftheplungerrangefrom0.6Nand0.25mmforaDCsolenoidwith1Wpowercon-sumptionand8.5Nand0.8mmforonewith11W.Therearespecificallypreferredapplicationsandoperatinglimitsduetothephysicallawsforthesetwotypes.OneessentialadvantageoftheDCsolenoidisthedelayed,gentlepick-upofthearmatureduetothedecliningcurrentriseandthesilentholdingfunctionofthesolenoidarmature.Sometypesofvalvespermittheexchangeofcoils.TheyaredesignedtoworkwithbothACandDCcoils.Acoilofanyvoltage,ACorDCofthesamepower,canbefittedorexchangedonthesamestem.However,lowandhighpowercoilscannotbeexchangedbecausetheinternaldesignofthevalve,i.e.orificediameterandspringstrength,mustmatchthecoilpower.Table7.1.VoltagesforsolenoidsDCAC12V24V50/60Hz24V48V50/60Hz110/120V50/60Hz220/240V50/60HzTable7.2.ComparisonofDCandACsolenoidsDCsolenoidACsolenoidQuieterDirectoperationwithlinevoltageLesswearofsolenoidcoreTendencytohumHighsolenoidholdingforceRiskofburn-outofthesolenoidcoilifSamepick-upandholdingpowersolenoidcoreisjammedSimplerdesign(Noshadingringre-FasterswitchingspeedthanDCsole-quiredtoavoidhumming)noids 847Electro-MechanicalConvertersTheavailablemagneticforceincreasesastheairgapbetweenironcir-cuitandplungerdecreases,bothforACandDCcoils.AnACsolenoidsystemhasalargermagneticforceavailableatagreaterstrokethanacomparableDCsolenoidsystem.OnACcoils,thecurrentisdeterminednotonlybytheDCresistanceofthewindingsbutalsobytheinductiveresistance(reactance)whichissig-nificantlyinfluencedbythepositionofthesolenoidcore.Ifthesolenoidcorehasdroppedout,theinductiveresistanceislowerandthecoilcurrentishigher.Thecurrentisthushigherinthepick-upphasethaninthehold-ingphase.Consequently,anACcoilshouldneverbeoperatedwithoutasolenoidcore.Otherwisethereisariskofthecoiloverheatingandburningoutafterafewminutes.Theinductiveresistanceandforcealsodependsonthemainsfrequency.Thereisalossofforceofabout10–30%dependingondesignandsizewhenoperatingacoildesignedfor50Hzat60Hz.Ifacoildesignedfor60Hzisoperatedat50Hz,thisresultsinanincreasedpowerconsumptionwithahighercoiltemperature.Thecoilcouldbedamagedundersuchop-eratingconditions.Theelectricpowerofacoilisconvertedvirtually100%tothermalen-ergy.Thismeansthatthecoilmayheatupconsiderably.Withcontinuousoperationandnormalambienttemperature(20°C)ofasolenoidvalve,temperaturesonthesurfaceof80–90°Corevenhighermaybeachieved.Atthemomentacoilisswitchedoff,thecollapsingmagneticfieldin-ducesacurrenttryingtokeepitenergised.Thisisseenashighnegativevoltageattheswitchthatcanproducearcingacrossswitchcontactsorde-stroythepowersupply.Iftheendsofthecoilwereconnectedatthemo-mentofswitchoff,theinducedcurrentwouldflowaroundthecoilatlowvoltagefadingtozeroinabout200ms.ForDCsystemsthisisachievedbyfittingadiodeacrossthecoil.Adiodeallowscurrenttoflowinonedirec-tiononlyandneedsjust1.5Vpotentialdifference.ForACcoilsadiodewillshortcircuitandthereforeavoltagedependentresistor(VDRorvaristor)shouldbeconnectedacrossthecoil.WhenthevoltageacrossaVDRisbelowagiventhreshold,thereishighresistancepreventingcurrentflow.Forvoltageabovethethresholdtheresistanceislowallowingcur-rentflow.CurrentisblockedwhenthecoilisenergisedasthethresholdoftheVDRisabovetheworkingvoltage.Onswitchoff,theinducedvoltagewillriseabovethethresholdandflowaroundthecoilandVDRuntilitfades. 7.1Solenoids857.1.2ProportionalSolenoidsProportionalsolenoidsareusedinpressureregulatorsandalsoinpropor-tionaldirectionalcontrolvalvestogenerateaforcethatisideallypropor-tionaltotheelectricDCcurrentandindependentofthestrokeofthear-mature.Toachievethis,aspecialdesignofthesolenoidcoreandironcircuitisusedwhichiscalledprofiledarmature.Ringswithatrapezoidalcross-sectionmadeofnon-magneticmetalsarecommonforsolenoidsforhydraulicvalves(Hong1986)wheretheentranceofoilunderhighpres-sureintothecoilhastobeavoided.AgroovecanbeusedforthesolenoidinapneumaticvalveasshowninFig.7.3.Unlikevoice-coilsystems,theproportionalsolenoidinFig.7.3hasnopermanentmagnet.Itsforceisthereforegeneratedinonlyonedirection,regardlessofcurrentdirection.Whileinthepastthedevelopmentofproportionalsolenoidswasbasedontrialanderrorandlaboratoryexperiments,itcannowbedoneonacomputerusingFEMsoftware(MacBain1985;Lequesne1987;Schultz2004).Thedifficultpointsarethenumerousnon-linearitiesinthemagneticcircuit,thebackelectromagneticforce,eddycurrentsandthemovingofthearmature.Fig.7.3.Cut-awayviewofaproportionalsolenoid 867Electro-MechanicalConverters7.1.3Pulse-WidthModulationFigure7.4givestheforceofaproportionalsolenoidasafunctionoftheelectriccurrentifastabiliseddirectcurrentamplifierisusedtoprovidethevoltage.Thehysteresiswhichistypicalofmagneticcircuitscanbeseenclearly.Theforcehysteresisiscausedbyfrictionandthemagnetichys-teresisoftheusedironmaterials.Atypicalvalueisabout4.5%(Hong1986).However,typicallypulsemodulatedsupplyvoltagesareusedthatcanbegeneratedbycheaperamplifierswhichalsohavebetterefficienciesthanstabilisedDCamplifiers.Thesimplestformofpulsemodulationispulse-widthmodulation(PWM)wherethesignalisswitchedoninequi-distanttimestepsandheldaslongasnecessarytoreachtherequiredmeanvalueoftheelectriccurrent.Thisapproachcanbeappliedbothtopressurecontrolvalvesanddirectionalcontrolvalves,seeChaps.14and12,re-spectively.Togeneratethesignal,acomparisonwithaperiodicsaw-toothsignalisused:thePWMsignalis“on”whenthecommandsignalislessthanthesaw-toothsignal,otherwiseitis“off”.Thepercentageofthecycleforwhichthesignalisoniscalleddutycy-cle“D”.Thedutycyclecanbefrom0(thesignalisalwaysoff)to100%(thesignalisconstantlyon).A50%“D”resultsinaperfectsquarewave.7060504030Force(N)201000100200300400500600700800Current(mA)Fig.7.4.Solenoidforceasafunctionofelectriccurrent(stabilisedDCsignal) 7.1Solenoids87Fig.7.5.Pulse-widthmodulationschemeEarlyhydraulicservo-valvesusedtorquemotorsthatrequiredonlysev-eralmA.Thislowcurrentcaneasilybesuppliedbyananalogueelectroniccircuit.However,researchersfoundthatthissmoothactuationcouldnotovercomehighstaticfrictionforces,andthevalvesthereforehadaconsid-erablehysteresis.Toimprovethebehaviour,ahigh-frequencysignalwasadded,e.g.asinewavewithafrequencyof100Hzandanamplitudeof10%ofthemaximumamplitude.Thisditherkeepsthemechanicalpartsinpermanentmotion,reducesthestaticfrictionforcesandthusthehysteresiswithoutaffectingtheloadthatcannotfollowduetoinertiaandcompressi-bilityofthefluid.Asimilareffectcanbenoticedforpneumaticvalvesdrivenbypulse-widthmodulatedamplifiers.Thepermanentswitchingofthesupplyvolt-agegeneratesaperiodicvaryingforce.Thiscanleadtoeithernoiseoranunacceptablebehaviourwithveryhighdisplacementsifthefrequencyisneartotheresonantfrequencyofthedrivensystem,e.g.aspring-poppetsystem.Ifahigherfrequencyischosen,theseexcitationskeepallme-chanicalpartsinpermanentmotionandthusavoidstickingfrictionandav-erageouthysteresis.However,ditheramplitudemustbelargeenoughandthefrequencyslowenoughtoenablethearmaturetorespondandyetsmallandfastenoughnottocausearesultingpulsationinthepneumaticoutput. 887Electro-MechanicalConvertersTheserequirementsaresometimesinconflict,andthegoalistoprovidejustenoughdithertoovercometheproblemwithoutcreatingotherissues.Thisditherisgeneratedasaby-productofthePWMprocessforlowfrequencyPWM,typicallylessthan400Hz.ThePWMfrequencyislowenoughthatthecurrenthastimetodecaybeforethenextrisetakesplace.However,theamountofditherchangesastheaveragecoilcurrentchanges;theditherismaximumat50%D,decreasingtozeroat0and100%D.Thiscanresultintoomuchditheratsomecurrentlevelsandnotenoughatothers.ThefrequencyforhighfrequencyPWMistypicallyabove5,000Hz;inapressureregulatorwefoundavalueof16kHzandthusbeyondthefre-quencyrangethatcanbeheardbyadults.ForhighPWMfrequencythecoilcurrentforallpracticalpurposeswillbeconstant.Noby-productditherwillbeproducedbyhighfrequencyPWM.Dithercanthenbegen-eratedseparatelyandsuperimposedontopoftheoutputcurrentaswithearlytorquemotors.Thisallowstoindependentlycontrolthecurrentlevel,aswellastheditherwaveform,frequencyandamplitudewhichcanbesettooptimisethefunctionoftheparticularapplication.Typicalvaluesarebetween70and350Hzfortheditherfrequencyandupto10%ofthemaximumcurrentfortheamplitude.Figures7.6and7.7showmeasurementsmadeonaproportionalsole-noidfromapressureregulator.ForseveralPWMfrequenciesanddutyratestheresultingmeancurrentsandforcesareshown:thefrequencyhasnosignificanteffectontheshapeofthecharacteristics;thedesignoftheelectronicamplifiercanhavemoreeffects(PrestandVaughan1987;Ka-jimaetal.1992).OthermodulationstrategiesthanPWMhavebeenstudiedinresearchprojects(Lü1992;Czinki2001).Iftheyarefine-tunedtotheparticularsolenoidandapplication,betterperformancecanbeachievedthanwithconventionalPWM.Figure7.8showstheresultingvoltagesandcurrentsofamoreelaboratemodulationschemeusedinacommercialpressurecontrolvalve:theon-phasesuse3.3kHzpulseswhilethemainswitchingfrequencyis115Hz.APWMsignalof115Hzcannotbeusedforthispressurecontrolvalvebecausethisfrequencyisneartothenaturalfre-quencyofthepoppet-spring-systemandwouldleadtopoorperformanceduetoresonanceandtonoise. 7.1Solenoids89700606005050040400Force(N)30Current(mA)3002020010100000204060801000200400600Dutyrate(%)Current(mA)Fig.7.6.MeancurrentasafunctionofFig.7.7.Meanforceasafunctiondutyrate;PWMfrequency200,500,ofmeancurrent,PWMfrequency1500,4000Hz200,500,1500,4000HzFigure7.9showstheoutputpressureoftheregulatorasafunctionofthesolenoidcurrent;bothforacontinuous,i.e.DC,voltageandamodulatedvoltageaccordingtoFigure7.8.Thepressurehysteresisismuchsmallerinthecaseofthemodulatedvoltagewhichjustifiesthecomplexityofthevalve’samplifier.Inadditiontothewaveformofthesupplyvoltage,thepositionofthearmatureandthevelocityofthemovementofthearmaturehaveconsider-ableeffects.Figure7.10showsthehysteresisoftheforcewhenthearma-turestrokeisfirstreducedandthenincreased.Figure7.11showsmeanvaluesoftheforceasafunctionofthestrokewhendifferentvaluesofthecurrentareused.Solenoidmanufacturersrecommendtouseonlyapartofthetotalstrokeasworkingrangetoreducethenon-linearities. 907Electro-MechanicalConverters2010Voltage(V)0400200Current(mA)115Hz3.3kHz00481216Time(ms)Fig.7.8.Voltageandcurrentasafunctionoftimeforpressurecontrolvalvesole-noid(meancurrent280mA)64modulatedDCPressure(bar)2000.20.40.60.8Current(A)Fig.7.9.OutputpressureofaregulatordrivenbyamodulatedoraDCsupplyvoltage,respectively 7.1Solenoids913020Force(N)10000.40.81.21.62.0Stroke(mm)Fig.7.10.Solenoidforceasafunctionofstroke,velocity20mm/min,stabiliseddirectcurrent300mA50Force(N)000.40.81.21.62Stroke(mm)Fig.7.11.Solenoidforceasafunctionofstroke,velocity20mm/min,parameterstabiliseddirectcurrentfrom100mAto700mAin100mAincrements 927Electro-MechanicalConvertersvoltageon800400Current(mA)0806040Force(N)20000.050.10.150.20.25Time(s)Fig.7.12.Stepresponseofaproportionalsolenoid,armaturefixedForvalvesusedinclosed-loopsystemsthedynamicresponseisimpor-tant.Figure7.12givesthestepresponseofaproportionalsolenoid.Att=0.05sthesolenoidisconnectedtotheDCpowersupply.Thenthecur-rentandforceatthearmaturebuildup.Att=0.175stheelectriccircuitisopened.Afterthatthecurrentandforcebreakdownrapidly.Theforcebuild-upcanberoughlymodelledbyafirstordersystemwithatimecon-stantofτ=20ms.The-3dBfrequencyofthismodelisaround8Hzwhichcorrespondstomeasurementsusingasmallinputsignalandavolt-agecontrolledamplifier.Thisfrequencyrangeistypicalofelectricallyop-eratedregulators.Usingacontrolledovervoltagetechniqueindrivingthesolenoid,considerableimprovementsintransientresponsearepossible.Detailedmathematicalmodelsofsolenoidsandamplifiersareusedforthedesignofthesecomponents(Sethson1993;KajimaandKawamura1995;VaughanandGamble1996)butnotforsimulationsofcompletepneumaticdrives.Thenumerousnon-linearitieswouldrequireadetailedknowledgeoftheparticulardesignandmaterialproperties;theinclusionofthebackelectromotoricforceduetothemotionofthearmaturewouldleadtoacloselycoupledelectro-mechanicalsystem.Thevalveactuationisthereforeusuallymodelledasaloworderlinearsystemdescribedbyanequivalenttimeconstant. 7.3PiezoelectricActuators937.2VoiceCoilandPlungerTypeSystemsToconvertelectricalenergyintoacousticalsignals,manyaudioloud-speakersuseatubularcoilofwiremovinginastrongradiallyorientedmagneticfield.Permanentmagnetsliningtheinsidediameterofaferro-magneticcylinderproducethefield.Whencurrentflowsthroughthecoil,itgeneratesanaxialforceonthecoilandproducesrelativemotionbe-tweenthefieldassemblyandthecoil.Duetoitsoriginthiskindofelectro-mechanicalconverterisoftencalledvoicecoilormovingcoil.Thecoilcanbedirectattachedtotheload,e.g.avalvespool,orcontrolanozzle-flappersystemviaaleverandaspringasshowninFig.19.4.Theforceonthecoilisdirectproportionaltothecurrentinthecoilandachangeinpolarityresultsalsoinachangeofforcedirection,makingvoicecoilsmoreattractivethansolenoidsforcontrolapplications.Leufgen(1992)measuredvaluesof4%forlinearityand2%forhysteresis.Thevoicecoilfordirectspoolactuationdesignedbyhimhasasizeof45*45*33mm,weighs350gandgeneratesamaximumforceof4.5Nfromamaximumcoilcurrentof500mA.Duetothelowresistanceandinductanceofthecoil,thedynamicresponseismuchbetterthanthoseofproportionalsolenoids.Aninversedesignisusedinsomecommercialdirectionalcontrolvalveswherethespoolisfixedandthepermanentmagnetisdirectattachedtothevalvespoolleadingtoasuckingcoilandaplungertypearmature,alsocalledmovingmagnetsystem.StationaryIronCoreDisplacementPermanentMagnetCoilFig.7.13.Schematicviewofavoicecoilsystem 947Electro-MechanicalConverters7.3PiezoelectricActuatorsThephenomenonthatapressureactingoncertainnonconductingcrystalsorceramicmaterials(likebariumtitanate)generatesanelectricchargeiscalledthepiezoelectriceffectanditisappliede.g.inpressuresensors.Theinversepiezoelectriceffectalsoexists:whenanelectricfieldorapotentialdifferenceisapplied,amechanicaldeformationisproduced.Thematerialexpandsinonedirectionandcontractsinotherdirections.However,theextensionisverysmall.Toincreasetheextensionandfacilitatetheuseoflowervoltages,twodesignalternativesareavailable:buildastackofseveralcrystalsforalin-earactuatororbondtwoormorelongnarrowstripstogetherasabenderandmountthemasacantilever,similartoabimetallicstripinathermo-stat,seeFig.7.14.7.3.1StackTranslatorsStacktranslatorshavebeenusedinseveralresearchprototypesofhy-draulicvalves.Themaindifficultyisthesmalldeflectiongeneratedbythepiezoelectricstackofabout1/1000ofthestacklengthrequiringamecha-nismorhydrostatictransformertoachievethenecessarydisplacementofthespool.Oneofthedesignchallengesistheeffectthattheforcede-creaseswiththeelongation:thehighestforcecanbegeneratedatthebeginofstroke,whileattheendofstrokenoforceisgenerated(AthertonandUchino1997;Kingetal.2000;Murrenhoff2002;Hantke2003).LiuandHiguchi(2001)reportforatypicallead-zirconate-titanate(PZT)actuatoranelongationof16µm/100V,amaximumforceof850N,anaturalfre-quencyof50kHzandacapacitanceof1.5µFforastackwiththedimen-sions5*5*20mm.7.3.2BendersInpneumaticvalves,wheretherequiredforcesaremuchsmallerthaninhydraulicvalves,bendershavebeenused,bothinexperimentalandcom-mercialvalves(IkebeandNakada1974;TaftandHerrick1981;AmsandOehrle1995;Yunetal.2005).Bimorphelementsconsistoftwothince-ramicsheetsbondedtogetheroronanintermediatemetallicplateorcar-bonfibretoimprovethestability.Thebendingresultsfromoppositepo- 7.3PiezoelectricActuators95larisationdirectionsofthetwopiezoelectricplatesandensuingexpansionofoneandcontractionoftheotherplate.Theworkingprincipleissimilartobimetals,butthedrivingforceisthepiezoelectriceffect.Displacementisgreaterthanthatofalineartranslator,butthedevelopedforcesaremuchsmaller.Somemanufacturersuseamulti-layertechnologytoobtainlargerstrokesandtoreducetherequiredvoltagefromover100Vtoonly20to40V.Formodelsandexperimentalresultssee(GoldfarbandCelanovic1997;Uchino1997;Belforteetal.1998,2002).Whilethedisplacementsofbendersaremuchhigherthanthoseoflinearstackactuators,theirmaximumforcesandresonancefrequenciesaremuchlower.ChoiandYoo(2004)reportforabimorphpiezoceramicflapperwithawidthof7.4mm,alengthof36mmandathicknessof1mmanaturalfrequencyof253Hzandadisplacementof200µmat300V.Völker(1999)givesforaslightlysmallerbimorphandamaximumvoltageof60Vadisplacementof±0.4mm,acapacityof1.4µFandamaximumforceof0.7N.FFFFig.7.14.Designalternativesforpiezoelectricactuators:linearstack,bimorphbender,multilayerbender.Thedeformationduetoanelectricfieldandthereac-tionforceisshownbythedashedlines.7.3.3PiezoelectricElementsinPneumaticValvesAnalternativetoelectro-magneticactuatorsarepiezoelectricactuators:theyhaveminutepowerdemand,highelectro-mechanicalefficiency,theycanbeextremelyfastinoperationand,withpracticallynomovingparts,areveryreliable.Inpneumaticvalvesbendersareusedthatdeflectindi-rectproportiontotheappliedvoltage,sothatthevalvemayeitherbeusedinasimpleon/offswitchingfunctionorinaproportionalmode.Inexplosiveenvironments,remotecontrolledorbatteryoperatedsys-temsenergyconsumptionisparticularlyimportantanddemandscarefulconsiderationwhenspecifyingcomponents.Withapowerratingofseveral 967Electro-MechanicalConvertersmWandnon-sparkingcharacteristic,piezoelectricvalvesareidealforus-ageinexplosiveenvironmentswherenosparkcaneverbetolerated.Thereisalsonoriskofoverheating,nomagneticfieldandnorequirementforthecircuitprotectionnormallyassociatedwithsolenoidvalves.Piezoelectricvalvescanbeactuateddirectbydatasignalsusingsimpletwo-wiretech-nologyandarecompatiblewithmosttypesoffieldbussystems.Switching,proportionalandpulse-widthmodulationcontrolmodesareallpossible.Thedisadvantagesofpiezoelectricvalvesaretheverylimitedflowareasthatcanbeopenedandconsequentlytheriskofcontamination.Thedi-ameteroftheflowareamaybearound0.3mmleadingtoverylowflowrates,typicallysomel/min(ANR)atupto8bar.Somemanufacturersofferthereforethepiezoelectricvalveaspilotvalveandintegrateitwithalargervalveinonehousing.Ortheyuseitasavariablenozzleinapressuredi-viderorinanozzle-flappersystemwhichactsasapilotstage,e.g.inapressureregulator(Paulsen1989).Someswitchingpiezoelectricvalvesareofferedwithintegratedelec-tronicsthatrequireonlyalowoperatingvoltage,e.g.6V.Thiskindofvalvecanbeusedinpositionersintheprocessindustryandcanbedrivendirectfromthe4–20mAdatasignal,withouttheneedforaseparatepowersupply.Thedrawbackofvalveswithintegratedelectronicscanbealongswitchingtime,upto760ms.Ontheotherhand,valveswithreactiontimesoftypicallylessthan2msandapowerratingaslowas0.014mWarealsoavailable.Proportionalpiezoelectricvalvesrequirehighersupplyvoltagesandhaveaconsiderablehysteresiswhichisduetocrystallinepolarisationef-fectsandmolecularfriction.Somemanufacturersspecifyhysteresisvaluesofupto15%.Hysteresiscanbereducedbyadedicatedamplifierorclosed-loopcontrol(Spanner2000;Belforteetal.2002).Figure7.15givesthemeasuredflowrateofaproportionalpiezoelectricvalvewith7barsupplypressureanddischargeintotheatmosphere.Thiskindofvalveshowsanimportantdifferencetosolenoiddrivenvalvesthathastobetakenintoaccountwhendesigningtheelectronicdrivecircuit:piezoelectricmaterialexhibitscharacteristicssimilartoaca-pacitor.Thereforeitdoesnotsufficetodisconnectthevalvefromthepowersupplytocloseanopenedvalve.Inadditionthetwoconnectionsofthepiezoelectricelementhavetobeshortcircuitedtoremovetheelectriccharge.Otherwisethevalvewillstayopen.Thisrequirementhasledtothedevelopmentofdedicatedamplifiers. 7.3PiezoelectricActuators974321MassFlowRate(l/minANR)001510250Voltage(V)Fig.7.15.Measuredflowratethroughaproportionalpiezoelectricvalvewithasupplypressureof7barasafunctionofcontrolvoltage 8CylindersCylindersconvertpneumaticenergytomechanicalwork.Theyusuallyconsistofamovableelementsuchasapistonandpistonrod,orplunger,operatingwithinacylindricalbore.Cylindersareoftendouble-sided,i.e.pressurisedaircanworkonbothsidesofthepistontoextendorretractit,andtheyhavemostlyasingle-endedpistonrod.AtypicaldesignisshowninFig.8.1.Thepistonrodiscasehardenedandchromeplatedwhilethebarrelismadeofstainlesssteelor–fortie-rodcylinders–ofanaluminiumprofile.Mostcylindershaveabandofmagneticmaterialaroundthecircumferenceofthepistonandarefittedwithanon-magneticcylinderbarrel.Themagneticfieldwilltravelwiththepistonasthepistonrodmovesinandout.Byplacingmagneticallyop-eratedswitchesontheoutsideofthebarrel,electroniccontrolofthepistonmovementwithaPLCispossible.Somemeansofstrokecushioning,i.e.gradualdecelerationofthepistonneartotheendofitsstroke,areprovidedbycushioningringsintheendpositionorelaboratepneumaticvalvesys-tems,seeChap.8.1.PistonwithSealsandMagnetCushionSpearPortCylinderCushionBarrelSleeveRodSealandWiperCushionSealPistonRodFig.8.1.Cut-awayviewofsinglerodcylinderaccordingtoISO6432andsymbol 1008CylindersThesymbolinFig.8.1showsbarrel,pistonandrod.Thearrowandthetworectanglesbesidethepistonsymbolisetheadjustablecushioning.Inthe1950s,thefirststandardswereissuedtofacilitateaninterchangebetweendifferentmanufacturers,e.g.intheUSAafteraconferenceinDe-troit1951asthe“JointIndustryConferenceStandardsforIndustrialEquipment”(StewartandJeffereis1955).Today,therearethreeISOstandardsthatcontainmountingdimensionsforcylinderswithpistondiametersfrom8to320mmandamaximumpressureof10bar.Cylinderscomplyingwiththesestandardscaninmostcasesbereplacedbythosefromothermanufacturers.Thesestandardsdonotincludetheattachmentofthemountingstothecylinderorthemount-ingofproximitysensors.Themountingsfromonemanufacturermaythereforenotfitwiththecylinderfromanothermanufacturer.However,evenifthedimensionsareidenticalthereremainalotofdif-ferencesbetweenmanufacturers:especiallytheinternaldesignmaybesig-nificantlydifferent.InFig.8.18thedeadvolumesof10differentcylinderswithpistondiametersof32mmand40mm,respectively,areshownwhenthestrokehastheminimumorthemaximumvalue:thesmallestdeadvol-umeisonly13%ofthelargestvalue.Ifthedynamicresponseofacylin-derhastobetakenintoaccount,thisdifferencecanbecomeimportant.Thestandardsalsocontainnospecificationsaboutpropertiesthatareimportantduringoperation,likefrictionorleakage.Wheninstallingacylinder,careshouldbetakenthatnounnecessarylat-eralforcesactonthepistonrod,e.g.byanunsupportedloadtothepistonrod.Themaximumpermissibletorque,i.e.theproductoflateralforceandstroke,dependsonthecylinderdesignandsize,e.g.0.8Nmforacylinderwith32mmboreor25Nmforacylinderwith250mmbore.Anexcessiveloadwillincreasethewearandcanleadtoanearlyendoftheservicelife.Ifpossible,self-aligningrodcouplingsshouldbeusedtominimisetheforcesfromradialandangularmisalignment.ThecylindershowninFig.8.1isadouble-actingsinglerodcylinderac-cordingtothestandardISO6432.Largertie-rodcylinderswithpistondi-ametersfrom32to320mmarestandardisedinISO15552.Thesecylin-derscanalsobeorderedwithadoublesidedrod,e.g.toachieveasymmetricaldriveortoallowahigherradialloadbecausetherodissup-portedatbothends.Sometimessingle-actingcylindersareusedwheretherodisextendedbyairandretractedbyaspringlocatedinthecylinder,orviceversa.Mostsingleactingcylindersareinthesmallboreandlightdutymodelranges.Typicalexamplesarestoppercylindersthatareusedtostopapalletonaconveyorbyextendingaroundbarorroller.Byapplyingpressure,thebarcanberetractedandthepalletwillmoveon. 8.1StrokeCushioning101Forspeedsofupto1m/s,thebasiclubricationprovidedinthefactoryissufficientandthecylindercanbeoperatedwithoil-freeair.Forspeedshigherthan1m/s,cylindersmayrequirelubricatedcompressedair.Theminimumvelocitywherethepistonmoveswithoutstickingandslippingisusuallynotspecifiedanddependsontheload.Anindicationthatstandardpneumaticcylindersarepoorlysuitedforlowspeedoperationisgivenbythespecificationsof“slowspeed”cylinderswhichstateminimumspeedsintherangefrom5mm/sto50mm/sforameter-outcircuitandhorizontalno-loadoperation.Standardsealscanbeusedfrom+2°Cto+80°C.Forlowertemperatureapplicationsdownto-20°C,sealsmadeofsoftlowtemperaturenitrileorPTFEcanbeused.Toavoidcondensationofwaterfromthecompressedairandsubsequentfreezing,thecompressedairhastobedriedtoadewpointoflessthantheambienttemperature.Otherwise,theiceinsidethecylindermaytearthesealsandblocksmallflowpaths.Forhightempera-tureapplicationscylindersfittedwith“Viton”sealsareavailablethatcanbeoperatedatambienttemperaturesofupto150°C.Animportanteffectthathastobeavoidedduringtheoperationofacylinderisbucklingoftherod.Topreventthis,themaximumpermissiblecompressiveaxialloadforapistonrodhastobecalculatedbyEq.(8.1).Thisvaluemaybelowerthanthevaluesuggestedbythemaximumper-missibleworkingpressureandthepistonarea,especiallyforcylinderswithaverylongstroke.Usingresultsfrommechanics,thepermissibleloadFKcanbecalculatedasafunctionofstrokeanddiameterofthecircularrod,π3⋅E⋅D4(8.1)F=Rod,K64⋅l2⋅SKwhereFKpermissiblebucklingforceinN,EmodulusofelasticityofrodinN/mm²,e.g.210,000N/mm²forsteel,DRodroddiameterinmm,lKeffectivelengthinmm,Ssafetyfactor,oftenusedvalueis5.ThevalueoftheeffectivelengthlKdependsonthemountingtypeandthestroke.Figure8.2givestheexampleofaswivel-mountedcylinder,whichisthemostcriticaltypeofmounting.Othercasesandtablescomputedforparticularcylindertypescanbefoundinmanufacturers’catalogues. 1028CylindershSwivelMountingLoadl≈2hKFig.8.2.Effectivelengthofrodforswivelmountingofpneumaticcylinder8.1StrokeCushioningPneumaticcylindersareverywellsuitedtorapidlyaccelerateamasstoahighspeed,e.g.1m/s.Attheendofthecylinderstrokethismasshastobedeceleratedgentlytopreventdamagefromtheloadandthemachine,andavoidexcessivenoise.Duetothetendencytouselightweightaluminiumprofilesevenforextendedassemblylines,thepermissibleimpactatstrokeendhasbecomesmallerthaninthepastwhenheavyironbeamswereused.Smalllightdutycylindershavelittlemassintheircomponentsandloadandthereforefixedcushioningwithshockabsorbingpadssetintotheendcoversisinmanycasesadequatetosolvetheproblem.Forlargercyl-inderswithmoremassindustrialshockabsorbersareoftenused.Arecentalternativeistheuseofelectroniccontrol(anon.2003).Forsomeapplicationsadjustablecushionedcylindersaresuitedthatcontainapneumaticdampingsystem1.Asshownbelow,thissystemcanonlyworkiftheloadandthevelocityduringthefinalpartofthestrokeremainthesameduringthecompleteservicelifebecauseachangeoftheseparametersputsthesystemoutoftune.Anadjustmentrequiresthoroughunderstandingofthedynamiceffectsandrequiressuitablemeasurementequipment,e.g.anaccelerationsensor.Thereareseveralcushiondesigns,buttheprincipleofoperationisthesame.Figure8.3showsatypewiththecushionsealsplacedintheendcovers.Theoperationisasfollows:thepistonismovingfastfromlefttorighttowardstherearendcover.Airisventingthroughthecentreofthe1Pneumaticdamperscanbefoundinanumberofdeviceslikecamerasormeas-urementinstruments.DesignandmodellingisdescribedbyGrassl(1981). 8.1StrokeCushioning103Fig.8.3.Principleofcushioningsealtheseal.Thisflowpathissuddenlyclosedwhenthecushionspearentersthecushionseal.Theexhaustingaircanonlyescapethroughamuchsmallerpathwhichisseverelyreducedbytheadjustablecushionscrewwithitstaperedrestrictor.Therapidlymovingpistonisdisplacingmoreairthancanexhaustpastthecushionscrew,sothebackpressurebuildsupanddeceleratesthepiston.Earlycylindersusedcushioningdeviceswithmetal-to-metalfitcon-sistingofamachinedcushionboreintheendcapandataperedsleeveorspearonthepiston,andanadditionalballnon-returnvalve(StewartandJeffereis1955:16-9to16-12).Today’scushionsealsperformadualroleofsealandnon-returnvalve,sealingontheinsidediameterandontherightfaceonly.Whenthespearenterstheseal,thisispushedtotherighttomakecontactagainsttheedgeoftheendcapthusblockingflowpastthespear,seeFig.8.4.a.Whentheendcapispressurisedforthereturnstroke,theairpushesthesealtocon-tacttheleftedge.Thenaircanflowaroundtheoutsideofthesealandthroughthegroovesontheleftsidetopressurisethepistonoverthefullareawithnormalstartingthrust,seeFig.8.4.b.Withoutthesegrooves,fullareapressurebuild-upwouldbeslowastheonlypathwouldbepastthecushionscrew.Inanidealcase,thecushioningissaidtobringthepistonandloadtoasmoothgentlehaltagainsttheendcover.Afirstassessmentwhetherthisisfeasiblecanbemadebyequatingthekineticenergyoftheloadtothecompressionworkinthecushioningchamber(Graham1965). 1048CylindersGrooveSealingEdgeFig.8.4.a.PrincipleofFig.8.4.b.PrincipleofFig.8.4.c.Cut-awaycushioningseal:block-cushioningseal:allowviewofacushioningingtheairflowfromtheairtoflowtothepistonsealwithgroovesandchamberwhilethepistonthatismovingleftsealingedgeismovingtotherightAssuminganadiabaticprocessandneglectingfrictionforces,thepressurep2attheendofthecompressionprocessisgivenbyγ(8.2)⊬m⋅v2·γ−1p2=p1∆∆1+(γ−1)⋅¸¸«2⋅p1⋅V1¹wherep2pressureatendofcushioninginPa,p1pressureatbeginningofcushioninginPa,γratioofspecificheatcapacities,mmassofloadinkg,vpistonspeedatbeginningofcushioninginm/s,3V1airvolumeatbeginningofcushioninginm.Thispressurep2mustbelessthanorequaltothemaximumpermissiblepressureforthecylinder.Theactualmaximumpressurewillbedifferentbecausetypicallyadrivingpressureintheotherchamberactsonthepistonasdofrictionforces,andairleavesthecushioningchamberthroughtheneedlevalve.However,theapproachinEq.(8.2)hasaseriousdrawback:itisastaticviewonlyanddoesnottakeintoaccountthatthecompressedairwillactasaspringleadingtopronouncedoscillations.Forinstance,ifthecushionscrewistooseverelysetthepistonbouncesbeforecompletingthestrokeornotcompletesthefinalpartofstrokeatall.Ifthecushionscrewisnotsetatall,thepistoncomestoastopbyhittingtheendcoverwithoutanypneumaticcushioning.Thisisnoisyandwillfatiguethepistonandendcovermaterial.Asanextensiveliteratureresearchshowedanumberof 8.1StrokeCushioning105simulationstudiesbutdidnotproduceaclosedtheoryorguidelineshowtosetthecushionscrew2,atestrigwasbuiltinthecontrollaboratoryinSoestandexperimentscarriedout(Voss2002;Beater2003).Figure8.5givesaschematicviewofthetestringandFig.8.6thelayoutofthepneumaticcircuit.Fig.8.5.Schematicviewofatestrig,consistingofcylinderwith63mmbore,mass,positionsensorandaccelerationsensor1ChargeValve542DirectionalControlValve3Volume4Load65CylinderwithCushionScrew6One-WayFlowControlValve27Silencer731Fig.8.6.Pneumaticcircuitofthetestrig2forexample(AdamsandBonnell1968;Backéetal.1974;Ballard1974;Jebar1977;Eschmann1994:120;anon.1996a) 1068CylindersTohaveconstantoperatingconditions,firsttheloadismovedthreetimesfromlefttorightandback,volume3chargedwith10barandthenvalve1closed.Valve2isoperatedandthepressures,theaccelerationandtheloadpositionarerecordedwhilethepistonextends.Fromthemeasureddatathemaximumaccelerationonimpactandtherequiredtimeforthecompletestrokeareobtainedandplottedasafunctionofthesettingofthecushionscrewandtheone-wayflowcontrolvalve6,seeFigs.8.7and8.8.Duetothelimitationoftheaccelerationsensor,valuesgreaterthan200m/s²cannotbemeasuredandtheplotsarethereforeclippedatthisvalue.TheinterestingobservationfromFigs.8.7and8.8isthatthereisaset-tingwherethestroketimeisshortandtheaccelerationonimpactlow.Thiscombinationismarkedwithanarrowinbothfigures.Theexplanationofwhathappensduringthecushioningiseasyifthere-sultsofadigitalsimulationareavailablethatshowthepistonvelocity.ThesimulationmodelisshowninFig.8.9.Oneofthemainreasonsfordigitalsimulationofpneumaticsystemsistheeasewithwhichparameterscanbechanged,eithermanuallyorprogrammed.Inthiscasethesettingofthecushionscrewwaschangedfromfullyopentopartiallyopen,fromasonicconductanceof100%to40%.200150100500AccelerationonImpact(m/s²)86674534One-WayFlowControl2012CushionScrew(rev)(rev)Fig.8.7.Maximumaccelerationonimpactasafunctionofthesettingofthecushionscrew(opening)andtheone-wayflowcontrolvalve(opening)inrevolu-tions.Favourabletuningismarkedwithanarrow. 8.1StrokeCushioning1071.81.410.6Time(s)0.224681032107654One-WayFlowControlCushionScrew(rev)(rev)Fig.8.8.Totalstroketimeasafunctionofthesettingofthecushionscrewandtheone-wayflowcontrolvalveinrevolutions.Favourabletuningismarkedwithanarrow.Fig.8.9.Digitalsimulationmodel,implementedwiththepneumaticslibraryPneuLibforthesimulationpackageDymola(modifiedscreenshot) 1088CylindersInFig.8.10thepistonvelocityduringthecushioningphaseisshown.Forafullyopenedcushionscrewthepistonstartsoscillatingwhenthecushionsleeveentersthecushionseal,butthevelocitydoesnotdropsig-nificantly.Asaconsequence,thepistoncollidesathighvelocitywiththeendcoverandthesimulationrunisautomaticallyterminated.Thesituationchangescompletelywhenthecushionscrewisclosed,i.e.ifthesoniccon-ductanceinthemodelhasavalueofonly50%,seeFig.8.10thirdplot.Asbefore,thepistonstartsoscillatingwhenthecushionsleeveentersthecushionseal.However,thisoscillationleadstoamuchlowervelocitywhenthepistonreachestheendcoverthaninthe100%case.TheconclusionfromFig8.10isthatthereisanoscillatingspring-masssystemwhenthecushionsleevehasenteredthecushionseal.Themassisgivenbythepistonmassandtheload,thespringisduetothecompressi-bilityoftheairtrappedbetweenpistonandthetaperedrestrictor.Ifthissystemistunedinsuchawaythatonimpactthevelocityislow,thenanacceptablecushioningcanbeachieved3.Alowimpactvelocitymeansinthiscaseavaluebelow5cm/s,ahigherspeedmaydamagethecylinder(Müller1998:153).Butthissystemisveryparametersensitive:ifthemasschanges,theeigenfrequencyofthissystemchangesandthereforethere-sponse.C_Cushion=100%0.2v(m/s)0C_Cushion=60%0.2v(m/s)0C_Cushion=50%0.2v(m/s)0C_Cushion=40%0.2v(m/s)00.70.740.780.820.86Time(s)Fig.8.10.Calculatedpistonvelocityvduringthecushioningphase.Avalueof100%forC_Cushioncorrespondstoafullyopenedcushionscrew3AsimilarapproachwaschosenbyKawakamietal.(1991)tosizeameter-outcircuit. 8.1StrokeCushioning109Theoscillatorymovementduringthecushioningphase,varyingac-cordingtocylindersizebetween10and30mm,cancauseproblemswhenelectroniccylinderswitchesandcontrollersareused.Dependingonthepositionoftheswitch,therecanbeasignalindicatingtheendofstrokethoughthepistonisactuallystilloscillatingandhasnotreachedthefinalposition.Whenthefastelectroniccontrolleractivatesthenthemovementofanothercylinder,acollisionmayoccur.Inthesecases,ashockabsorbershouldbeusedthatguaranteesagentlestopwithoutoscillations.InFig.8.11thekineticenergybeforeimpactisplottedasafunctionofthesettingoftheone-wayflowcontrolvalveandthecushionscrew.ThisfigurecorrespondstoFig.8.7butwasautomaticallygeneratedinthecom-puterwhileFig.8.7tookmanyhoursforthemeasurementsinthelab.Figure8.12showsthatcushioningdoesnotrequireaslowdrive,i.e.asignificantlyclosedone-wayflowcontrolvalve.Forthreedifferentset-tingsofthisvalve,whereasettingof40%correspondstoalowspeedand400%toahighspeed,asettingforthecushionscrewcanbefoundthatleadstoalowvalueofthekineticenergyonimpact.Figure8.13showsthesensitivitywithrespecttothepistonmass.Ifthemassischanged,adifferentsettingforthecushionscrewhastobefound.Thisisthereasonthatpneumaticcushioningcanonlyworkwellinsys-temsthathavealwaysthesamemassattheendofstroke.Figures8.14and8.15showcorrespondingresultsfrommeasurements.InFig.8.14theac-celerationonimpactisshownasafunctionofthesettingofthecushionscrewwhichwasopenedinstepsof1/8revolution.0.250.20.150.10.050KineticEnergy(Nm)0.50-814x103.531.5-92.52x101.512C_One-WayC_CushionFig.8.11.Calculatedkineticenergybeforeimpactasafunctionofthesoniccon-ductanceofthecushionscrewandtheone-wayflowcontrolvalve 1108Cylinders0.25C_Throttle=40%0.20C_Throttle=125%C_Throttle=400%0.150.1KineticEnergy(Nm)0.050-94x3.532.522.510ConductanceofCushionScrew(m³/s/Pa)Fig.8.12.Calculatedkineticenergybeforeimpactasafunctionoftheconduc-tanceofthecushionscrewforthreesettingsoftheone-wayflowcontrolvalve0.40.30.20.1030KineticEnergy(Nm)25201510Mass(kg)532.5-92x1001.51C_Cushion(m³/s/Pa)Fig.8.13.Kineticenergybeforeimpactasafunctionofthemassandthesonicconductanceofthecushionscrew 8.1StrokeCushioning11120016012080Acceleration(m/s²)40000.40.81.21.62CushionScrew(rev)Fig.8.14.Measuredaccelerationonimpactasafunctionofthesettingofthecushionscrew(opening)Figure8.15showsmeasuredaccelerationsforonecylinderwithamassof100%andareducedmassof78%.Thisdifferenceof22%leadstosignificantchangesandrequiresanothersettingofthecushionscrew.InFig.8.16fourcylindersofdifferentmanufacturersarecompared.Thoughthemountingdimensionsareidentical,thedynamicresponsevarieswidely.Whileanopeningof1.5revolutionsofthecushionscrewisopti-malforonecylinder,thevalueistoolowforanotherandtoohighforathird. 1128Cylinders1601601201208080Acceleration(m/s²)4040highermass0001230246CushionScrew(rev)CushionScrew(rev)Fig.8.15.EffectofamassreductionofFig.8.16.Comparisonoffourdif-22%ferentcylinders8.2MathematicalModelInordertosimulatethepistonmovementandchamberpressures,amathematicalmodelisneeded.Asaresultoflargechangesinpressureanddensity,thetemperatureoftheairalsochangesandshouldbetakenintoaccountforadetailedmodel.Ageneralmodelofavolumeofgasconsistsofthreeequations:theenergyequation,theconservationofmassequationandanequationofstate,e.g.theidealgasequationofstate.Alumpedparameterapproachwillbetakenassumingahomogeneousgastemperature4.Becauseofthelowheatcapacityoftheairandthehighheatcapacityofthesurroundingmaterialofthebarrelandrod,thetem-peratureofthemetallicpartscanberegardedasconstant.Thepressurechangerateisrathersmallcomparedwiththevelocityofsoundandthere-4Foraverylongcylinderwithastrokeof4.23madistributedparametermodelhasbeenproposedbyLinetal.(1996). 8.2MathematicalModel113forethepressureinthechamberisassumedtobeuniform.Kineticandpotentialenergytermswillbeneglected(Ballard1974:2.10–2.11).Thederivationofthemathematicalmodel5startswiththechangeoftheinternalenergyUoftheairinonechamberofthecylinder,givenbyd••(3.6)U=m⋅c⋅T+m⋅c⋅T.dtvvThechangeoftheinternalenergyisalsogivenbytheenergychangeof•theenteringandleavinggasmass,cp.Ti.m,themechanicalwork,p.dV/dt,andtheheatflowfromthegastothecylinderwall,h.a.∆T,whichleadstod•••(8.3)U=cp⋅Tin⋅min−cp⋅Tout⋅mout−p⋅V−h⋅a⋅∆T.dtCombiningEqs.(3.6)and(8.3)leadstothemathematicalmodelofthegastemperatureTinthechamber:••••p⋅A⋅vh⋅a⋅∆Tm⋅T+m⋅T=γ⋅T⋅min−γ⋅T⋅mout−−(8.4)inccvvwherem&=m&in−m&outnetairmassflowrateinkg/s,mgasmassinkg,TabsoluteairtemperatureinK,TinabsolutetemperatureofenteringairinK,γratioofspecificheatcapacities,pabsoluteairpressureinPa,2Apistonareainm,vpistonvelocityinm/s,cvspecificheatcapacityatconstantvolume,2hheattransfercoefficientinJ/(m⋅K),2aheattransfersurfaceareainm,∆TtemperaturedifferencebetweenairandbarrelinK.5Morethermodynamicallyinclinedreadersmightpreferforthefollowingderiva-tionthermodynamicpropertiesbut,asdelasHeras(2003)putsit,“inpracticeitisbettertoworkwithtemperaturesandpressuresinsteadofenthalpy”.Therearemorestrictderivationsofthecylindermodel,see(Maréetal.2000). 1148CylindersF,x,vm,Textininm,p,T,VmToutBarrelFig.8.17.Co-ordinatesystemforEqs.(8.4)–(8.6)Thisapproachneglectsthekineticenergyoftheairbecauseitistypi-callyverysmallcomparedtothethermalenergy,asbecomesapparentwhenconsideringthatataroomtemperatureof20°C,i.e.293K,thespe-5cificthermalenergyiscp⋅293K=2.94⋅10Nm/kgwhileforahighgas23velocityof100m/sthespecifickineticenergyisonlyw/2=5⋅10Nm/kgorlessthan2%ofthethermalenergy.InEq.(8.4)convectionisassumedasmodeofenergytransferbetweentheairinthechamberandtheinsideofthebarrel.AsimplemodelofthatisNewton’slawofcooling6whichstatesalinearrelationshipbetweentherateofheattransferandthetemperaturedifferential∆T,theheattransfercoefficienthandthesurfaceareaa.ThemovementofthepistonisdescribedbyNewton’ssecondlaw•M⋅v=A⋅p+Fext(8.5)whereMlumpedpistonmassinkg,vpistonvelocityinm/s,2Aeffectivepistonareainm,FextnetexternalforceinN.SolvingEq.(8.5)givesthepistonpositionandbymultiplicationwiththepistondiameterthechambervolumetowhichthedeadvolumehastobeaddedtoarriveattheeffectivevolumeV.Theadditionofthedead6ThismodellingapproachwasintroducedbyJebaretal.(1975);adetaileddiscus-sionisgiveninChap.8.3.3.Foranintroductiontoheattransferseeatextonthermodynamics,e.g.(Çengel1997). 8.2MathematicalModel115Fig.8.18.Relativedeadvolumesofdifferentcylinders,withrespecttothemaxi-mumvaluefoundvolume,i.e.thegasvolumewhenthepistonhascontactwiththeendcover,isimportantbecauseotherwiseaneffectivevolumeofzerowouldoccurduringsimulationswhenthepistonhasreachedanendposition.Un-fortunately,thedeadvolumeisnotgiveninthedatasheetsandcanvarysignificantlyfromonecylindertypetotheother.InFig.8.18thedeadvol-umeof10differentcylinderswithapistondiameterof32mmand40mm,respectively,isshownwhenthestrokehastheminimumorthemaximumvalue.Thesmallestdeadvolumeisonly13%ofthelargestvalue.Thegasmassinthechambercanbecomputedfromtheinitialgasmassm0andthenetflowrate,••(8.6)m(t)=³(min−mout)dt+m0.Thepressurepisgivenbytheidealgasequationofstate,Eq.(2.1),whereTisknownfromthesolutionofEq.(8.4),VcanbecalculatedbasedonEq.(8.5)andmfromEq.(8.6).Dependingonthepurposeofthemodel,therearevariations,especiallywithrespecttothegastemperatureinEq.(8.4).Insomecasestheassump-tionofanisothermalprocessisjustifiedandthenthemodelreducestop⋅V=m⋅R⋅T,(8.7) 1168Cylindersseee.g.Chap.18.1or(Németh2004:38)whereasasimplificationofacomplexmodelthegastemperaturedynamicsisneglected.AnalternativetothecomplexmodelofEq.(8.4)andthesimplemodelofEq.(8.7)istheassumptionofapolytropicprocess(Andersen1967;Sorlietal.1999).Ikeoetal.(1993)showthatthewaveformofpressureanddisplacementagreewellwithexperimentalresultswhenapolytropicindexofn=1.1isselected.However,thesteady-statetemperaturedifferedconsiderablyfromthemeasureddata.Otherstudiesshowthatforexpansionavalueofn=1.11andforcompressionavalueofn=1.18forthepolytropicindexleadstothebestagreementofmeasurementandsimulation(Bialas1973;Göttert2004:18–24).Anotherapproachisthedefinitionofathermaltimeconstantthatde-scribesthetemperaturebehaviourwithalinearfirstordermodel,seeChap.8.3.3.8.3CylinderParametersTwoparametersofacylinderareusuallyknown:thepistonorboredi-ameterandthemaximumstroke.Forstandardisedcylindersthediameterofthepistonrodandthemostrelevantmountingdimensionsaregivenintheparticularstandard.Butthereistypicallynoinformationavailableabouttheparametersandcharacteristicsneededformathematicalmodel-ling,e.g.friction,leakageorheattransfer.8.3.1SealFrictionWhenselectingormodellingacylinder,thedeliveredforceattherodhastobeknown.Itisgivenbythediametersofthepistonandtherod,thepressuresinbothchambersandfrictionforcesofthepistonandrodseals.Thegeometricaldataisusuallygiveninthedatasheetandthenominalpressurespecifiedduringthedesignofthedrive.Unknownarethefrictionforces.Theyoccurbecauseaslidingpistonsealhastopushoutwardsagainsttheslidingsurfacewithenoughforcetopreventcompressedairfromescapingfromthecylinder.Inanumberofcases,frictionforcesofthecylindersealsaremuchsmallerthantheforcesrequiredtomovetheloadortheforcegeneratedbythebackpressureofafastmovingpistonwhentheresistanceattheex-haustportishigh.Inthosecasesitismoreimportanttocalculatetheop- 8.3CylinderParameters117posingforcescarefullythantospendmuchtimeondeterminingthefric-tionforces.Buttherearealsosituationswhenthefrictionforceshavetobetakenintoaccountbecausetheydominatetheresponse(BownsandBallard1972;Jebaretal.1975).Unfortunately,thephenomenonfrictionisverycomplexandthereexistsnouniversalmodeltodescribeit.Manyap-proachesstillrefertotheworkbyStribeck(1903)whostudiedfrictionandrollerbearings.Anearlymathematicalmodelsuitablefornumericalanaly-sisanddigitalsimulationisgivenbyTustin(1947),Eq.(8.8).Thisequa-tioncanbeusedforpneumaticdrivesiftheinfluenceofthechamberpres-suresissmall,e.g.becausethefrictionforcesaredominatedbytheload.-fexp⋅vF=F+K⋅v+F⋅e(8.8)FrictionCoulombpropStribeckwhereFFrictiontotalfrictionforceinN,FCoulombconstantorCoulombfrictioninN,KpropvelocitydependentfrictioninN⋅m/s,FStribeckStribeckeffectinN,fexpcoefficientofexponentialdecayins/m.Therearealsodynamicfrictionmodels(Olsson1996;Göttert2004:38–48;Guentheretal.2006)thatcanleadtoabetterdescriptionofthiscom-plexprocess.Duetotheirincreasedcomplexity,theyrequirenoteasilydeterminedparametersandcanleadtoadifficulttosolvesimulationmodel(AbergerandOtter2002).Todescribetheeffectsofsealfrictioninpneumaticcylinders,differentmodescanbedefined:Staticfrictionforceorbreak-awayfrictionforcehastobeovercometobeginarelativemovementofthepiston.Sealsinherentlyneedtoexertaforceradiallyoutwardtomaintainaseal7.Thisforcegraduallysqueezesoutanylubricantsbetweenthesealandthebarrelwallandallowsthesealtosettleintothefinesurfacetexture.Afterthepistonhasbeenstandingforawhile,thepressurerequiredtostartmovementisthereforehigherthanitwouldbeifitismovedagainimmediatelyafterstopping.Likeallfrictionforcesinpneumaticcylinders,staticfrictionforcede-pendsonthepistondiameterandthepressuresactingontheseals.Andonthetypeofsealsandlubricantusedbythemanufacturerwhomayofferdifferentversionsofacylinderforstandardapplications,lowvelocityap-plicationsorwithreducedfriction.7Forsmallcylindersseallesspistonsmightbeanalternative(Belforteetal.2005). 1188CylindersThereareconsiderabledifferencesbetweenthebreak-awaypressuresofastandardcylinderandthelowfrictionvariant.Dependingonthesizeandthemanufacturer,theminimumoperatingpressurecanbeaslowas0.1–0.2barforalowfrictiontypewhileitmaybeabove0.5barforstandardcylinders.Themaximumspecifiedleakageforalowfrictionvariantismuchhigherthanthoseforstandardcylinders,e.g.0.5l/min(ANR)forpistondiametersof20to40mm,andtheadmissiblevelocitymuchlower.Animportantobservationisthefactthatthevaluesheavilydependonthedwelltime,i.e.thetimebetweentwomovementsofthepiston.Arakietal.(1982)foundthat“maximumstaticfrictionforceincreasedapproxi-matelyinproportiontothelogarithmofthedwelltimewhichwaslessthanonehour”.Andthefrictionforcesarealsoinfluencedbythetemperature,boththeambienttemperatureandthegastemperaturesinthechamberswhichdependontheloadcycle(Pasieka1991).AllthisshowsthattherearemanyfactorsthatinfluencethefrictionforcesinpneumaticcylindersandcautiousscientistsstopherebecauseapreciseandrepeatablemodellingisbetweenverycomplexandimpossibleandonlyagreewithBelforteetal.(1989):“Allofthecurvesobtaineddemonstratethatthefrictionforcealwaysgrowswhenthepressureinoneofthetwochambersincreases,whenthetranslationspeedofthepistonin-creases,andwhenthediameterofthecylinderundertestisincreased.”Itcouldbeadded“andwhenlubricationgetspoorer”followingresultsfromRaparellietal.(1997)whofoundasixfoldincreaseofthefrictioncoeffi-cientofapistonsealwhenrunningitdryinsteadoflubricated.Allfol-lowingmodelsormeasurementsareonlyvalidforthespecifiedsystemcondition:abitoflubricationcanchangeeverything!Dynamicfrictionforceoccurswhenthepistonismovingarbitrarily.Todescribesuchamovementandtocalculatefrictionforcesfromsuchameasurement,accuratevaluesofthepistonaccelerationareneededwhicharenoteasilyobtained.Therefore,typicallysteady-statefrictionforcesarerecordedwherethepistonmoveswithconstantvelocity.Thereareseveralwaystomodelfrictionforceinpneumaticcylinders.Thesimplestwayistoignorethemcompletelyandassumeidealcondi-tions.Fleischer(1995:72)givesasaruleofthumbthefollowingequationsforstaticanddynamicfriction:N(8.9)F=0.67⋅dstaticmmwhereFstaticstaticfrictionforceinN,dcylinderboreinmm, 8.3CylinderParameters119N(8.10)F=0.4⋅ddynamicmmwhereFdynamicdynamicfrictionforceinN.SomemanufacturersdefinetheefficiencyηofacylinderbyFRod(8.11)η=p⋅APistonandsuggesttouseavaluebetween0.85and0.90forηwhichmaybelessforverysmallborecylindersandmoreforverylargeones.Forsomecyl-indersgraphsareavailablethatgivetheefficiencyasafunctionofpres-sure,seee.g.Fig.8.19.Ifagoodmodelofthefrictionisimportant,theactualvalueshavetobemeasuredtoformatablefortheefficiencyasafunctionofbothchamberpressuresandpistonvelocity.Therehasbeenconsiderableeffortspenttofindmathematicalmodelsofthefrictionforcesinpneumaticcylinders.SchroederandSingh(1993)givealistofmodelsandthecoefficientofdeterminationasameasureof“thegoodness-of-fit”oftheirmeasureddataandtheequations,seeTable8.1.WiththeexceptionofmodelA,theyincludethepressuredifferencebetweenthecylinderchambers,∆p.ForaslightlymodifiedversionofmodelD,Belforteetal.(1989)givecoefficientsforcylinderswithpistondiametersrangingfrom32to100mm.However,thesesmodelsarenotverywellsuitedforthedigital100ø40ø209080ø8ø127060Efficiency(%)5013579WorkingPressure(bar)Fig.8.19.Efficiencyasafunctionofpressureandpistondiameter(anon.2000e) 1208CylindersTable8.1.EmpiricalfrictionmodelsstudiedbySchroederandSingh(1993)ModelDynamicfrictionmodelCoefficientofnumberdeterminationAFf=Fo+c⋅v0.90BF=F⋅e−ξv+c⋅v+β⋅p0.91fo∆CFf=Fo+c⋅v+β⋅∆p0.93D(n)()Ff=Fo+1+c⋅v⋅β⋅∆p+β1⋅p10.94EFFcvnpmf=o+⋅+β⋅∆0.91FFFe−ξvcvnpm0.95f=o⋅+⋅+β⋅∆GF=F⋅e−ξv+c⋅v+β⋅∆p+β⋅p0.93fo22simulationoftransientresponsesbecausetheyarevalidonlyforvelocitiesabovethecriticalspeed,inBelforte’spapergreaterthanorequalto0.1m/s.AndthismeansthatFoinTable8.1representsanasymptoticorbiasvalue,notthe“break-away”frictionforce.Ballard(1974:2-13)usesthefollowingmodelforhisdigitalsimulation:Ff=Fc+CA⋅v+CB⋅∆p,(8.12)whereFffrictionforceinN,FcconstantfrictioninN,vpistonvelocityinm/s,CAandCBparameters,∆ppressuredifferentialacrossthepiston.AmorerefinedmodelsuitedfordynamicsimulationsisgivenbyEsch-mann(1992):2(8.13)F−F⊬v·°1+statmin∆1−¸forv≤v°F−F∆v¸critmin0«crit¹°°α(v)=®Fref−Fminv°1+−1forv≥v,vvcrit°refcrit()F−F−1°min0v°¯critDPiston»Fmin−F0º(8.14)FFriction(v,∆p)=…F0+∆p⋅α(v)»Dref¬∆pref¼whereα(v)auxiliaryfunction, 8.3CylinderParameters121FstatstaticfrictionforceinNforv=0and∆pref,seeFig.8.20,FminminimumfrictionforceinNfor∆pref,seeFig.8.20,F0frictionforceinNforv=0and∆p=0,seeFig.8.20,FreffrictionforceinNforv=vrefand∆pref,Dpistonpistondiametermm,Drefreferencediameterinmm,∆ppressuredifferentialinPa,vcritcriticalspeedinm/s,seeFig.8.20.Hestatesthatthereisanalmostlinearrelationshipbetweenthepistondi-ameterandthefrictionforce,asalreadyBelforteetal.(1989)hadfound:“Ascanbenoted,apracticallylinearincreaseofthefrictionforceisreg-isteredwiththeboringofthecylinders,whichconfirmstheconclusionsofotherauthors,whoconsidertherelationoftheproportionbetweenfrictionandboringtobevalid”.Theabovegivenphysicalmodelsrequiredetailedknowledgeofthecylinderandload,likemassofthemovingparts,diameterorfrictionforces.Analternativeisadata-drivenmodelfromShoukatChoudhuryetal.(2005)forprocesscontrolvalvesthatusestwoparameterswhichcanbedeterminedfrommeasuredplantdata.FrefFstatFrictionForce(N)∆p=6bar=∆prefFmin∆p=0barF0vvcritrefVelocity(m/s)Fig.8.20.FrictioncharacteristicsaccordingtoEqs.(8.13)and(8.14).Therefer-encecondition,∆prefandvref,canbearbitrarilychosentogiveFref. 1228Cylinders300200FrictionForce(N)100000.040.080.120.16Velocity(m/s)Fig.8.21.ModelledfrictionforcesaccordingtoEqs.(8.13)and(8.14)fordiffer-entialpressuresof0,2,4,6,8and10bar,respectively8.3.2CylinderLeakageWhenmodellingacylinder,leakagebetweenthechambersandtotheat-mospherehastobeconsidered.Datasheetsdotypicallynotcontaininfor-mationaboutthiseffectbecausemostcustomersdonotrequirethisinfor-mation.Andduringourlaboratoryworkwefindthattheleakageofmostcylindersissosmallthatwecannotdetectit.Inordertodoso,weconnectapressuresensortothecylinderchamber,pressurisethechamberandthenclosethevalvetotheairsupply.Butafterseveralhoursorevendaysthereadingofthesensorhasnotchanged.ThestandardISO10099(anon.2001a)definesatestprocedureandamaximumleakrate.Aftercyclingafewtimes,airat1.5andthen6.3barisappliedatthefrontortherearport,respectively.Theadmissibleleakratesrangefrom0.6l/h(ANR)foracylinderboreof8mmto5l/h(ANR)foracylinderboreof320mm.Forcylinderswithnon-roundpistonrodsnoval-uesaregiven,buttheleakratecanbesignificantlyhigher;especiallyatlowpressure.Thesameistrueforcylindersforlow-speedorlow-frictionapplications.Excessiveleakageduetowearusuallyendstheservicelifeofacylin-der.Aswithfriction,thisisacomplexmechanismandanumberofpa- 8.3CylinderParameters123rametersareresponsibleforthewear.Manufacturerstypicallydonotgiveinformationaboutthistopicintheirdatasheets.ThestandardCNOMOE06.22.115.Nrequiresadistanceof3,000kmpistontraveldistanceastheminimumpermissiblelimit,anumberthatisalsogivenbysalesrepresen-tatives(e.g.anon.2004b).DuringtestsdescribedbyBelforteetal.(1999)allcylindersexceededthisdistancebyalargemargin.Forcylinderswith8metalsealsalongservicelifeofupto10,000kmor10fullcyclesisad-vertised.Fleischer(1995:264)describestestsofcylinderswitha1inchrod(25.4mm)whichafter50,000cyclesundersomereasonablesideloadinginamoderatelycleanfactoryenvironmenthadthesameamountofleakageasapinholewith1.14mmdiameter.Thisleakageoccurredprimarilyaroundtherodbushingandiscomparabletoa0.013mmradialclearancearoundtherod.Whendesigningasystem,theeffectsofthisleakageshouldbein-cludedbecauseitleadstoahigherdemandflowduringpressurisationoftherodendchamberandalowerbackpressureduringextension.Whenusingmeter-outcontrol,thecylinderspeedduringextensioncanthereforeincrease.8.3.3CoefficientofHeatTransferThereisnodoubtthatthetemperatureinacylinderchamberchangeswhenthepistonismoving,chargingordischargingtakesplace.However,theactualtaskdetermineshowmucheffortshouldbespentondetailedmod-elling.Foranin-depthanalysisofheattransferseeBackéandOhlig-schläger(1989).Ohligschläger(1990:60–65)pointsoutthatheattransferleadstoadampingofthepistonmovementandhecalculatesadampingcoefficientof(only)0.092duetothiseffect.However,ifallotherdampingforcesaresmallthemodellingofthiseffectmaymakeadifference.AcriticalviewisgivenbyKawakamietal.(1988)whostudydifferentmod-elsandendwith:“Fromtheseresults,itcanbeconcludedthatundernor-malcylinderusageconditions(whenthepressurechangeisrelativelysmall)heattransferneednotbeconsidered,andasimpledynamicmodelassuminganisothermalchangecanbeemployedforanaccurateapproxi-mation.”ThisisconfirmedbyGöttert(2004:18–24)whocalculatesachangeofthemassflowrateofonly3.5%ifthetemperaturedeviates20°Cfromroomtemperature.Earlyresearchersstudiedthefillingandemptyingofvesselsandusedfirsttheassumptionofanadiabaticoranisothermalprocessandlaterofapolytropicprocess(Bialas1973;Ballard1974;Grassl1981).However,Jebaretal.(1975)statethat“thepolytropicequationisvalidforsystems 1248Cylinderswithfixedmassbutitcanbeshownthatwithconstantpolytropicindexnitisinvalidinvariablemasssystems”.Thisledtoseveralproposalsforavariablepolytropicindex.Mostofthesestudieslookedatvesselswithaconstantvolume.Theirresultscannotbeusedforthemodellingofacyl-inderbecausethevolumeofacylinderchamberisnotconstantbutvariesaccordingtothedisplacementandbecausethepistonmovementstirsthegas.KagawaandShimizu(1988)conductedanexperimentwheretheyusedavesselinwhichasmallfanwasinstalledtodisturbthethermalboundarylayeroftheairinthechamber.Ithadaconsiderableinfluenceontheheattransfer;amovingpistonmightbesimilar(seealsoBurrowsandPeckham1977).Thecharginganddischargingprocessesofairinsideanactuatingcylin-derwerestudiedexperimentallyforaworkcyclewhichraisedandloweredamassbyAl-IbrahimandOtis(1992).Theairtemperatureduringcharg-inganddischargingprocessesvariedconsiderably;theprocesseswereneverisothermaloradiabatic.Theyfoundthatduringthechargingprocessthetemperaturehistoryfollowedcloselytotheadiabaticprocesswhiletheadiabaticprocessoverestimatedthetemperaturehistoryduringthedis-chargingprocess.Inthiscasethemeasuredairtemperaturefollowedmostcloselytotheisothermalprocess.Amorefundamentalapproachtothemodellingofheattransferwasin-troducedbyJebaretal.(1975)who,basedonanapproachtakenbyRey-noldsandKays(1958),modelleditassimilartoheatconductioninasolidbyusingNewton’slawofcooling.Settingtheparameterfortheheattransfertoh=0inthemodelequation(8.4)givesanadiabaticprocess;settinghtoahighvaluegivesanisother-malprocess.Thevalueofhdependsonanumberofparameters,e.g.loadandpistonspeedandchangesduringaloadcycle.Detaileddiscussionsare2givenbyPasieka(1991)whogiveslimitsof5W/(m⋅K)0,themassflowratefrominlet1ofthestatortoinlete,m&from_1,isgivenby•p1⋅ω⋅Vdisp,nom(11.16)mfrom_1=.2⋅π⋅R⋅T0Forthemassflowratem&to_2,itfollowsaccordingly:•pe⋅ω⋅ε⋅Vdisp,nom(11.17)mto_2=.2⋅π⋅R⋅T0Themassflowratem&to_ecanbecalculatedfromthefactthatthenetmassflowmustbeequaltozero:•••(11.18)mto_2+mto_e=mfrom_1.Thismathematicalmodeloftheidealmotorlacksimportantfeaturesthatareneededfortime-domainsimulations,likeconsiderableleakageflows,forexamplebetweenthevanesandthestator,inertiaormechanicalfriction.Therefore,additionalflowpathswithnozzles,inertiaandbearingfrictionhavetobeaddedtothisidealmotor,seeFig.11.13fornozzles.Tofindparametervaluesforthenozzlesandthefrictionmodel,anumericalestimationschemecanbeused(Beater2004). 11.1VaneMotors163.p..pm.mto_vol_22mto_2mfrom_11to_vol_1ωvolume2volume1..mto_2mfrom_1volumeepeto_e.mFig.11.12.MassflowintheidealmotorFlowpathwithnozzletomodelleakageIdealmotorpp21port2port1ω..mmto_2from_1peporteFig.11.13.SimulationmodelconsistingofidealmotorandleakageflowpathsThemathematicalmodelcanbeusedtoanalysethelossesinthevanemotorasafunctionofspeed,Fig.11.14.Therearemechanicallossesduetobearingfrictionandfrictionbetweenthevanesandthecylinder.Thisfrictioncouldbereducedbyusinglubricatedairwhich,however,cancauseenvironmentalproblems.Therearealsolossesthroughleakageinthemotor.Thebiggestpartarethelossesintheductsleadingtoandfromtheworkingcompartments.Theselossescouldbereducedsignificantlyby 16411AirMotorsandAirTurbinesductswithagreaterdiameter.Inthatcasesomespeedcontrollerwouldbeneededtolimitthemaximumspeedofthemotorwhichwouldotherwiseoverspeedanddestroyitself.Thisisduetothefactthatpneumaticsystemsarefedbyapressuresourcethatcanprovidealmostunlimitedmassflow.Hydrostaticmotorsaresuppliedbyaflowsourceandthereforedonotfacethisproblem.Pneumaticvanemotorswithintegratedcentrifugalcontrol-lersaredescribedintheliterature(Hansson1975;Sbahi1992)andareusedforsafetyreasonsingrindingmachinesbutareusuallynotofferedforstandardapplications.Exhaust8060DuctLosses40LeakagePower(%)Mech.Losses20ShaftPower04000800012000RotationalSpeed(rpm)Fig.11.14.Shaftpowerandlossesasafunctionofangularvelocity11.1.3SpeedControlVanemotorsareusuallyselectedinsuchawaythattheyoffertherequiredtorqueattherequiredspeedforthegivensupplypressure.Oftenmechani-calgearsareusedtoreducethehighspeedandtypicallythemanufacturersofferawiderangeofintegratedgearboxes.Ifnecessary,thespeedcanbereducedbyloweringthesupplypressure,anozzleattheinletoroutletport,oranelectro-pneumaticcontrolsystem.Figure11.15showstheeffectsofreducingthesupplypressure.Theinputmassflowrateandthemaximumspeedgodown.Thisalsolimitsthestalltorqueandreducesthestartingtorque.Section11.1.3.1showstheeffectsofthrottlingand11.1.3.2givesresultsforanelectro-pneumaticcontrolsystem. 11.1VaneMotors16542MassFlowRate(g/s)01351200080004000n(rpm)0135PressureatPort1(bar)Fig.11.15.Characteristicsofflowrateandfreespeed;pressureatport1varied11.1.3.1ControlbyThrottlingOnewaytoreducethespeedisbythrottlingattheinletoroutlet.Usuallythisisdoneatport1,althoughitcanbefittedatport2.Thefollowingfig-uresshowtheeffectonthesteady-statespeedandthetorque-versus-speedcharacteristicsforthrottlingattheinputusingthemotormodelfromSect.11.1.2.120008000n(rpm)40000-800.40.81.21.6x10ConductanceofInletNozzle(m³/s/Pa)Fig.11.16.Steady-statespeedasafunctionofconductanceofinletnozzle 16611AirMotorsandAirTurbines80increasingconductance60(lessthrottling)40Torque(%)20020406080RotationalSpeed(%)Fig.11.17.Characteristicoftorqueversusspeedfordifferentinletnozzles11.1.3.2Electro-PneumaticControlLoopAmuchmoreprecisespeedcontrolcanbeachievedwithanelectro-pneumaticcontrolsystem.Intheliteraturetherearesomereportsaboutelectronically-controlledairmotorsbuttypicallyinpositioncontrolloopsforrobotapplications.Ioannidis(1987)andScholzandSchabbel(1987)usemodifiedvanemotorsandelectro-pneumaticservovalves.Noritsuguetal.(1987)andTokhietal.(2001)useradialpistonairmotors,electro-pneumaticdirectionalvalvesandPID-controllers.Takemuraetal.(2000)useaslidingmodecontrollerandfindsignificantsteady-stateerrorsandtrackingerrorsandthereforeproposeanadditionalelectricdrivetobuildahybridpneumatic/electricmotor.ThesysteminthecontrollabinSoestconsistsofanelectro-pneumaticpressurecontrolvalve,thevanemotorwithelectricspeedpick-upandadigitalPI-controller.Theelectro-pneumaticpressurecontrolvalvewaschosenforthisstudybecauseitismuchlessexpensivethanahigh-quality4/3-waydirectionalcontrolvalvewhichcancostmuchmorethanthevanemotoritself.Forthedesignofthecontrolloopthefrequencyresponseofthemotorwascalculatedfromthemathematicalmodelandcouldbedescribedastwofirstordersystemsinseries.Theimplementationofthedesignedcon-trollershowed,however,thatthedynamicresponseoftheusedvalvecan-notbeneglected.ThisprompteddetailedstudiesofelectricallycontrolledpressureregulatorswhicharedescribedinChap.14.Theyshowthatal-mostallofthesevalvesareratherslowandnotintendedtobeusedwithrapidlychanginginputsignals.Therefore,thefrequencyresponseofthe 11.1VaneMotors167valveandthemotorwasmeasuredandusedforthedesignofaPI-controller.Theopen-loopfrequencyresponseisgiveninFig.11.18.Figures11.19and11.20showtheresponseoftheclosed-loopsystem.Fig.11.19givestheresponsestoastepinputofthereferencesignalwhileFig.11.20showsthereactiontoaloaddisturbance.Theelectro-pneumaticspeedcontrolhastheunexpectedside-effectthattheminimumspeedofthemotorwithoutloadisreducedfrom2,900rpmwithoutcontrolto365rpmwithcontrol.200Gain(dB)-20-90-180Phase(°)-270-101101010ω(rad/s)Fig.11.18.Frequencyresponseofopenloopsystem,i.e.pressurecontrolvalveandmotor4000n(rpm)02p(bar)00.2ValveInput000.511.52Time(s)Fig.11.19.Stepresponseofcontrolledmotor:rotationalvelocity,pressureatmotorinletandcontrolvalveinputsignal,normalisedfrom0to1;stepappliedatt=1s 16811AirMotorsandAirTurbines60002000n(rpm)4p(bar)00.40.2ValveInput000.511.52Time(s)Fig.11.20.Disturbancerejectionofcontrolledsystem:rotationalvelocity,pres-sureatmotorinletandcontrolvalveinputsignal,normalisedfrom0to1;distur-banceappliedatt=0.5s11.2AirTurbinesWhilepneumaticmotorsworkdiscontinuouslybecausetheychargeanddischargeworkingcompartmentsorcylinders,airturbinesuseacurrentofsteam,waterorairuponbladesofawheeltosteadilyrotateashaft.Theirtheoreticalanalysisiscoveredbythedisciplineofturbomachinesandre-quiresabroaderbasisinthermodynamicsandfluiddynamicsthangiveninthefirstchaptersofthisbook,see(Csanady1964;Falkmann1975c;Dixon1978;Balje1981;Kentschke2004:66-80).Inoperation,low-velocity,high-pressuregasexpandsthroughanozzle,creatingalow-pressure,high-velocitygasjet.Thejetimpingesonatur-binewheel,producingtorque.Generally,theturbinespinsinonlyonedi-rection.Theirefficiencycanbeintherangefrom65to75%whilevanemotorstypicallyhaveonly25to35%(Graham1981).Dependingonthenatureoftheflowpaththroughthepassagesoftherotor,turbomachinesarecategorisedasaxialflowturbomachineswhenthepathofthethrough-flowiswhollyormainlyparalleltotheaxisofrota-tion.Or,whenthepathofthethrough-flowiswhollyormainlyinaplaneperpendiculartotherotationaxis,thedeviceistermedaradialflowtur-bomachine.Therearealsomixedflowturbomachineswherebothradial 11.2AirTurbines169andaxialvelocitycomponentsofthethrough-flowattherotoroutletarepresent.Airturbinesaretypicallyusedformachinesrequiringalowlevelofpower,butinwhichthehighspeedofrotationcanbeutilizedwithoutre-ductiongears,e.g.forsmallgrindingmachines,becauseduetothehighspeedthedesignofthesegearsposesanumberofproblemslikenoiseorhighmassforces.Airturbinescanhaveaverygoodpower-to-weightratioandrunatveryhighspeeds.Theformerpropertycanbeseenathand-heldgrinders:atur-bine-poweredtooldelivers2kWandweighs1.7kg,doublethepowerofacomparablepneumatictoolwithavanemotor.Ahighfrequencyelectrictoolprovides0.385kW,auniversalelectricgrinderonly0.265kWperkg.Alargerturbine-poweredgrinderevenoffers4.5kWwhilethetoolmassisonly4.0kg(Tormen1998).Thesegrindersneedaflowrateof15l/sper1kWshaftpower,comparedwith22l/kWsforthevanemotortype.Torunthegrindingdiskattheoptimalspeed,thereareintegratedmechanicalcentrifugalcontrollersthatalsoactassafetydevicestopreventover-speeding.Theabilitytorunatveryhighspeedsisthereasonthatturbinesareusedindentalhandpieces.TheBelgiandentistEmilHuet(1874–1944)isre-gardedtohavebeenthefirsttoobservethatathigherspeedsthetoolsworkedmoresmoothlyandwithlessdiscomforttothepatient(DysonandDarvell1993).In1945theAustralianJohnWalsh(nowSirJohn)discov-eredthatatafrequencyof256Hzthemaximumunpleasantnessofvibra-tionsensationoccurredwhileat1024Hznovibrationswerenoticeable(Cherryetal.1974).Thefirstcommerciallyavailablehandpiecewithaturbineinthehead,runningupto300,000rpm,wasintroducedin1957.Theballbearingsusedinthoseearlyairturbineswerenoisy,requiredoilmistlubricationandhadashortservicelife.Intheearly1960s,theywerereplacedbyairbearingswhichdominatedforthenextdecade.InalatersurveybyDysonandDarvell(1999a,1999b)almostallhandpiecesusedballbearings.Theyfoundsixtoninerotorbladesfortheradialflowtur-bines,rotordiametersintherange3.5–5.6mm,recommendedsupplypres-suresfrom1.86to2.7bar.Theymeasuredstalltorquesfrom0.7to1.5mNm,outputpowerintherangeof5–10Wandefficienciesaround25%,increasingwithhighersupplypressure.Inadditiontotheuseashigh-speedgrindingtools,dentalhandpiecesorasmeasurementdevices,therearenotmanyapplicationsofairturbinesre-ported.Someareusedasstartersforreciprocatingengines(Graham1981;BeckerandMueller1993)orauxiliarydrivesinaircraft(UpchurchandVu2000). 17011AirMotorsandAirTurbinesRotorBearingRotorBearingBurFig.11.21.Schematicviewofarotorandcut-awayviewofadentalturbine 12DirectionalControlValvesDirectionalcontrolvalvesareprimarilyusedtocontrolthedirectionofflowbetweenthecomponentsofapneumaticcircuit.Duetotheirinternalresistancetheyalsothrottletheairflow,aneffectthatisusuallynotwel-come.Thereareseveralwaystodistinguishbetweendirectionalcontrolvalves:-numberofports,-numberofswitchingpositionsorinternalstablestates,-internaldesign,e.g.spool,poppetordiaphragm,-typeofoperation,e.g.electrical,pneumaticalormanual.Directionalcontrolvalvesareoftencharacterisedbytwonumbers:thenumberofmainportsfollowedbythenumberofswitchingpositions.Theworkingportsareusuallylabelled2and4,theexhaustports3and5andthesupply1;thecontrolportsofpneumaticallyoperatedvalvesarenotcounted.ThevalveinFig.12.2has5portsand2distinctpositionsofthespoolandisthereforecalleda5/2-waydirectionalcontrolvalve.SometypicalvalvesandtheirsymbolsareshowninTable12.1.Theswitchingpositionsarerepresentedbysquares.Internalconnectionsbetweenportsforagivenpositionareindicatedbyanarroworarrowswithinthesquare.Incircuitdiagramsvalvesarealwaysshownintheneutralposition,i.e.valvenotactuated.For2/2-wayand3/2-wayvalvesitisimportanttodistinguishbetweenthosethatareopenfromthesupplytotheworkingportwhennooperatingsignalisappliedandthosethatareclosed.Thisinformationisoftenap-pendedtothevalvedescriptionas2/2-wayvalveNCfornormallyclosedor3/2-wayvalveNOfornormallyopened.Thereareseveralconfigurationsforthemiddlepositionof5/3-wayvalves.Allportsmaybeclosed,asinTable12.1,ortheworkingportsmaybeconnectedtoeachotherandthepressuresupplyortotheexhausts.Forcylinderdrives,5/2-wayvalvesareoftenused.Thetypeofactua-tion,monostableorbistable,dependsontheapplicationandtherequiredbehaviourofthedriveduringandafteranemergencyshut-down.Ifthedriveistoremaininthestatebeforetheemergency,bistablevalvesare 17212DirectionalControlValvesused.Theyrequireanelectricalsignaltochangethepositionofthespoolwhichremainsotherwiseunchangedduetofrictionforces.Thisisatypicalsituationforhorizontaldrives.Inassemblyapplicationsverticaldrivesarerequiredtomoveupwardstocleartheworkingspacewhenanemergencyoccurstoavoidcollisionswithpartsthatareoutofcontrol.Inthiscase,monostablevalvesareusedthatrequireanelectricalsignaltomovethedrivedownwards.Theoperatingsignalcanbeaforceappliedthroughalever,apressurefromanothervalveoranelectricvoltage.Thissignalactseitherdirectonthemainstageorwiththehelpofanadditionalvalve,calledpilotvalve.Thisvalveisapoweramplifierandreducestherequiredenergyofthecommandsignal,e.g.theforceofamanuallyoperatedvalveortheelectriccurrent.Inthemainstagetheflowcontrolisoftendonebyaslidingspoolthatopensorblockspassagesbetweentheports.Otherdesignsusepoppetsordiaphragms,especiallyforvalveswithtwopositionsandtwoorthreeports.Table12.1.TypicalconfigurationsofdirectionalcontrolvalvesSymbolName/Description22/2-wayvalveNC,directelectricallyoperated123/2-wayvalveNC,electricallyoperated,withpilot13valve23/2-wayvalveNO,electricallyoperated13424/2-wayvalve,mechanicallyoperated13425/2-wayvalve,electricallyoperated,monostable513425/2-wayvalve,electricallyoperated,bistable5134214125/3-wayvalve,pneumaticallyoperated513 12.1DesignofDirectionalControlValves17312.1DesignofDirectionalControlValvesAtypicaldesignofapilotvalveisshowninFig.12.1.Itisa3/2-waydi-rectionalcontrolvalvethatconnectstheworkingport2eithertothecom-pressor1ortheexhaust3.Ifthecoilofthisvalveisnotenergised,themainspringpushesthemoveablearmaturetotheright,thepoppetclosesthepathfrom1to2andopensthepathfrom2to3.Ifthecoilisenergised,thearmatureisdrawntotheleft,thepoppetopensapathfrom1to2andclosesthepathfrom2to3.Thiskindofvalve,apoppetvalve,hasalowflowresistancewhenopenandgivesagoodsealwhenclosed.Itisrobustanddoesnotrequirelubri-cationbecauseithasnoclosetolerancegapsormovingparts.However,itrequiresalargeoperatingforcebecauseoftheinherentpressureunbalanceonthepoppetwhichinpracticelimitsitsusetosolenoidorpilotoperation.Figure12.2givesacut-awayviewofa5/2-waydirectionalcontrolvalve.Ontherightsidethepilotvalveisshownthathasasocketfortheelectricconnectorandabuttonforthemanualoperationofthevalve.Thispilotvalvesallowsairtoflowtothepistonsideofthespoolwherepres-surebuildsupandsubsequently-eitherdirectorbytheuseofapiston-aforcethatmovesthespool.Thenotshownpilotvalveontheleftsideworksinthesameway.Theremayalsobespringstoguaranteeacertainpositionwhennopilotvalveisactivated.Thespoolopensflowpathsfromthecompressortothecylinderandthesilencersandhasinthiscasetwodistinctpositions.Fig.12.1.Schematicviewofapilotvalveas3/2-waydirectionalcontrolvalve,di-rectoperated,normallyclosed 17412DirectionalControlValvesManualOverridePort4SocketPort2PilotSupplyPistonPilotValveSealPort1SpoolFig.12.2.Cutawayviewofa5/2-wayvalve,leftpilotvalveremovedTherearedifferentdesignsforthespool.Sometimessteelspoolsinsteelhousingsorsleevesareusedwhichofferlowfrictionandrequirenoelas-tomericseals.Thisdesignisalsocalledglandlessorlappedspoolvalvebecausespoolandsleevearelappedtogiveclearanceoflessthan2µm(PinchesandCallear1999:66).Becauseoftheairflowthroughtheannulibetweenspoolandhousing,thesevalvesarenotcompletelyleak-freeandcontaminationoftheairsupplymayleadtostickingofthespoolwhenthevalveisnotoperatedforaperiodoftime.However,theirservicelifeiswithmorethan200millionswitchingcyclesmuchlongerthanthatofvalvesusingelastomericseals.Thedefinitionofcyclelifevariesfromonemanufacturertoanother,butfailureisnowadaysdefinedasunacceptableleakage,e.g.18to60l/h(ANR),andnotanylongerasfailuretofunction(Metzner1993:183;HallerandLatio1996).Valveswithelastomericsealsareguaranteedforonly50millionswitchingcyclesbutarecompletelyleak-freeandhavealowerinternalresistanceandthereforeahigherflow.ThevalveinFig.12.2hastheelastomericdiscsealsmountedonthespool.Thereareotherdesignswherethesealsaremountedinthehousing.Mostspoolvalvesaredesignedwithapositiveoverlap.Whenthespoolchangesfromonepositiontoanother,onesetofportsareclosedbeforetheotherareopened.Thisavoidsa“shortcircuit”betweenthecompressorportandtheexhaustports.Amainadvantageofspoolvalvesisthefactthattheforcesonthespoolarebalancedwheninoneofitsstablepositionsandthusrequireonlysmallactuatingforcesthatarenotpressuredepend-ent.Theforcesmaybeunbalancedwhenthespoolmovesfromoneposi-tiontoanother.Thehighfrictionforcesofthesealsusuallyrequireapilotvalve.Directoperationofthespoolbyasolenoidwouldrequireanover- 12.2OperationofDirectionalControlValves175sizedsolenoidwhichiscostlyandrequiresahighelectriccurrentthatpro-ducesalargeamountofheat.Thesevalvesrequirethereforeaspecifiedminimumsupplypressuretooperatethepilotvalve;anelectricvoltageconnectedtothesolenoidalonecannotoperatethevalve.12.2OperationofDirectionalControlValvesThesteady-stateoperationofadirectionalcontrolvalveissimple:ifoper-ated,theportsareconnectedaccordingtothevalvesymbol.Forsimpletasksthisinformationissufficient.Whenusingadriveinatime-criticalapplicationorbuildingadynamicvalvemodel,thetransientsbetweentheswitchingpositionshavetobetakenintoaccount.Figure12.3givesthestepresponseofthecurrentthroughaDCsolenoidwhenthearmatureisfixed.Att=5msthesupplyvoltageof24Visswitchedon.Asdescribedbyafirstorderdifferentialequation,thecurrentirisesaccordingto⊬t·i=imax⋅∆1−eτ¸∆¸«¹(12.1)whereicurrentinA,imaxmaximumcurrentinA,ttimeins,τtimeconstantins.24SupplyVoltage(V)020015010050Current(mA)00102030Time(ms)Fig.12.3.StepresponseofcurrentthroughDCsolenoid,armaturefixed 17612DirectionalControlValvesStartofEndofMovementMovement200150100Armaturemoving50Current(mA)Armaturefixed00102030Time(ms)Fig.12.4.Stepresponseofsolenoidcurrent,armaturemovingThecurrentinducesaforceinthearmature.Ifthisforceisgreaterthantheopposingspringforceandfriction,thearmaturemoves.Thismove-mentinducesanadditionalvoltageinthecoilwhichchangesthecurrentcurve.AmeasurementisshowninFig.12.4andcomparedwithasolenoidwherethearmatureisfixed.Atthemomentwhenthearmaturestartstomovethetwocurvesseparate.Whenthearmaturehasreacheditsfinalpo-sition,thecurrentincreasesagain.Thistypicalresponseofthecurrentcanbeusedtodetectthemovementofthearmatureandhasevenbeenimple-mentedinelectronicvalvecontrollersforpowersaving(reducevoltageifarmaturehasbeenpulledin)orsafetychecks.Thisresponsecurvecanalsobeusedfortheidentificationoftheproc-essesinavalve:noadditionalsensorsareneededwhichwouldbedifficulttoapplywithoutconsiderablychangingthedynamicsofthevalve.Onlythemeasurementoftheelectriccurrentthroughthesolenoidandthepres-sureattheblockedoutputportareneeded.Figure12.5showsanexperimentalset-up.Att=20msastepinputisappliedtoaswitchthatconnectsthesolenoidtoapowersupply.Thecur-rentincreasesasinFig.12.4.Afterthearmatureandwithitthepoppethasreachedthefinalposition,thecurrentincreasestothesteady-statevalue.ThelowerpartofFig.12.5showsthemeasuredpressureattheblockedport2.Thepoppethastomovebeforeanyaircanflowfromthesupplyport1throughthevalvetoport2.Oncethepoppethasstartedmoving,thepressurebuildsuprapidly,dependingonthesupplypressure,theresis-tanceofthevalveandthevolumeconnectedatport2.Inthiscase,ittakes15mstoreachthesteady-statevalueofthepressure. 12.2OperationofDirectionalControlValves1771024VCurrent5Current(mA)0Pressure42Pressure(bar)001020304050Time(ms)Fig.12.5.Solenoidcurrentandpressureatport2whenoperatinga3/2-wayvalveatt=20msandset-upforswitchingtimemeasurement2015105Current(mA)08642Pressure(bar)00102030405060Time(ms)Fig.12.6.Solenoidcurrentandpressureatport2whenoperatingapiloted5/2-wayvalveatt=20msThistimecanbemuchlongerifthevalveisbiggerbecausemoreairisneededtofilltheinternalvolumesofthevalve,thefittingandthepressuresensor.Andifthevalveispilotoperated,ittakesevenlongerbecausethepilotvalvehastoopenfirst.Theresultingairflowwillbuildupapressurethatmovesthemainspooltoopenapathfromthesupplyporttotheworkingport.Fig.12.6showsthemeasurementofsolenoidcurrentand 17812DirectionalControlValvespressureatthevalveportofapilotoperatedvalvewithanominalflowrateof750l/min(ANR).Theelectriccurrentincreasesuntilthearmaturemovesandthenafterthefinalpositionisreacheditrisestoitssteady-statevalue.Thenittakesmorethan15msforthepressureatthepistonendofthespooltobuildup,exceedthespringforce,movethespool,openapathtotheworkingportandincreasethepressureatthatport.Thesteady-statevalueofthepressureisreachedforthefirsttimeafterlessthan5msafterthebeginningofthespoolmovement,amuchshorterdurationthanthetimeneededtomovethemainstageatall.Theoscillationsaftert=40msaretypicalofthisset-up.Iftheoperatingvoltageisswitchedoff,thepoppetorthespoolmovebacktotheneutralpositionand-forthetwovalvesstudied-connecttheworkingporttotheexhaustport.AsFig.12.7shows,ittakessometimeforthepressuretofalltoatmosphericpressure.Theactualdepressurisationtakeslongerforthesmallpilotvalvethanforthebigger5/2-wayvalvebe-causetheresistanceofthepoppetismuchgreaterthanthatofthespoolwithrespecttothevolumeconnectedattheworkingport.Typically,directionalcontrolvalvesworkinabinarymode:theirflowpathiseithercompletelyopenorcompletelyclosed.However,usingapulse-modulatedvoltage,directactingvalvescanbeusedasnozzleswithvariableconductance.Thesimplestmodulationschemeisapulse-widthmodulationwithaconstantrate,seeChap.7.1.3.6420642Pressure(bar)Pressure(bar)00102030405060Time(ms)Fig.12.7.Pressureatport2whenclosinga3/2-wayvalve(top)or5/2-wayvalve(bottom)att=20ms,respectively 12.2OperationofDirectionalControlValves179-9x1030Hz1.650Hz70Hz100Hz150Hz1.2200Hz300Hz0.830Hz0.4SonicConductance(kg/s/Pa)100Hz300Hz0060801002040RelativeDuty(%)Fig.12.8.SonicconductanceofasmallpoppetvalveasafunctionofrelativedutyfordifferentmodulationfrequenciesandaPWMschemewithaminimumon-timeof0.24msTheeffectiveconductanceofthevalveisthengivenbytherelativeduty,i.e.theratiobetweenthetimethevoltageisonandthetimethevolt-ageisoff.Figure12.8showstheconductanceofasmallpoppetvalvewhentherelativedutyisvariedforsevendifferentmodulationfrequenciesandwhenaminimumon-timeof0.24msisused.For30Hztheconduc-tancechangesalmostlinearlywiththerelativeduty,whilefor300Hzthecurveisalmostflatupto75Hzandbecomesverysteepupto85Hz.ThisbehaviourcanbeexplainedwithFig.12.9whichshowsthecurrentasafunctionoftimeforfourrelativeduties.Whentheon-timeisshort,themaximumforceonthearmatureisnothighenoughtomovethepoppetandopentheflowpath.Thisisthecaseforarelativedutyof15%inFig.12.9.Forthisoperatingcondition,noairflowsthroughthevalve,butthecoilheatsup.ThismeansforPWMschemesthatthemodulatingfrequencyshouldbeselectedsuchthatevenforsmallvaluesoftherelativedutythevalveopenstemporarily.Increas-ingtherelativedutyto25%inFig.12.8leadstoamovementofthepop-petandashortopeningofthevalve.Anyrelativedutythatishigherkeepsthevalveopenforalongertime. 18012DirectionalControlValves1.275%50%25%15%0.8Current(A)0.4002468Time(ms)Fig.12.9.CurrentasafunctionoftimefordifferentrelativedutiesofapurePWMschemeWhiletherelativedutyinfluencesthemeanvalueofthesonicconduc-tance,thegeneralshapeofthecharacteristicflowratevspressureratiore-mainsalmostunchanged,seeFig.12.10.100%6075%50%25%4020FlowRate(l/min(ANR))00.10.30.50.70.91p/p21Fig.12.10.Flowratevspressureratiofordifferentrelativeduties 12.3SimulationModelofDirectionalControlValves181HighfrequencyPWMisnotsuitedforthiskindofvalvewithswitchingcoils.Thismodulationschemecancontrolthesolenoidcurrent,butduetothehighlydisplacementdependentsolenoidforcethereisnolinearde-pendencyofthepoppetpositiononthedutyandoftenalsonolinearrela-tionbetweenpositionandconductance.Thiscanresultinunstableopera-tionandaverysmallrangeofthedutywheretheopeningofthevalvecanbeinfluenced.Duetothedelaysinpilotoperatedvalves,lowfrequencyPWMshouldonlybeusedfordirectactingvalves,e.g.poppetvalves.However,thenumberofswitchingcyclesandthereforethewearofthesealingelementsismuchhigherthaninasimpleon-offsituationandthevalveselectionhastoreflectthis.12.3SimulationModelofDirectionalControlValvesAdetailedsimulationmodelofadirectionalcontrolvalvehastoincludethesolenoid,themechanicalpartsofthefirstand-incaseofpilotoperatedvalves-secondstage,e.g.massesandfriction,theinternalvolumesandtheresistancecharacteristicsoftheopenedflowpaths.Thesedataareverydifficulttomeasureandusuallynotavailable.Asimplermodelisthereforeproposedthatneedslessparameters.ThestructureisgiveninFig.12.11.ForfurthermodelsseePawlakandNehl(1988)orMäkinenetal.(1993).Forthemodelfourparametersareneeded:thetwovaluesforthenozzlemodelandtwotimeconstants.Ifmeasurementsareavailable,thetimeconstantscanbeestimatedbycomparingtheresponseofthemodelwiththatoftherealvalve.However,inmostcasestheseparametershavetobeestimatedfromthemanufacturer'sdatasheet.Thetimeconstantofthefirstfirstordersystemcanbecalculatedfromtheshiftingtime.TheshiftingtimeaccordingtoISO12238isdefinedastherequiredtimefortheworkingpressureoftheblockedporttoreach10%ofthesupplypressure(anon.2001b).2ValveInputPilotMainSpoolSignalPressureStagePositionResistance1Fig.12.11.Simulationdiagramofa2/2-wayvalve 18212DirectionalControlValvesFigures12.5and12.6indicatethatthisisapproximatelythetimeneededtooperatethesolenoidandmovethemainstage.ThesecondfirstordersysteminthesimulationdiagramFig.12.11describesthetimeneededtofullyopenthemainstageanditisusuallymuchfasterthanthefirstone.Thetwovaluesforthenozzlemodel,criticalpressureratiobandsonicconductanceC,arediscussedinChap.5.Forpoppetvalves,theoreticalmodelsaregivenbyAndersen(1967:77–88),includingtheeffectiveareaandforcesactingonthepoppet.However,inasystemsimulationcaremustbetakenthatthesevaluesbandCincludeallrelevantlossesoftheconnectorsandmountingsofthevalve.Thesepartsmayreducethemaxi-mumflowrateconsiderablyandmaynotbeincludedinthemanufacturer'sdata.Fortypicalvalvesitisnotnecessarytoincludeinthemodeltransientfloweffectslikepropagationandreflectionofpressurewavesorvortexdevelopments(Leonhardetal.2006).Thereasonisthatthesetransientsaremuchshorterthantheswitchingtime.Thedeviationduetotheseef-fectsaregivenas14%fortheflowrateand4%fortheactuatorforce.FormostsimulationsthemodelinFig.12.11iswellsuitedbecauseitdescribestheresistanceofthecompletelyopenedandclosedvalveandthetimedelaybetweentheelectriccommandsignalandthepressurebuild-upattheport,seeFig.12.12.Butthemodelalsohasitslimitations.Onlythesteady-statebehaviourofdirectionalcontrolvalvesisdefinedinthedatasheets.Thereisnoinformationaboutthetransitionbetweentwospoolpo-sitions,especiallyasthesolenoidsaredesignedtohavetwostableposi-tions.Inthesimulationmodelthedynamicbehaviourofthespoolisde-scribedbyalineartime-invariantfirstordersystem,therestrictionisdescribedasanozzlewhosesonicconductanceCdependslinearlyonthespoolposition.Buttheseareassumptionsthatmaynotbevalidforapar-ticularvalve.Forasmallspoolvalvewithmetal-to-metalsealFig.12.13showsthattheopeningtakesplaceonthelast10%ofthetotalspooldis-placementonly.Ifadefinedresistanceatintermediatespoolpositionsisrequiredbytheapplication,aproportionalvalveshouldbeused.Thesimulationmodelisalsooflimiteduseiftheelectricinputsignalchangeswithahighfrequencycomparedtothecornerfrequencyofthefirstordersystem.Thiscanhappenifadirectionalcontrolvalveisusedinaclosed-looppositioncontrolloopusinganon-offcontroller;especiallyiftheelectricpowersupplydoesnotconsistofaconstantvoltagesourcewiththeratedsupplyvoltagebutusessomeelaboratecurrentorpowercontrolschemewithahighersupplyvoltageforpartsoftheswitchingpro-cess. 12.3SimulationModelofDirectionalControlValves18364simulatedPressure(bar)2measured0864Pressure(bar)2001020304050Time(ms)Fig.12.12.Comparisonsofmeasuredandsimulatedpressureresponse:directactingpoppetvalve(top)andtwostagevalvewithmodelofshortlinebetweenvalveandpressuresensor(bottom)5432MassFlowRate1032.510.820.6SpoolDisplacement1.50.40.2p/p121Fig.12.13.Massflowratethrougha3/2-waydirectionalspoolvalveasafunctionofspoolpositionandpressureratio 13Shut-OffValvesAshut-offvalveisdefinedinISO5598asa“valvewhosemainfunctionistopreventflow”.Examplesarenon-returnvalves,shuttlevalvesortwinpressurevalves.Inmostcircuitstheycanbetreatedassimpleswitchesthatworkinbi-narymode:eithercompletelyopenorcompletelyclosed.Incriticalappli-cationstheinternaldesignhastobetakenintoaccountandleakage,over-orunderlapandinertiahavetobeconsidered.Anexampleisanon-returnvalvewhenthesupplyandloadpressurearealmostidentical:asituationthatcanleadtosevereoscillations,seeFig.13.5.Intheserarecasesaverydetailedmathematicalmodelhastobebuilt,usingtheequationsfromthefirstpartofthisbook.Fordigitalsimulationstheinertiaofthemovingelementsshouldbein-cluded.Inmostcasesasimplefirst-ordermodelissufficient.Thisalsoprovidesasmoothtransitionofsystemvariableswhicheasestheburdenforintegrationalgorithmswithvariablestepsize.13.1Non-ReturnValvesNon-returnvalvesorcheckvalvesallowtheflowofairinonedirectionalmostunrestrictedwhileblockingitintheoppositedirection.Asonemanufacturerstates,“freeflowinonedirection;fullyleak-proofshutoffintheother”.AtypicaldesignisshowninFig.13.1,thesymbolforapneu-maticcircuitdiagraminFig.13.2.Inadditiontoballsotherelementslikecones,diaphragmsordisksareused. 18613Shut-OffValves21Fig.13.1.Cut-awayviewofanon-returnFig.13.2.Symbolofanon-returnvalve,partiallyopenvalveAsimplemathematicalmodelofanon-returnvalvedescribesthestaticrelationshipoftheflowrateasafunctionoftheup-anddownstreampres-sures.InFig.13.3themeasuredmassflowrateasafunctionofthepres-sureratiop2/p1isshown.Thischaracteristicisverysimilartothatofanozzle.ThedifferencebetweenthemeasureddataandthecomputedmassflowrateforanozzleisgiveninFig.13.4.Thisdifferenceisdividedbythemaximummassflowrateandgiveninpercentasrelativeerror.Withtheexceptionofapressureratioofalmostunitytheerrorisverysmall:anozzleisavalidmodelofanopennon-returnvalve.Thissimplemodelisusuallysufficientunlessthespringforceinthenon-returnvalveissignificant.Inthatcasethecharacteristiclooksdiffer-entforapressureratioofaboutunity.Theforceduetotheupstreampres-surep1mustbegreaterthanthosefromthedownstreampressurep2andthespringforce.Forthenozzlemodeltheupstreampressurep1hastobere-ducedbythevalueoftheopeningpressure,oftenbetweenalmost0and0.5bar.74523Errorin%MassFlowRate010.20.40.60.810.20.40.60.81p/pp/p2121Fig.13.3.MeasuredmassflowrateFig.13.4.Modellingerrorwhenusingasafunctionofpressureratiop2/p1nozzleasmodelofnon-returnvalve 13.2Non-ReturnValveswithOverride1877.56.52p(bar)5.54.50481216Time(ms)Fig.13.5.Measureddownstreampressure(theresistanceofthenozzleissohighthatp1isalmostequaltop2)andexperimentset-upHowever,itispossiblethatoscillationsoccurwhichcannotbeex-plainedbythismodel.Figure13.5.ashowsthemeasureddownstreampres-sureforthecircuitinFig.13.5.b.Severeoscillationsarepresentthatcanbeheardandmayreducetheservicelifeofthecomponent.Theyareduetothelowlydampedspring-masssystemandthesmalldifferencebetweenupstreamanddownstreampressurethatdoesnotallowastablepositionoftheball.Thissituationcanoccurinapistonpumpwhenthepistonextendsandthecompressedairinthecylinderhasapressureonlyslightlyhigherthanthatonthedownstreamsideofthenon-returnvalve.Wemeasuredaswitchingfrequencyaround15Hzonthehighpressuresideofapressureboosterwhichhasadesignlikeasingle-pistonpump. 18813Shut-OffValves13.2Non-ReturnValveswithOverrideNon-returnvalveswithoverridecombineinparallelanon-returnvalveanda2/2-waydirectionalcontrolvalvethatcanbypassthenon-returnfunction.Thebypassisusuallyoperatedpneumaticallyatport12.Therequiredpressuresto“switchon”orto“reset”thebypassdependontheoperatingpressureandareshownforaparticularvalveinFig.13.6.Non-returnvalveswithoverridecanbeusedasasafetydevicetoholdacylinderinpositionevenifthereisaleakageinthesupplyline.Thevalveshouldbeconnecteddirecttothecylinderportinordertoprotectagainstanypipingfaultsbetweencylinderanddirectionalvalve.Thuscompressedaircanflowfreelyintotherespectivecylinderchamberwhileabackflowispreventeduntilthepilotport12ofthevalveispressurisedthusinter-ruptingthenon-returnfunction.Throttlingoftheconnectionbetweenvalveandcylinderchambershouldbeavoided.Ifnecessary,aone-wayflowcontrolvalvecanbebuiltintothelinefromthecylindertothedirec-tionalvalve.Theoverridingsignalatport12couldalsobegeneratedfromanothervalve.Non-returnvalveswithoverridecanalsobeusedforshort-durationpositioningorbrakingfunctionsofcylinders.1081262tching"on"4SwiResettingControlPressurep(bar)212100246810OperatingPressureatPort1(bar)Fig.13.6.Switchingcharacteristicsofanon-Fig.13.7.Symbolofreturnvalvewithoverrideanon-returnvalvewithoverride 13.3ShuttleValves18913.3ShuttleValvesTheshuttlevalve1selectsthehigheroftwoinletpressures.Itallowsairtoflowfromoneofthesupplyports,port1orport12,toport2dependingonthepressurelevelsatport1andport12;theairflowfromtheothersupplyportisblocked.Assumingthatp12p1,theaircanflowinbothdirections,e.g.from1to2or2to1,dependingonthepressuresattheports1and2.2112Fig.13.11.SchematicviewofatwinpressureFig.13.12.Symbolofavalve,drawnforp12>p1twinpressurevalve2accordingtoISO5598lowerpressurepriorityshuttlevalve“” 13.5QuickExhaustValves191841p(bar)08122412p(bar)01842p(bar)00200400600800Time(s)Fig.13.13.Pressureatport2ofatwinpressurevalvewhenpressurep1isheldconstantwhilepressurep12isvariedTwinpressurevalvesoftenhaveanunderlap,meaningthatinthemiddlepositionofthepoppetthereisanairflowfromport1andport12toport2.Intheendpositionsofthepoppetitisnotguaranteedthattheairflowtothethirdportiscompletelyblocked.13.5QuickExhaustValvesSomecylinderapplicationsrequireaveryhighpistonspeedwhichcanbeachievedbyusingaquickexhaustvalve.Thisvalveallowsexhaustingtheairfromthecylinderthroughalargeexhaustportandsilencerthusby-passingthetubingandmaincontrolvalve.Asaresult,thebackpressureislowerthaninacircuitwheretheairhastopassthroughtubingandthedi-rectionalcontrolvalvetobeexhausted.Increasesofupto50%ofthepis-ton’sspeedcanbeachieveddependingonthecylindertypeandloading.Especiallythelongretractiontimeofsingle-actingcylindersresultingfromthethrottlingofthesuppliedairtoslowtheextractionandtheweakreturnspringcanbereduced.However,built-incushioningwillbelesseffectivewhenaquickexhaustvalveisusedinsteadofaone-wayflowcontrolvalvebecausetheformerwillgivealowerbackpressure. 19213Shut-OffValvesFig.13.14.Quickexhaustvalve;leftp1>p12andcharging;rightp122->504->31->41SonicConductance(m/s/Pa)2-80-4004080SpoolDisplacement(%)Fig.16.9.Sonicconductancesofthefourmeteringnozzlesasfunctionsofspoolposition;thesymbols*and+markmeasureddatapoints 16.3SimulationModelofProportionalControlValves231ForonevalvetheresultsaregiveninFig.16.9.Thespoolpositionisshownonthex-axisandthevalueofthesonicconductanceConthey-axiswhichhasonlypositivenumbers.Themeasurementpointsaremarkedandasplineisusedtogiveacontinuouscurve.Thisvalvethrottlestheairfromthecompressortotheworkport(1→2or1→4)morethanthatfromtheworkporttotheexhaust(4→3,2→5).Figure16.10showsthemeasuredsonicconductanceofFig.16.9again.Thecurveisnotsymmetricalwithrespecttotheorigin.Toshowthedif-ferencesbetweenpositiveandnegativespoolposition,thegraphhasbeencopiedandrotatedby180degreesandtheareabetweenthetwocurveshasbeenmarkedgrey.Thisareacanbeconsideredtogivetypicaltolerancesforthiskindofvalveandthemeasurementmethodsused.Aloworderpolynomialcanbeusedtodescribetheconductanceasafunctionofthespoolposition,e.g.-8-102-83C=-2.5472⋅10⋅s+5.2631⋅10⋅s+1.1812⋅10⋅s(16.2)where-1b(s)•°°1Spool0T1-b(s)pSpool∆¸m=®1Spool1∆¸(18.1)°«¹°°Tpp⋅C(s)⋅ρ⋅0for2≤b(s)°1Spool0TpSpool°¯11wherem&massflowrateinkg/s,p1pressureupstreaminPa,C(sspool)sonicconductanceasafunctionofspooldisplacement3inm/(s⋅Pa),ρ0densityofairatreferenceconditionsinkg/m³,T0temperatureofairatreferenceconditionsinK,T1upstreamtemperatureofairinK,p2pressuredownstreaminPa,b(sspool)criticalpressureratioasafunctionofspooldisplacement,sspooldisplacementofspoolinm.Todescribeallflowpathsina5/3-wayvalve,fourmeteringnozzlesarerequired.Equation(18.1)isnon-linearandnotsuitedforcontrollerdesign.Inafirststeptowardsasimplemodel,Schwenzer(1983:29–42)replacesitbyalinearrelationshipbetweenmassflowrateandpressure:•(18.2)∗()m=C(s)⋅ρ⋅K⋅p−pSpool0112∗whereK1valvecoefficient.Hegivescomparisonsbetweenmodelandmeasurementsofasuddenpressurebuild-uporpressurereleaseinapneumaticcircuitandshowsthatthedifferencesaresmall.TheconductanceC(s)dependsinmanycaseslinearlyonthespooldisplacements.Ifafastservovalveisused,thedynamicsofthevalvecanbeneglectedandtheinputsignaltothevalveuValvecanbeusedinsteadofthespooldis-placement.Schwenzerreportsthatthisconditionismetifthecornerfre-quencyofthevalveis40to50timeshigherthanthatofthecylinderunlessveryhighfeedbackgainsareusedinthepositioncontroller.Inthatcase 18.3PressureDynamics253theinfluenceoftheneglecteddynamicscanbenoticed.Equation(18.2)canthenbesimplifiedtoyield•()(18.3)m=K⋅p−p⋅u112ValvewhereK1valvecoefficientinkg/(s⋅Pa⋅V),uValveelectricvalveinputsignalinV.ThenextstepisafurthersimplificationofEq.(18.3)byassumingthepressuresp1andp2tobealmostconstant.Thisleadsto•(18.4)mA=K⋅p⋅u.2AValve18.3PressureDynamicsThegeneralequationsforthepressurebuild-upinacylinderhavebeende-rivedinChap.8.2.Forthedesignofcontrolsystemstheseequationsarenotwellsuitedbecauseofthenon-linearitiesandlargenumberofparame-ters.Manystudieshaveshownthatforadequatemodelsforcontrollerde-signanisothermalbehaviouroftheairmaybeassumed(Kawakamietal.1988;Göttert2004).Startingwiththedefinitionofthedensityρ,m=ρ⋅V,(18.5)themassflowrateisgivenby•••(18.6)m=ρ⋅V+ρ⋅V.uValvemmABABsPistonFig.18.3.CylinderwithchambersAandB,meteringnozzlesandco-ordinatesystemforpistondisplacementandmassflowrates 25418PositionControlofPneumaticSystemsUsingtheidealgasequationofstate(2.1)andassuminganisothermalpro-cess,thedensityρasafunctionofpressurepisgivenwithrespecttoref-erencevaluesp0andρ0byρ(18.7)ρ=0p.p0Thereferenceconditionscanbechosenarbitrarily,e.g.standardambientpressurep0anddensityρ0accordingtoISO6358.Thederivativewithre-specttotimeofEq.(18.7)isgivenby•ρ•(18.8)ρ=0p.p0ForasymmetriccylinderthevolumesinthechambersAandBaregivenbyV0(18.9)VA=+APiston⋅sPiston2andV0(18.10)VB=−APiston⋅sPiston2whereVA,BvolumeofchamberAorB,respectively,V0totalairvolumeofcylinder,includingfittingsandlines,APistoneffectivepistonarea,sPistonpistondisplacement,sPiston=0isthemiddleposition.DifferentiatingEqs.(18.9)and(18.10)withrespecttotimeandusingEqs.(18.5)–(18.8)givesthedifferentialequationsforthepressuredynam-icsinthetwocylinderchambersasafunctionofthemassflowrates:••ρ⊬V·ρ•(18.11)m=p0∆0+A⋅s¸+p⋅0⋅A⋅s,AA∆PistonPiston¸APistonPistonp2p0«¹0••ρ⊬V·ρ•(18.12)m=p0∆0−A⋅s¸−p⋅0⋅A⋅s.BB∆PistonPiston¸BPistonPistonp2p0«¹0Rearrangingthesetwoequationsgivesthedifferentialequationsthatde-scribethepressurebuild-upinchamberAandB: 18.3PressureDynamics255•••mpp⋅A⋅sPistonp=A0−APiston,(18.13)AV/2+A⋅sρV/2+A⋅s0PistonPiston00PistonPiston•••mpp⋅A⋅sPistonB0BPistonpB=+.(18.14)V/2−A⋅sρV/2−A⋅s0PistonPiston00PistonPistonThefirsttermontherighthandsideofEq.(18.14)describestheeffectofthemassflowrateonthepressurechangeandthesecondparttheeffectofthevolumechange.Assumingforthemomentapositiveinputsignaltothevalve,ashortlinebetweenvalveandcylinderandchamberpressuresofabouthalfthesupplypressure,Eq.(18.4)becomes•p(18.15)SupplymA=K⋅⋅u,22Valve•p(18.16)SupplymB=-K⋅⋅u.22ValveSubstitutedintoEqs.(18.13)and(18.14),respectively,gives•(18.17)•K⋅p⋅upA⋅sPiston⋅p/22SupplyValve0PistonSupplyp=−,AV/2+A⋅s2⋅ρV/2+A⋅s0PistonPiston00PistonPiston•(18.18)•-K⋅p⋅upA⋅sPiston⋅p/22SupplyValve0PistonSupplyp=+.BV/2−A⋅s2⋅ρV/2−A⋅s0PistonPiston00PistonPistonNeglectingthepositiondependenceofthevolumeandassumingamid-strokeposition,sPiston=0,leadsto•(18.19)•K⋅p⋅upp⋅A⋅sPiston2SupplyValve0SupplyPistonp=−,AV/22⋅ρV000•(18.20)•-K⋅p⋅upp⋅A⋅sPiston2SupplyValve0SupplyPistonp=+.BV/22⋅ρV000 25618PositionControlofPneumaticSystemsDefiningadifferentialpressurep=p−p(18.21)ABABleadsto•p⋅A•K⋅pp(18.22)SupplyPiston2Supply0pAB=−⋅sPiston+⋅u.V/2V/2ρValve000Thisdifferentialequationdescribesthepressurebuild-upinthecylinder.Multipliedbythepistonarea,thedifferentialpressureisthedrivingforceforthepistonmotion.18.4EquationofMotionTheaccelerationofthepistoncanbedescribedbyNewton’ssecondlaw••A⋅(p−p)−F−F(18.23)s=PistonABLoadFricPistonMPistonwhereFLoadexternalloadforce,FFricfrictionforce,MPistonpistonmass.Theloadforcedependsontheparticularapplicationofthedrive.Itcanbeconstant,e.g.aweightforce,orpositiondependent,e.g.aspringforce,orhaveanothercharacteristic.Itwillnotbeconsideredinthefollowingderivation.DetailedmodelsofthefrictionforceofpneumaticcylindersaregiveninChap.8.3.1.Both,theloadandthefrictionforce,canhavesig-nificanteffectonthedynamicbehaviourandtheaccuracyofthesystemmodeldependsheavilyonagooddescriptionofthoseforces.Forthederi-vationofamodelsuitableforcontrollerdesigntheassumptionofspeedproportionalfrictionandnoloadforceswillbeused.Thisleadsto:••p⋅A•(18.24)s=ABPiston−.PistonKsPistonM3PistonDifferentiatingEq.(18.24)withrespecttotimeandusingEq.(18.22)leadstothelinearsystemmodel.Forthefollowingcontrollerdesignitiswritteninstatespaceform: 18.4EquationofMotion257•(18.25)x=Ax+buwiththestatevectorx»xº»positionº(18.26)…•»…»x=x=velocity,…»…»••…¬x»¼…¬acceleration»¼thesystemmatrixA»º(18.27)…010»…»…»…»A=…001»…»…2⋅p⋅A2»…SupplyPiston»0−−K…M⋅V3»…¬Piston0»¼andtheinputvectorb»º(18.28)…0»…»…»…»b=0.…»…»…2⋅K2⋅pSupply⋅APistonp»0…»M⋅Vρ…¬Piston00»¼Thisthirdordersystem2canbesplitintoasecondordersystemandanintegratorinseries.Thesecondordersystemischaracterisedbynaturalfrequencyωn,dampingratioζandgainK.Whenneglectingtheeffectofthesmalldampingduetothefrictiononthenaturalfrequency,ωnisgivenby2(18.29)2⋅p⋅ASupplyPistonω=.nM⋅VPiston02Foramodelwithfourstates[pps&s]'seeFerraresietal.(1994)orGöttert12(2004:76–83) 25818PositionControlofPneumaticSystemsAsshowninChap.3.5,thenaturalfrequencyofacylinderdependsonthepistonpositionandhasitslowestvalueatmid-stroke.Equation(18.29)thusdescribesthemostcriticalpositionforcontrollerstability.“Thechangesofsystemnaturalfrequencyisquitestronglyneartothecylinderends.Thatiswhyitiscommonpracticetoleaveabout5–10%ofthecyl-inderstrokeunusedinordertoavoiddynamicdifficulties”(Virvalo1995b).ToderiveEq.(18.29),severalassumptionsweremade.Ifanadiabaticprocessinsteadofanisothermalprocessisassumed,thetermunderthesquarerootinEq.(18.29)hastobemultipliedbytheratioofspecificheatcapacities,γ=1.4(RusterholzandWidmer1985;seealsoVirvalo1989).Ifthemeancylinderpressureishigherthanhalfthesupplypressure,thenaturalfrequencyωnishighertoo.Thepressurelevelinthecylinderchambersdependsonthesupplypressureandvalveoverlaps,butnotthevalvesize.Atypicalvalvedesignhaspressuresintheoutputportsthatareatleastslightlyhigherthan50%andcanbeashighas80%ofthesupplypressure(Moore1986;Nguyen1987;Virvalo1995a;Göttert2004:31).ThevelocitygainKisgivenbyK⋅p(18.30)K=20.A⋅ρPiston0ThedampingratioζisgivenbyK2⋅M⋅V(18.31)ζ=3Piston0.4p⋅A2SupplyPistonVirvalo(1995a)givesarangebetween0.02and0.15forthedampingratioζofpneumaticcylinderdrives.Thevaluedependsheavilyonthefriction,hererepresentedbyK3,andisdifficulttoevaluateexactly.TheparametersdefinedinEqs.(18.29–31)cangiveanestimateofthesystemdynamics.However,duetothemanyassumptionswhenderivingthemodelthecalculatednumbersmaydiffersignificantlyfromthosethatcanbeobtainedbyanalysingthesystemoutput.Thismaybeacomparisonofameasuredstepresponsewiththesolutionofthedifferentialequationoranidentificationscheme(LuandHong1988). 18.5ControlLaws25918.5ControlLawsControlofpneumaticdrivesismuchmoredemandingthanforinstancecontrolofelectricDCmotorsbecauseofthecombinationofacompressi-blemediumandadifferencebetweenstaticanddynamicfrictionforce.Thismakesastepresponsewithoutovershoothardtoachieveandoftenleadstostick-sliposcillationsandlimitcycles.Inthepastalargenumberofapproacheshavebeenstudied.Presentedaresomecontrollawsthathavebeendevelopedespeciallyforandtestedonactualpneumaticdrivesandarenotprimarilyapplicationsofmathematicaltheory.18.5.1SingleLoopControllersThesimplestcontrollawisaproportionalfeedbackofthepositionerror:u=K⋅(s−s)(18.32)ValverefPistonwhereKcontrollergain,srefreferenceposition,sPistonpistonposition.Duetotheintegrationofthevelocitytotheposition,anidealdrivewouldhaveapositionerrorofzero,evenforsmallvaluesofK.Butthefrictioninthecylinderandtheloadaffectthevelocitysuchthattheintegratingbe-haviouroftheplantdoesnotguaranteeazeropositionerror.Tokeepthaterrorsmall,anintegratingcontrollercouldbeused.How-ever,thisisnotrecommendedforthiskindofdrivebecausetheunsteadynatureoffrictionwillleadtolimitcycles.Instead,aconsiderablegainKisneeded.Butthishasadestabilisingeffectwhichleadstoapoorlydampedsystemresponsethatcannotbeacceptedforaservodrive.Severalothersingleloopcontrollershavethereforebeentested,likePI,PIDetc.,butnonecouldachieveagooddynamicandsteady-stateresponse3(Schwenzer1983).3Therearefewexceptions:FokandOng(1995)designasingle-loopproportionalanddifferentialcontrollerandachievenominal10%overshoot,ameansettlingtimeoflessthan2sfor900mmdisplacement,aloadbetween40and80kg,andarepeatabilityofbetterthan±0.3mm. 26018PositionControlofPneumaticSystemsFrictionControllerModelledRealVelocityVelocityPosition-Fig.18.4.Blockdiagramofapositioncontrolloopwithfrictionasdisturbance18.5.2AdditionalLoopsToimprovetheresponsegivenbyaproportionalfeedbackoftheposition,additionalsignalshavebeenused.MiyataandHanafusa(1988)useforex-amplesensorsforpressure,velocityandposition.18.5.3StateFeedbackControlChapter18.5.1showsthatitisgenerallynotfeasibletocontrolapneu-maticdrivewithaclassicalsingleloopcontroller.AbettersuitedtoolisthemoderncontroltheorythatwasdevelopedbyKalmanandLuenbergerinthe1960s.Theylookatmultivariable,multi-input,multi-outputsystemsthataredescribedbyvectordifferentialequationsinstatespaceform:•(18.33)x=Ax+Bu,y=Cx+Du(18.34)whereAsystemmatrix,dimensionn.n,Binputmatrix,dimensionn.m,Coutputmatrix,dimensiono.n,Dfeedforwardmatrix,dimensiono.m,nnumberofstates,mnumberofinputs,onumberofoutputs.Forpneumaticdrivesthisgeneralformreducesto•(18.25)x=Ax+bu 18.5ControlLaws261withthestatevectorx»xº»positionº(18.26)…•»…»x=x=velocity,…»…»••…¬x»¼…¬acceleration»¼thesystemmatrixA»º(18.27)…010»…»…»…»A=…001»…»…2⋅p⋅A2»…SupplyPiston»0−−K…M⋅V3»…¬Piston0»¼andtheinputvectorb»º(18.28)…0»…»…»…»b=0.…»…»…2⋅K2⋅pSupply⋅APistonp»0…»M⋅Vρ…¬Piston00»¼Inordertosuccessfullycontroltheoutput,thesystemmustbecom-pletelystate-controllableandcompletelystate-observable.Thenanystatecanbereachedbyasuitableinputsignal(controllability)andanyinitialconditionreconstructedbylookingattheoutputsignal(observability).Theseconditionscanbemathematicallyformulatedasrankconditionsforthecontrollabilityandobservabilitymatrices.!(18.35)[]2n−1rankbAbAbKAb=n,!(18.36)[]2n−1rankc'A'c'A'c'KA'c'=nwhere'denotesthetransposeofavectorormatrix,respectively. 26218PositionControlofPneumaticSystemsForthemodelinEqs.(18.27)and(18.28)thisleadsto»00bº(18.37)…3»[]2rankbAbAb=rank…0bab»=33333…2»…¬b3a33b3a33a32b3+a33b3»¼fornonzeroelementsb3,a32,a33.Theobservabilitymatrixistheidentitymatrixandhasalsofullrank.Thesystemisthereforecompletelystate-controllableandcompletelystate-observable.Itcanbeshownthatthedynamicsofthesystemcanbearbi-trarilymodifiedbyasuitablechosenstatefeedbackcontroller,u=−k⋅x(18.38)wherekvectoroffeedbackcoefficients.Thepolesoftheclosedloopsystemaregivenbyeigenvalues()A−b⋅k(18.39)andapopularmethodtodeterminetheelementsofthevectorkisAcker-mann’sformula.Whiletheclosedlooppolelocationscanbearbitrarilychosenforalin-ear,time-invariantsystem,therearerestrictionsfortechnicalsystems.Thepositionsignalismeasuredandthereforenotideal,butnoisy.Typically,theresolutionislimitedifanincrementalmeasuringsystemisusedoradigitalcontrollerwithananalogue-to-digitalconverter.Themaximumpowerthatcanbetransmittedthroughthevalveislimitedbecausethereisamaximumflowareathatcanbeopenedandagivensupplypressure.Thevalvedynamicscannotbeneglectedifthepolesofthecontrolledsystemsaremovedveryfartotheleftside.Inearlystudies,statecontrollersandobserverswereusedbutnodesignmethodsgiven.Insomecasestheupto16parametersseemtohavebeenfoundbymanualfine-tuning,i.e.timeconsumingtrialanderror.Inthelaterliteraturetherearesomeguidelineshowtoplacethepoles.Rusterholz(1986:170–176)suggeststocomputeωn,thenaturalfre-quencyofthedrivegiveninEq.(18.29),andtousetherootsλ1,2ofasec-ondordersystemwiththisnaturalfrequencyandadampingratioofζ=1.5.22λ+2⋅ζ⋅ω⋅λ+ω=0,ζ=1.5.(18.40)nnThethirdpoleistobeplacedat 18.5ControlLaws263λ3=λ1+λ2.(18.41)Hestatesthatthesystembecomeshighlyoscillatoryifthepolesofthecontrolledsystemaremovedfurthertotheleft,λ=−3⋅ζ⋅ω,λ=−4⋅ζ⋅ω.(18.42)1,2n3nRusterholzseesthereasonforthisbehaviourinthehighnumberofnon-linearitiesthatbecomemoreimportantforhigherfeedbackgains.Healsoshowsthatthecontrolleddriveissensitivetoachangeinmassandrec-ommendstodesignthecontrollerforthelowestmass.Chen(1995:76)andVirvalo(1995a)alsogivedesignrules.18.5.4ReconstructionoftheVelocityandAccelerationSignalToimplementthestatespacecontroller,thevelocityandaccelerationsig-nalsareneeded.Inthebeginningofpositioncontrolofpneumaticdrives,thesesignalsweremeasured.Duetothecostofthesensorsotherap-proacheshavebeenstudied.Foranidealsystemthreeapproachescanbeused:-directdifferentiationofpositionsignal,-fullorderstateobserver,-reducedorderstateobserver.Thedirectdifferentiationoftheoutputsignalofthesystemseemstobethemostoftenusedmethodtoobtainthestates(Virvalo1995a).Thedirectdifferentiationhastheadvantagethatnoobserverdesignisrequiredandthereforenoparametershavetobeselectedandfinetuned.Butthealwayspresentnoiseonthepositionsignalrequiresfiltering.Schwenzer(1983)usedanaloguefirst-orderfilterswithacornerfrequencyof200Hz.Are-ductionofthecornerfrequencybelow200Hzinfluencedthecontrollerwhosegainshadthentobereduced.Hisexperimentstoreducepowerlineinducedsignalswithfrequenciesof50Hzand100Hz-whichareinafre-quencyrangewherethevalvecanstillreact-byfilteringwerenotsuccess-ful.AdifferentiatingelectroniccircuitthatincludesalowpathfilteringisgivenbyNguyen(1987:56).Inthe1980s,analogueelectronicsandanaloguepositionsensorswereused.Laterdigitalcomputersandincrementalsensorsbecameavailable.Especiallytheincrementalsensorsprovedtobebettersuitedthantheana- 26418PositionControlofPneumaticSystemsloguesensorsbecauseatstandstillthedigitalsignalisconstantwhiletheconvertedanaloguevoltageoftenresultsinanoscillatingleastsignificantbitthatgeneratesconsiderablenoiseafterthedoubledifferentiationandleadstoanoisyvalveinputsignalandunsteadypressuresignals.18.5.5Non-LinearControlLawsDuetothegreatnumberofnon-linearities,alinearcontrollawcanonlyachievealimitedperformance.Itcanbetunedforthestabilisationofthesystematoneoperatingpoint,butoftentheperformancewillnotbeade-quateforotheroperatingconditions.Thereforemanynon-linearcontrollawshavebeendesigned.Forasurveysee(Scavarda1993;Xiang2001;Göttert2004).Manypapersaboutcontrolschemesforpneumaticdrivesseemtohavebeenpublishedtoprovethatcertaintheoreticalconceptscanbeappliedtoactualhardware.ForinstanceHahn(2000)describesanon-linearcontrol-lerthatusesafeedbacklinearisation.Buthestatesthat“itisaschemethatrequiresamaximumofinformationandneedsamaximumofresourcesforimplementation”.Wikander(1988)usesgainschedulingtocompensateforthepositiondependenceofthepiston.Theideaofgain-schedulingistoselectanumberofoperatingpointswhichcovertherangeofthesystemoperation.Ateachofthesepoints,thedesignermakesalineartime-invariantapproximationoftheplantdynamicsanddesignsalinearcontrollerforeachlinearisedplant.Betweenoperatingpoints,theparametersofthecontrollerareinter-polated,orscheduled,thusresultinginaglobalcompensator.Wikander(1988)describesahybridcontrollerthatswitchesbetweenpositioncontrolandvelocitycontrol,wherethereferencevalueforthevelocityisneartothemaximumpistonvelocity.Bothcontrollersruninparallelandforlargedifferencesbetweenactualandreferencepositiontheoutputofvelocitycontrollerisused.Whenthepistoncomesclosetothereferencepositionandtheoutputsignalofthepositioncontrollerfallsbe-lowthatofthevelocitycontroller,theoutputofthevelocitycontrollerisfedtotheproportionaldirectionalcontrolvalve.Pu(1988)usesathreephasescheme.Afterchangingthepositionrefer-encesignal,thevalveiscompletelyopeninonedirectionsuchthatthedriveachievesthemaximumforwardvelocity.Thenthevalvereverses,i.e.thepistonisdeceleratedwithmaximumpressure.Finally,acontrollerisactivatedthatensurestheprecisepositioning.AveryapplicationorientedstructureisdescribedbyBauer(2002:89–94).Heusesamodifiedstatefeedbackcontroller 18.6PerformanceofaCommercialSystem265u=K⋅(s−s)−K⋅s&−K⋅&s&.(18.43)srefpistonvpistonapistonToachievetherequiredsteady-stateaccuracy,headdsanintegratorthatisswitchedoffifthepositionerrorexceedsacertainvalue.Insteadofadifferentiationofthepositionsignal,heusesthepressuredifferential.ThefeedbackgainKsisnotaconstantbutgivenbyapolynomial:2K=a+b⋅∆s+c⋅∆s.(18.44)sThisgivesthenon-linearcontrollerwhoseparametersaretunedbyanu-mericaloptimisationschemethatusesamathematicalmodelofthedriveandthecontroller.Thestatecontrollerfromchapter18.5.3isintendedforpoint-to-pointoperation;differentstructureshavebeenproposedforcontinuouspathcontrol(ScholzandMostert1988;Göttert2002).Allsystemsinthischapteruseacontinuouslyworkingproportionaldi-rectionalcontrolvalve.Toreducethecomplexityofthecircuitandcosts,theuseofswitchingvalveshasbeenproposed,oftenforrotatorydrivesinrobots(e.g.Noritsugu1986;Hippe1988;NoritsuguandWada1989;Laietal.1990;Miyataetal.1991;LeufgenandLü1991;Virvalo1993;Pauletal.1994;Shen2006).Theselectionofasuitablevalverequiresalotofconsiderationbecauseithastobefastandhastoopenalargeflowarea.Thesecontradictoryrequirementsledresearchestocircuitswhereseveralvalvesofdifferentsizeswereused.Manydesignmethodshavebeenstud-ied,fromasimplereplacementofthedirectionalcontrolvalvebyapulsewidthmodulatedswitchingvalvetoneuro-fuzzycontrol.AcomparisonofsixdifferentcontrolmethodsisgivenbyChillarietal.(2001).18.6PerformanceofaCommercialSystemSincethemid1990s,thereisasimpletoinstallpneumaticservodrivecommerciallyavailablefromFestoAG.Zhou(1995),Latino(1996)andGöttertandNeumann(1997)giveadescriptionofthissystem,consistingofthecontrollerSPC,theproportionaldirectionalcontrolvalveMPYEandacylinderwithpositionmeasurementsystem.InourlabweusethecontrollerSPC11andarodlesscylinderwith400mmstroke.Thesystemhasanelectronicstrokecushioningfortheendpo-sitionsandcanstopattwomidpositionsthatcanbetaughtbytheuser.Theprecisionofthein-betweenpositionsisspecifiedas±0.25%ofthelengthofthemeasurementsystem,butnotbetterthan±2mm.Themini- 26618PositionControlofPneumaticSystemsmumdistancebetweenanin-betweenpositionandtheendofstrokemustbe10%ofthetotalstroketoensureasufficientaircushion.Thisrequire-mentalsorelievesthecontrollerfromthedifficulttasktopositionattheendofstrokewherethenaturalfrequencyofthedriveismuchhigherthanatamidstrokeposition.Thedistancebetweenthetwoin-betweenposi-tionsistobegreaterthan20mm.Thebigadvantageofthissystemistheeaseofinstallation:afterthemechanicalinstallationonlythreeparametershavetobeenteredinthecontroller.Thesevaluescanbeobtainedfromta-blesoracomputerprogramanddependonthediameterandstrokeofthecylinderandthemassoftheload.Afterkeyinginthesevalues,anauto-maticinitialisationandfinetuningtakesplacethatalsodeterminesthepo-sitionofthemechanicalendstops.Thein-betweenpositionscanthenbetaughtbypositioningthedriveattherequiredpositionsandenteringacommand.Figure18.5showsposition,pressureandvalvecommandtrajectoriesforaworkingcyclewith4stops.Thoughtheaccelerationsarehigh,thedriverunsverysmoothlyandquietly.Figure18.6zoomsinonFig.18.5anditcanbeseenthatthevalvesignalandthepressuresarealwaysfluctuating.Figure18.7givesthedeviationsfromanin-betweenstopwhichweremeasuredbyhandwithamechanicaldialgauge.Itcanbeseenthatthepo-sitionrepeatabilityismuchbetterthanspecifiedbythemanufacturer.Theonlydrawbackmightbetheoperatingcostswhichcanoutweighthelowercapitalinvestmentwhencomparingthisdrivewithanelectricdrive(OhmerandNeumann2004). 18.6PerformanceofaCommercialSystem267600400Position400200Velocity20000Position(mm)Velocity(mm/s)-2008Supply6PressureA4PressureBPressures(bar)210.50-0.5-1Rel.ValveOpening01234Time(s)Fig.18.5.Position,pressureandvalvecommandtrajectoriesforaworkingcyclewith4stopsSupply76A5Pressures(bar)B40.20.0-0.2Rel.ValveOpening0.60.650.70.750.8Time(s)Fig.18.6.Pressureandvalvecommandsignalswhenholdingaposition 26818PositionControlofPneumaticSystems2010NumberofSamples0-0.4-0.200.20.4PositionError(mm)Fig.18.7.Histogramofpositiondeviations 19ControlofActuatorsforProcessValvesThegreatmajorityofpneumaticdrivesareoperatedinabinarymode:therodiseithercompletelyextendedorretracted.Systemsthatcanstopatin-termediatepositionsormoveinacontrolledwayhavebeentopicofre-searchprojectsinthe1980sandarecommerciallyavailabletodaybutveryrarelyusedforindustrialautomation.However,thereisoneareaofautomationwherepositioncontrolledpneumaticactuatorshavebeenusedfordecades:fortheactuationofproc-esscontrolvalves.Thesevalvesmanipulateaflowingfluidsuchasgas,steam,waterorchemicalcompoundstokeepprocessvariablessuchaspressure,flow,levelortemperaturewithinarequiredrange.Fig.19.1.Schematicviewofaprocesscontrolvalve 27019ControlofActuatorsforProcessValvesThecontrolvalve,ormoreexactlythecontrolvalveassembly,consistsofthevalvebody,theinternaltrimparts,anactuatortoprovidetheforcetooperatethevalveandavarietyofadditionalvalveaccessories,amongthemapositioner.Thisisaservomechanismthatismechanicallycon-nectedtothevalvestemandthatautomaticallyadjustsitsoutputtotheactuatortomaintainadesiredstempositioninproportiontotheinputsig-nal.Avalvewithoutpositionerexhibitsaconsiderabledeadzonewhichistherangethroughwhichtheinputsignal,i.e.thepressureintheactuatorchamber,canbevaried,uponreversalofdirection,withoutinitiatinganobservablechangeintheoutputsignal,i.e.thepositionofthevalvestem.Adeadzonehasmanycauses,butbacklashinthecontrolvalveandfric-tion,especiallyinthepacking,areusuallydominant.Therearefewstudiesofpackingfriction:Jeschke(1968)statesthattheinfluenceofthestemspeedissmall,Lloyd(1968)givesaruleofthumbthat5%ofthefullthrustarerequiredtoovercomethepackagefriction,Mundry(2000:56–60)comparesdifferenttypesofpackages.Becausemostcontrolactionsforregulatorycontrolconsistofsmallchanges,acontrolvalvewithanexcessivedeadzonemightnotevenrespondtomanyofthesesmallchanges.Awell-engineeredvalveshould1008060Positioner40Electro-OutputSignal(%)PneumaticConverter200020406080100InputSignal(%)Fig.19.2.Characteristicsofaprocessvalvedrivenbyaninputpressurefrom0to1bar,generatedbyanelectro-pneumaticconverterorapositioner,respectively.Inputisanelectricsawtoothsignal,outputthepositionofthevalvestem:there-ductioninhysteresisisobvious. 19.1CharacteristicsofProcessControlSystems271respondtosignalsof1%orlesstoprovideeffectivereductioninprocessvariability.However,itisnotuncommonforsomevalvestoexhibitadeadzoneasgreatas5%ormore.Inaplantaudit,30%ofthevalveshaddeadzonesinexcessof4%;over65%oftheloopshaddeadzonesgreaterthan2%(anon.1999).Theactualselectionofaprocesscontrolvalveoritsdesignisbeyondthescopeofthisbook;see(ChampagneandBoyle1996;Wilke1999;Bel-forteet.al2000;Monsen2000).19.1CharacteristicsofProcessControlSystemsTheoperatingconditionsofactuatorsforprocessvalvesdifferinsomere-spectsfromthosefordrivesusedforfactoryautomation.Therequiredforceishigh,speedislow,inertiaofmovingpartsandcompressibilityofairnegligible.Thepressureisoftenlimitedto1barandtheelectriccurrentintherangefrom4to20mA.Thiscurrentisusedbothtotransmitinfor-mation,e.g.asetpointofacontroller,andtopowerinstruments.Togeneratetherequiredhighforces,spring-and-diaphragmactuatorsareused.Thespringsguaranteethedesiredactionintheeventofsupplypressurefailure,e.g.valvetoopen.Generally,diaphragmactuatorscon-tributelessfrictiontothecontrolvalveassemblythanpistonactuators.Anadditionaladvantageofspring-and-diaphragmactuatorsisthattheirfric-tionalcharacteristicsaremoreuniformwithage.PistonactuatorfrictionoftenincreasessignificantlywithuseasguidesurfacesandtheO-ringswear,lubricationfailsandtheelastomerdegrades.Mostpneumaticspring-and-diaphragmactuatorsuseamouldeddiaphragm.Itpermitsgreatertravelthanaflatsheetdiaphragmandprovidesarelativelyuniformeffec-tiveareathroughoutvalvetravel.Taubitz(1978:110)givesforanactuator2withanominaleffectiveareaof570cmadeviationof+12%forastrokeof0mmandadeviationof-5%forastrokeof30mm.Solenoidsareveryseldomusedtoconvertelectricsignalstopneumaticpressurebecausetheyrequirehighcurrentsandcangenerateheatorsparks,afactthatcangenerallynotbetoleratedinanexplosivearea.Typi-callyvoicecoilsareusedthatcanbeoperatedat10Vandupto20mA.Thegeneratedforceisproportionaltothecurrent.However,thisforceissmall,e.g.0.1N,andhastobeamplified.AtypicalwaytodothatisshowninFig.19.3thatgivesaschematicviewofanelectro-pneumaticconverter.Thepermanentmagnetliningtheinsidediameterofaferromagneticcylinderproducesamagneticfield.Whencurrentflowsthroughthecoil,it 27219ControlofActuatorsforProcessValvesgeneratesanaxialforceonthecoilandproducesatorqueonthebalancearm.Insteadystatethisforceisbalancedbytheforcegeneratedbythepressureactingonthediaphragm.Thispressureresultsfromthenozzle-flappersystemthatisusedtodetectthedeflectionofthebalancearm.Ifthedistancebetweennozzleandflapperistoogreat,theresistanceofthenozzleissmallandthepressurefallsalmosttoatmosphericpressure.Ifthedistanceisverysmall,thepressurereachesthesupplypressure.LeafSpringBalanceArmElectricalInputNCoilNozzleSDiaphragmPermanentAmplifierMagnetOutputSupplyFig.19.3.Schematicviewofanelectro-pneumaticconverter;inputistheelectriccurrent,outputthepressurefromtheamplifierCoilMagnetLeafSpringDiaphragmNozzleArmCylinderFig.19.4.Cut-awayviewofanelectro-pneumaticconverter 19.2Positioners273Themeasuringrangeofanozzle-flappersystemistypicallysmall,e.g.lessthan0.05mm,andthepressuredependsnon-linearlyonthedistance,seeFig.19.5.Themassflowratethatcanbetakenfromthenozzleislim-itedbecauseotherwiseanadditionalpressuredropwouldoccur.Thereforeanamplifierorboosterisrequired.Thisoperatesinthesamewayasapressurecontrolvalvewiththedifferencethatthereferencepressureisnotgivenbyaspringforcebutbythebackpressurefromthenozzle.Nozzle-flappersystemsareoftenusedasmeasurementdevicesandaredescribedby(WincklerandKramer1960;Colinetal.1996;Crnojevicetal.1997).Adetailedstudyofamplifiersisgivenin(Taubitz1978:65-107).Toavoidanyfrictionorwear,thebalancearmhasnobearing,buttwoperpendicularmountedleafspringsareusedthatpermitarotationofseveraldegrees.Theseconvertershaveahighaccuracywithahysteresisoflessthan1%,adeadzoneoflessthan0.1%andalinearitydeviationoflessthan2%.Toachievethisquality,preciselymanufacturedandassembledpartsarerequiredwhichareexpensive.Pressure0DistanceFig.19.5.Characteristicofanozzle-flappersystem,backpressureasafunctionofdistancebetweennozzleandflapper19.2PositionersApneumaticallyactuatedvalvedependsonapositionertotakeaninputsignalfromaprocesscontrollerandconvertittoproportionalvalvetravel.Thedevelopmentoftheseinstrumentsstartedwithpneumaticpositionerswhereapneumaticreferencesignal(usually0.2–1baror3–15psi)issup-pliedtothepositioner.Firstinstrumentswerepresentedinthe1950s 27419ControlofActuatorsforProcessValves(Young1955;anon.1956).Thepositionertranslatestheinstrumentpres-suretoarequiredvalvestempositionandsuppliesthevalveactuatorwiththerequiredairpressuretomovethevalvestemtothecorrectposition.Inthe1960sinstrumentswerepresentedthatuseanelectriccurrent(usually4–20mA)insteadofairasinputsignal.Thereareseveraldesignsoftheseanalogueelectro-pneumaticpositioners,rangingfrompneumaticpositionerswithintegratedelectro-pneumaticconverterstoinstrumentswithanalogueelectronicsandpiezoelectricpilotvalves.Today,manypo-sitionersusedigitalelectronics.Themostimportantcharacteristicofagoodpositionerisahighpropor-tionalgain.Additionally,itmustbecapableofsupplyingsufficientairtotheactuatortomakethevalveclosuremembermoverapidly.Thefirstpropertyissometimescalledstaticgainandthelatterdynamicgainofthepositioner.Theserequirementsusuallyleadtotwostages:apreamplifierwithhighstaticgain,e.g.anozzle-flapper-system.Thisistypicallyfol-lowedbyahighdynamicgainpoweramplifier,e.g.arelayoraspoolvalve(Roth1972;Taubitz1978).Apositioneristypicallydesignedforeithersingle-actingordouble-actingactuators.Theformerareoftenspring-and-diaphragmdrivesforslidingstemvalveswhilethelatterarerotatorydrives,e.g.forbutterflyvalves.Mostpositionerscanbeconvertedfromdirecttoreversedfunctionwhereanincreasingcontrolsignalgivesadecreasingvalveopening.nozzle/0.2...1barforcebalanceamplifierflappervoicenozzle/4...20mAforcebalanceamplifiercoilflappertimeanaloguepoppet4...20mApiezoelectronicsvalvedigitalpoppet4...20mApiezoelectronicsvalveFig.19.6.Evolutionofdifferentconceptsforpositioners,roughlyfromthe1950stothe1990s 19.2Positioners275Typicalstemvelocitiesarebelow0.1m/mintoavoidthe“waterhammereffect”thatoccursifacontrolvalveisclosedtoorapidlyandthemomen-tumofthefluidinthelineleadstopressuresurges.Somepositionersofferthereforealimitationoftheminimumclosingtimeto2.5s,10soreven25s.Generally,actuatorswithpositionerscanberegardedaslow-frequencysystems.Their-3dBfrequencyliesoftenbetween0.1and1Hz.19.2.1PneumaticPositionersThepneumaticpositionerinFig.19.7hasasimilarprincipleofoperationastheelectro-pneumaticconverterinFig.19.3.Thecommandsignal,i.e.thedesiredpositionofthevalvestem,isgivenasapressureandactsonadiaphragmthatisconnectedtothebalancearm.Opposingtheresultingforceisaspringforcethatresultsfromthestemposition.Thedistancebetweennozzleandbalancearmdeterminesthebackpressurethatisampli-fiedandfedintothechamberofthevalveactuator.Thepositioner’sinput-outputbehaviourcanbestbedescribedashighproportionalgainwithsaturationandsomeadditionaldelay.Thereareseveralwaystotunethispositioner.Theelongationofthespringasafunctionofthevalvestemdisplacementcanbemodifiedbychangingthepositionwherethisspringisattachedtothelever.Differentspringsareavailabletotakedifferentstemdisplacementsintoaccount.Thereisalsoanotshownnozzlethatdelaysthepressurebuild-up.How-ever,tuningrequiressomepatienceasthezero,spanandgainadjustmentsareinterdependent.LeafSpringDiaphragmAmplifierCommandCenterofRotationSupplyAdjustableNozzleNozzleSpringBalanceArmProcessValvePneumaticPositionerFig.19.7.Schematicviewofapneumaticpositioner 27619ControlofActuatorsforProcessValvesThiskindofpneumaticpositionerwithforcebalancehasahighgaintoreducethedeadzoneoftheprocessvalve.Figure19.2showstwomeas-urementswherethecharacteristicsofthevalvewithoutandwithpositioneraregiven.Thedeadzoneisconsiderablyreduced.Toachievethisreduc-tionahighgainisnecessary,typicalvaluesarebetween100and200andcanevenexceed10001.Thesepurepneumaticpositionershavetypicallyahysteresisoflessthan0.6%,adeadzoneoflessthan0.1%andalinearitydeviationoflessthan1%offullscale.Somedesignsuseacamforthemechanicalconnectionfromtheactua-tortothebalancearm.Dependingonitsshape,thecamcanbeusedtoes-tablishvariousrelationshipsbetweentheinputsignalandtheactuatorpo-sition,e.g.asquarerootrelationship,alinearrelationshiporasquarecharacteristic.Thiscanhelptoadaptagivencontrolvalvetoaparticularcontrolstrategy.Importantisthesteady-stateairconsumptionbecausethatisansubstan-tialpartoftheoperatingcosts.Itdependsconsiderablyonthesupplypres-sureandthegainandcanvaryfrom15to400l/h(ANR).Today’sinstru-mentsareregardedasveryrobust,reliableandeasytohandle.19.2.2AnalogueElectro-PneumaticPositionersBeginninginthe1960s,analogueelectronicswasusedtogenerateandtransmitcontrolsignalsandgraduallyreplacedpneumaticallyworkingin-struments.Firstacombinationofelectro-pneumaticconverterandpneu-maticpositionerwasused.Laterelectronicinstrumentsforvalveserviceweregraduallyacceptedandinsomeinstrumentsprecisionpotentiometers,electronicamplifiersandelectricallyoperatedvalveswereusedinsteadofsprings,balancearmsandnozzle-flapper-systems.Thesesystemsareeas-iertoinstallbecauseallcontrolcharacteristicslikezero,rangeorsensitiv-ityareelectronicallyresettableusingtrimmersonthecontrolcard.Thesepositionersreceivea4to20mADCinputsignalthatisusedbothtotransmitthedesiredvalvestempositionandtopowertheinstrument.1Whilethereisanenormousnumberofpublicationsonthemodellingandcontrolofcylinderdrives,therehasbeenalmostnotheoreticalworkpublishedaboutpositioners.Thismaybeduetothemanysignificantnon-linearitiesthatrenderananalyticalcontrollerdesignimpossible.Amongtheexceptionsisthecom-prehensiveworkofTaubitz(1978)and(Kollmann1968;Roth1972;KoenigandOhligschläger1989;Pyötsiä1991;Fujitaetal.1994;KayihanandDoyle2000;Wakuietal.2003,2004;ShoukatChoudhuryetal.2005). 19.2Positioners27719.2.3DigitalPositionersInthe1990slow-powermicrocontrollersbecameavailablethatcouldbeoperatedonlessthan6mW(UlneandWase2000).Theymadeitpossibletodesignsocalled“smart”or“intelligent”positionersbecausethesein-strumentsofferfeaturesthatformerlyrequiredhumanoperators:automaticinitialisationwithself-calibration,self-diagnostics,status-anddiagnosticmessagesusinggraphicalLCDdisplays.Forexampletheauto-initialisationroutineautomaticallysetsthebasicfunctionssuchaszero,range,speedandsensitivity.Thisprocesstakesonlyacoupleofminuteswhilethemanualiterativetuningofapneumaticpositionercantakeanhour.Themicrocontrollercanalsobeusedtoimplementanon-linearcharacteristicbetweenreferencesignalandvalvestemposition,afeaturethatformerlyrequiredacam.Itcanalsocompensatefortheerrorduetotherotationofthemeasuringarmandreducethiseffectconsiderably(Schwarzetal.1990).Thebasicversionofadigitalpositionerisstillatrue2-wireinstrumentusingtheindustrystandard4to20mAsignal.Assuchthedigitalposi-tionerisplugcompatiblewithwhathasbecometheindustrystandardoverthelast30years.Moreadvancedinstrumentsarealsocapableoftwo-waydigitalcom-municationoverthesamewiresusedfortheanaloguesignalusingtheHARTprotocol(HighwayAddressableRemoteTransducer).Anall-digitalpositionerreceivesandtransmitsdataviaafieldbus,e.g.PROFIBUS-PA.Thisfieldbustechnologyfacilitatesimprovedcontrolarchitecturesandre-ducedwiring.Italsoallowsfortwo-waycommunicationforprocess,valveandinstrumentdiagnostics.Inthepastvalvesusedtobereplacedonaroutinebasis,nowitispossibletodetectthephysicalconditionofthevalveremotelyandreplaceitonlyifnecessary.Detailsaboutconditionmonitoringcanbefoundine.g.(KoŹcielnyetal.1999;Mundry2000).AtypicaldesignofadigitalpositionerisshowninFig.19.8.Apreci-sionpotentiometeroranoncontactmagneticcouplingusingtheHallEffectprincipledetectsthestemposition.Thisanaloguesignalisconvertedbyananalogue-to-digitalconverterwitharesolutionof0.1%(KoenigandOhligschläger1989).Theoutputvalueiscomparedwiththereferencepo-sitionandifthedifferenceisgreaterthantheneutralzoneofe.g.±0.2%ofthetotalrange,oneoftheswitchingvalvesisopened2.Theyaretwo-stagevalvesthatoftenusepiezoelectricpilotstagestominimisethere-quiredelectricpower.2Therearealsodigitalpositionersthatusecontinuouslyworkingspoolvalves(Mundry2002:17–20). 27819ControlofActuatorsforProcessValvesSupplyDisplayReference_++24V+5VAD+yMicro-controller_yADControlledVariableProcessValveFig.19.8.SchematicblockdiagramofadigitalpositionerTheleakageflowofthesevalvesisverysmallanddigitalpositionersthereforeoftenhaveonlyaverysmallpercentageoftheairconsumptionofpneumaticpositionersresultinginsignificantlyloweroperatingcosts.However,theactuallyneededamountofairdependsveryheavilyonthedemandedstemmovements(Coughran2003).Somemanufacturersusealargervalvefortheoutletthanfortheinlettocompensateforthesmallerpressuredifferentialwhenexhausting.Thepressureinthevalveactuatoristypicallyonly1to3barwhilethesupplypressurerangesfrom4to6bar.Usingidenticalvalvesthestrokingtimeswouldthereforebesignificantlydifferent.Thecontrolstrategyofdigitalpositionerscouldbecharacterisedas“bang-bangcontrolwithdeadzoneandpulse-width-modulatedtransitionarea”.ThecharacteristiccurveforonevalveisgiveninFig.19.9.Iftheer-ror,i.e.thedeviationbetweensetpointandactualposition,isgreaterthanthelinearzone,thevalveiscompletelyopened.Thislinearzonemayliebetween1.5and30%ofthefullrange.Iftheerrorislessthanthedeadzone,thevalveisclosed.Inbetween,aPWMsignalisusedtohaveapro-portionalmeanvalveopening.However,detailsoftheimplementedcodeareusuallynotavailable,especiallythealgorithmsforfrictioncompensa-tion.Tokeepthepositionerhousingfreefromharmfulvapoursandconden-sation,itcanbefilledwithexhaustedairfromthepiezoelectricpilotvalve.Thesamecanbedonewiththespringchamberoftheactuatortoavoidcorrosion.Digitalpositionershavetypicallyahysteresisoflessthan0.3%,adeadzoneoflessthan0.2%oralinearitydeviationoflessthan0.3%offullscale(RockstrohandHofmann1996). 19.2Positioners2791ValveInput0Deviation(0.2%)DeadZoneLinearZone(1.5-30%)Fig.19.9.Actuationstrategyforaswitchingvalve:valveopeningasafunctionofdeviationfromcommandedposition 20DigitalSimulationThedesignofquiteanumberofpneumaticdrivesisstillbasedononlysteady-statespecifications,e.g.therequiredforceatendofstroke.Givenasystempressurethatisoftendeterminedbytheexistingcompressorhard-ware,therequiredpistonareacanthenbecalculatedandacylindercho-sen.Thevalveandaccessorieslikefittingsandtubesareoftenselectedtohavethesameportsizeasthecylinder.Ifthespecificationsaremoredetailedandincludearequiredstroketimeandadescriptionoftheloadwhichhastobemoved,adedicatedcomputerprogramforthesizingofcylinderdrivescanbeused.Allmajormanufac-turersprovidethiskindofsoftwarewherethecustomercanselectallcom-ponentsfromacatalogue,simulatethesystemandplotpressures,velocityandpositionoftheload.Withinashortwhileandwithsometrialanderrorasuitablecircuitcanbedesignedwhichwillworksatisfactorilyinhard-ware.Withsomeexperienceerrormarginstoallowformodellingerrorscanbeestablished,e.g.todesignforastroketimethatis30%shorterthanrequiredbythespecificationandtousetheone-wayflowcontrolvalvesforafinetuningonthemachine.Athirdcategoryofdesignapproachesdealswithcomplexsystemswithinteractingelectroniccontrolsystems.Lorries,forinstance,havealwaysbeenaggregationsofseveralsubsystemslikeengine,gearbox,brakesandsoforth,andthusshowedacertaincomplexity.Butinformertimesthosesystemsworkedratherindependentlyandcouldthereforebedevelopedandtestedseparately.Nowadaystheyarestronglyinterdependent.Asaconsequence,subsystemscannolongerbedesignedandtestedseparately.Usingmodernsimulationtoolsitispossibletobuildamathematicalmodelofasystemandsimulatethebehaviourduringacoupleofhours.Anditdoesnotnecessarilyrequireanexpertinnumericalanalysis.InourDepartmentofMechanicalEngineeringthiskindofcomputeraideddesignispartofasetoflaboratoryexperimentsforthirdyearmechanicalengi-neeringstudents(Beater2000).However,digitalsimulationofcomplexsystemsstillrequiresanexperi-encedfluidpowerengineerwhohasathoroughunderstandingofthesys-temheisworkingon.Butwithmoderncomputersoftwarehisefficiencyisincreasedverymuch.Hecanmodifyandanalysesystemsinacoupleof 28220DigitalSimulationhourswithouthavingactuallytoinstallcomponentsordoingtests.Thischaptershowswaystomodelfluidpowersystem,pointsoutthehistoricaldevelopmentanddemonstratestheuseofaparticularsimulationenviron-ment.20.1ModellingApproachesInthepastdifferentapproacheshavebeendevelopedtomodelfluidpowersystems.Usuallythemathematicalanalogyisapplied,i.e.theuseofequa-tionstodescribeatechnicalsystem.Manyoftheseequationsormathe-maticalmodelsweredevelopedinthe1960sbutcouldnotbeusedthenbecauseofthelimitedcomputingpoweravailableatthattime.Figure20.1showsaverysimpleexamplethatwillbeusedinthischaptertodemon-stratevariousapproaches.Thesystemconsistsofvolume1,alreadychargedwithcompressedair,nozzle2andchamber3whoserodisconnectedtoslidingmass4.Thisisrestrainedbyspringanddamper5inparallelwhicharefixedattheirrightends.Initiallythepressureinthevolumeis0.15MPa,themassisatrest.Att=0thesimulationisstartedandairflowsthroughthenozzletothechamberwerepressurebuildsup.Thisgeneratesaforceandadisplace-mentoftherodwhichpushesthemasstotheright,workingagainstspringanddamper.Usingthemodelsfromthepreviouschapters,thegoverningequationscanbewrittendown.Tokeepthisexamplesimple,anisothermalprocessandsubsonicflowinthenozzleisassumed.34512Fig.20.1.Simpleexamplewithvolume1,nozzle2,chamber3,mass4andspringanddamper5(modifiedscreenshotfromDymola) 20.1ModellingApproaches283Volume1:p(t)⋅V=m(t)⋅R⋅T;V,R,Tareconstant(20.1)1111wherep1pressure,V1volume,m1(t)timedependentgasmassinvolume,Rgasconstant,Tgastemperature.t(20.2)m1=m1(t=0)−³m&1dτ0Nozzle2withsubsonicflow:2⊬p·∆2−b¸∆p¸(20.3)m&=p⋅C⋅ρ1−1∆¸110∆1−b¸∆¸«¹whereCsonicconductanceofnozzle,bcriticalpressureratio,ρ0densityatreferenceconditions.Chamber3:p(t)⋅V(t)=m(t)⋅R⋅T;R,Tareconstant,(20.4)222p⋅A=F(20.5)2whereApistonarea,Frodforce.t(20.6)m2=m2(t=0)+³m&1dτ,0V2=x⋅A+V0(20.7)whereV2(x)positiondependentvolumeofchamber,xpositionofpiston,V0chambervolumeforx=0. 28420DigitalSimulationSlidingmass4withspringanddamper5:M⋅&x&=F−cSpring⋅x−DDamper⋅x&(20.8)whereMmassofslider,cSpringspringconstant,DDamperdampingconstant.Thissimplesystemshowsthetypicalstructureofsimulationmodels.Theyconsistofbalanceequationsandconstitutiveequations.Themassbalanceequationforalumpedvolumeintegratesthenetflowintothisvol-umetogivethegasmassandsubsequentlythepressure,e.g.Eqs.(20.2)and(20.1).Ingeneral,abalanceequationisadifferentialequationinthethroughvariableyieldingtheacrossvariable.Thethroughvariablecanbecomputedwithatypicallynon-linearalgebraicequationfromtheacrossvariable,e.g.Eq.(20.3)wherethemassflowrateisgivenasafunctionofthepressure.Thissystemofalgebraicanddifferentialequations(20.1–8)cannotbesolvedanalyticallybuthastobeprogrammedforacomputer.Informertimesanaloguecomputerswereusedthatrequireda“translation”oftheseequationstoasignalorientedblockdiagram(Reethof1955).Typicallyananaloguecomputerhashardwaredevicesforintegration,amplificationandsummation,sometimesfornon-linearoperationslikemultiplicationorcomputationofsquareroots.Theseanaloguecomputersareusednowa-daysonlyforveryspecialwork,e.g.tostudytheeffectofnoise(Cellier1992).Today,digitalcomputersareusedandtheyofferarangeofdifferentsimulationapproaches.40yearsagotheengineerhadtohandcodethesimulationprograminahigherprogramminglanguagelikeFORTRAN(McCloyandMartin1963).Later,textbasedblockorientedsimulationprogramsweredevelopedthatmadethehandlingofdifferentialequationseasierandincludedsupportforintegrationorgraphicalrepresentationofresults(Straussetal.1967).Todaysomepopularprogramssimplysimu-lateananaloguecomputerbyofferingalltheblocksinagraphicalenvi-ronment.Table20.1showsasexamplesiconsfromtheModelicaStandardLibrarythatamongotherapproachesalsosupportsblock-orientedmodel-ling(anon.2006b).Fig.20.2givestheresultingblockdiagramoftheex-amplesystemfromFig20.1.DuringthetrainingofyoungengineersitmakessensetodevelopablockdiagramasinFig.20.2.Theylearnhowafluidpowersystemcanbemodelledandwhatkindoffeedbackpathsexistinsuchasystem.Forthedaytodayworkofanexperiencedengineerthisisaveryexpensivetask,however,becauseithastoberepeatedforeverynewsystemmoreorless 20.1ModellingApproaches285fromscratch.Itisalmostimpossibletobuildalibraryofreusablecompo-nents.Thereasonisthatforeverycomponenttheflowdirectionhastobetakenintoaccount.Thedrawingofblockdiagramsisthereforeveryerrorprone.Fig.20.2.BlockdiagramofthesimpleexamplemodelfromFig.20.1(modifiedscreenshotfromDymola)Table20.1.BlocksforsignalorientedmodellingfromModelicaStandardLibraryIconOperationEquationtintegrationy=³u(τ)dτ0gainy=k⋅uadditiony=u1+u2multiplicationy=u1⋅u2squarerooty=u 28620DigitalSimulation0.14Volume0.12Pressures(MPa)Chamber0.18642Speed(mm/s)0-20246810Time(s)Fig.20.3.SimulatedtrajectoriesofpressuresandvelocityforexampleHowever,notonlythecapabilitiesofcomputerhardwarehavebeenim-provedtremendouslyinthelastdecades,butthemethodologiesofmodel-lingcomplexsystemswerealsodevelopedfurtherandmakeittodaypos-siblethatanengineercanconcentrateontheengineeringpartofthetask,e.g.modelling,systemdesignandanalysis,whileadvancedcomputerpro-gramsdothemenialwork,likesymbolicformulamanipulationandpro-ductionofexecutablecomputercode.Thefirstdevelopmentswerealreadymadeinthelate1970sbutstoppedbecausecomputerswerenotpowerfulenoughthen.ImportantworkinthatareawasdonebyElmqvistwhode-scribedwaystomodellargesystemsin1978.Muchfurtherprogresshasbeenmadesincethenandtodaywecanusetheconceptsofobject-orientedmodellingtosolverealworldproblems.20.2PrinciplesofObject-OrientedModellingTheexampleinFig.20.1showsthatevenaverysimplemodelleadstoacomplicatedblockdiagramandthatitisalmostimpossibletobuildali-braryofreusablecomponentsusingablockorientedsignalflowapproach. 20.2PrinciplesofObject-OrientedModelling287Asolutiontothatproblemofreusabilityisobject-orientedmodellingwhichcanbedescribedas1:Object-orientedmodelling:“Abilityofacomputerprogramtorepre-sentaphysicalobjectregardlessoftheenvironmentwherethatob-jectisused.”Thisleadstosomeinterestingconsequences:-nocausality,i.e.nodefinitionwhetherportsorinputsareoutputs,-modelsareequationbased,-modelandsystemhavethesamestructure,-subsystemsareconnectedthroughflow-andacross-variables.Theideabehindobject-orientedmodellingistodescribeacomponentwithoutlookingattheenvironmentinwhichitwillbeusedlater.Fromthisapproachfollowsthatitisnotdefinedwhichvariableswillbeinputandwhichoutputvariables,ordependentandindependentvariablesinthecompletesystem.AsimpleexampleisanelectricDCmotorthatcanbeusedasamotorandthenhasvoltageandcurrentasinputvariableswhilethetorqueandtheshaftspeedareoutputvariables.InotherapplicationsthisDCmachinecanbeusedasageneratorandthenhastorqueandshaftspeedasinputvariableswhilevoltageandcurrentareoutputvariables.Asimilarexamplearehydraulicpumpsormotors:thecircuitdesigndeter-mineswhethertheyareusedaspumps,i.e.convertingmechanicalenergytohydraulicenergy,orasmotors.Becausetheinputandoutputvariablesarenotdefinedwhenwritingthecomponentmodelbutdeterminedwhenanalysingthecompletesystemmodel,thisapproachiscallednoncausal(oracausal).Theblockorientedwayiscalledcausalmodellingbecausetheinputandoutputvariablesaredefinedwhenwritingthecomponentmodel.1Thisisanengineeringdefinitionofobject-orientedmodelling.Amoreformal,computersciencedefinitionisgivenbyFritzson(2003:25):”Object-orientationisprimarilyusedasastructuringconcept,emphasizingthedeclarativestructureandreuseofmathematicalmodels.Ourthreewaysofstructuringarehierarchies,component-connections,andinheritance.Dynamicmodelpropertiesareexpres-sedinadeclarativewaythroughequations.Anobjectisacollectionofinstancevariablesandequationsthatshareasetofdata.However:Object-orientationinmathematicalmodelingisnotviewedasdynamicmessagepassing.” 28820DigitalSimulationTocouplecomponent-orsubsystem-models,aphysicalorientedap-proachisused.GeneralisingKirchhoff'slawsforelectricsystems,throughandacrossvariablesaredefinedwhoseproductispower.Forhydraulicsystemsthismeansthatthethroughvariableisthevolumeflowrate,Q,andtheacrossvariableisthepressure,p.Theproductofvolumeflowrateandpressureishydraulicpower.Inpneumaticsystemsthepressureinde-pendentmassflowrateisusedinsteadtosimplifycalculations.Equiva-lentlyintheelectricworldthevariablesforelectriccurrentandvoltageareused;inthemechanicaldomainforceandvelocity.Object-orientedmodelsareequationbased.Theydonotuseassignmentstatementslikemosthigherlevelprogramminglanguages.TheforceonapistoncanbethereforedescribedasF=p⋅A.(20.9)Thesymbolicformulamanipulationoftheobject-orientedsimulationpro-gramscansolvethisequationfortherequiredvariable,e.g.thepressurep:F(20.10)p=.AThisobject-orientedapproachmakesitpossibletobuildalibraryofre-usablecomponentsthatcaneasilybeusedtomodelpneumaticsystems.Thenumericalanalystcanspendhistimetomodelthesystemandanalysethesimulationdata.Graphicalsymbolsandagraphicaluserinterfacecanbeusedandthenapneumaticsengineerunderstandsthesimulationmodelatoncebecauseitusesthesamesymbolsasapneumaticcircuitdiagram.Thetediousanderrorpronepartofrearrangingandsolvingequationsisdonebyadvancedcomputerprograms.20.3TheObject-OrientedModellingLanguageModelicaBeginningin1996,agroupofpeoplegatheredtodevelopanewobject-orientedmodellinglanguage:Modelica.Theyhaddifferentbackgroundsandwereinterestedinobjectorientedmodellingeitherfromthecomputersciencepoint,asusersofsimulationpackagesorascommercialtoolsup-pliers(Elmqvistetal.1999).TheclaimofModelicais:aunifiedobject-orientedlanguageforphysi-calsystemsmodelling.ThatmeansthatModelicaisintendedtobealan-guageforallengineeringdomainsandthisisreflectedbytheModelicali-braries.Rightfromthebeginning,therewerelibrariesforblockorientedsignalprocessing–e.g.asSimulink–,mechanics,multibodymechanicsandsimpleelectronics.Todaythelistismuchlongerandincludesvehicle 20.4FluidPowerLibrariesinModelica289dynamics,airconditioning,powertrains,SPICElikeelectronics,thermalsystemsorsomeaspectsofthermodynamics.Therearetwolibrariesforthefluidpowerdomain:HyLibtomodelhydrostaticdrivesandPneuLibforpneumaticactuationsystems(Tiller2001;Fritzson2004;anon.2006b).OneoftheimportantfeaturesforusersoftheModelicalibrariesisthefactthattheyaretypicallyprovidedassourcecode.Abeginnercanpickandplacecomponentsfromthelibrarytobuildhismodelandbyinspect-ingtheequationsgainaninsightintothemodellingofthatdomain.Anex-pertcanusethelibraryasatemplateforhisowncollectionofmodels,e.g.bycopyingandmodifyingthelibraryequations.Asimulationofahydraulicsystemcanoftenstartat“zero”,i.e.allpres-suresareequaltotheambientpressureandallvelocitiesarezero.Thisleadstoaverysimpleinitialisationofthesimulationmodelbecauseallstatevariablesarezero.Thesituationismuchmoredemandingforpneu-maticsystems.Herethegasmassinalumpedvolumeisastatevariableandthisisnon-zeroatatmosphericpressure.Thesituationbecomesevenmoreinterestingforcylinderchamberswherethevolumedependsonthestrokewhichisitselfastatevariable.BeginningwithModelica2.0thisinitialisationproblemcanbesolvedveryeasilybysettingupsocalled“initialequations”thatareevaluatedbeforethesimulationrunstarts.Ate-diousandoftenalmostimpossiblemanualsettingofinitialvaluesisthere-forenotnecessary,butpossible.SeveralimplementationsofModelicaareavailable.Oneofthemostad-vancedandforindustrialapplicationmostsuitedtoolsisDymola(anon.2006a)fromtheSwedishcompanyDynasimAB,Lund.Theprogramwasoriginallywrittenin1978asaresearchtool(Elmqvist1978),becameacommercialproductin1992andsupportsModelicasince1999.Itoffersallstandardcapabilitiesofmodernsimulationprogramsandproducesveryefficientsimulationcode.20.4FluidPowerLibrariesinModelicaTherearetwofluidpowerlibrarieswritteninModelica.Thoughpneumaticsystemswereinthepastsometimesbuiltfromhydrauliccomponentsandtheirprinciplesofoperationseemidenticaltohydraulicsystems,thereareveryimportantdifferenceswhenmodellingthem.Amongthemostobvi-ousdifferencesisthefactthattheoilvelocityneverreachesthespeedofsound,buttheairvelocitydoes.Airis“very”compressiblewhencom-paredwithalmostincompressibleoil.Themomentumofamovingoilcol-umncanoftennotbeneglectedwhiletheinductanceofairisusuallyvery 29020DigitalSimulationsmall.Allthesedifferencesrequiredifferentmodelsanddedicatedlibrar-iesarethereforeavailable.ThepneumaticslibraryPneuLibisapowerfulyetsimpletousetoolforthemodellingofpneumaticsystemsusingairandpressuresbetweenat-mosphericpressureand10bar(anon.2006c).Itprovidesbasicmodelclassesforthemodellingofcylinders-bothstandardcylinderswithcon-stantpistonareaandbellowswhichhaveastrokedependentpistonarea-andmotors,valvesandrestrictions,lumpedvolumes,linesandsensors.Intotaltherearemorethan80models.If,however,speciallydesignedcom-ponentsareused,theycaneasilybemodelledbymodifyinglibrarycom-ponents.Allrelevanteffectsareavailableassubmodels.Thereisquiteanumberofsimulationpackagesforpneumaticcompo-nentsthatusesimpletextbookmodels.Forrealapplicationsthesemodelsusuallyfailbecausetheyleadtoseverenumericalproblems.Oftenthesourceofthemodelsisnotgivensothatamodellerhastomakeguessesaboutthem.InPneuLiballmodelscanbeexaminedatsourcecodelevel.Themodellingconceptallowspneumaticcomponentstobeconnectedinanarbitraryway,e.g.inseriesorinparallel,justbydrawingconnectionlines,nospecialcomponentsforsplitsormergersarerequired.TheadvancedfeaturesofModelica,e.g.theinitialequationsection,areusedtomaketheinitialisationofstatesuserfriendly.Inhydraulicsatmos-phericpressureisusedasreferencepressure.Thereforeasystematresthaspressurestatesofzero.Inpneumaticsthegasmassinavolumeisusedwhichisnon-zeroatatmosphericpressure.Thereforeanumberofcalcula-tionsareneededtocomputethegasmassesinalllumpedvolumeswhichmayincludethecalculationofthegeometricvolumes,e.g.ofcylinders,beforethesimulationcanbestarted.20.4.1ExamplesofLibraryModelsAsimplepneumaticcomponent,acapillary,willbeusedtodemonstratethelibraryconcept.Thisisatypicalexampleofacomponentthathastwoports.Tobuildastructuredlibrary,itmakessensetostartmodellingthiskindofcomponentwithageneralclassdescribingaport,i.e.alocationwhereairentersacomponent,andanotherclassdescribingacomponentwithtwoportsandnoenergystorage.TheModelicacodelookslike(graphicalinformationanddocumentationareomitted): 20.4FluidPowerLibrariesinModelica291connectorPort"Pneumaticport"Modelica.SIunits.Pressurep"pressureatport";flowModelica.SIunits.MassFlowRatem_dot"massflowrate";endPort_1;partialmodelTwoPortComp"Superclass."Modelica.SIunits.VolumeFlowRatem_dot;PneuLib.Interfaces.Portport_1;PneuLib.Interfaces.Portport_2;equationm_dot=port_1.m_dot;port_1.m_dot+port_2.m_dot=0;endTwoPortComp;TheclassTwoPortCompusestwoinstancesoftheconnectormodelandcanbeusedinacomponentmodelbyinheritancewhichmeans“automaticallycopying”thecodeintothenewmodel.Thekeywordforinheritanceisextends.Tomodelacapillary,onlythoseequationshavetobeaddedthatdescribetherelationbetweenpressuredropdpandflowratem_dot.Thetemperatureandpressuredependentviscosityisde-scribedbythefunctionnuAir.modelCapillaryNoStates"Capillary"extendsPneuLib.Interfaces.TwoPortComp;parameterModelica.SIunits.Lengthlength=10;parameterModelica.SIunits.Diameterdiameter=1.e-3;Modelica.SIunits.Pressuredp;Modelica.SIunits.KinematicViscositynu;equationdp=port_1.p-port_2.p;nu=PneuLib.Interfaces.nuAir(temp_surroundings,(port_1.p+port_2.p)/2);m_dot=dp*diameter^4*Modelica.Constants.pi/(128*length*nu);endCapillaryNoStates;Modelsofthiskindcanalreadybeusedtobuildasystemmodel.How-ever,whenconnectingthesemodelstoasystem,theuserhastoaddmanu-allylumpedvolumestomodelthepressuredynamics,i.e.theairpressurebetweentheparts.Theportsofarealcomponentareairfilledandthere-forealumpedvolumeisaddedateachportofthemainlibrarymodel.AnexampleisthemodelCapillarywhosecompositiondiagramisshowninFig.20.4.Thisapproachmakesitveryeasytobuildasystemmodelbecausetheuseronlyhastoselectcomponents,enterparametersandconnectthem.Theuserdoesnothavetoincludelumpedvolumesaswithotherfluidpowersimulationpackages.ThereasonthatitcanbedonethiswayisthataModelicatoolcancopewiththeresultingvolumesconnectedinparallel 29220DigitalSimulationwhichleadtoahigherindexsystem.Duringthesystemanalysisthetoolfindsvolumesthatareconnectedinparallelandlumpsthemtogether(Mattssonetal.2000).Fig.20.4.Capillary,mainModel,consistingofbasicmodelCapillaryNoStates1andtwovolumes(states)attheports(modifiedscreenshotfromDymola)20.4.2ComplexComponentModelofthePneumaticLibraryFigure20.5showsthecompositiondiagramofapneumaticcylinder.Theleftandtherightchambersaremodelled.Inbetweenisthepistonandattheendsarenozzlesforthestrokecushioningandhardstops.Thismodelhasthusthesamestructureasthecomponentwhichisanotheradvantageofobject-orientedmodelling.Ifthestrokecushioningisnotrequired,itcaneasilybeerasedfromthemodel.Inblockdiagramorientedsimulationlan-guagesthemodelandthecomponenthaveverydifferentstructuresandmodificationofthemodelismuchmoredifficult.Figures20.4and20.5showthetypicalmodellingapproachinModelica.Basicmodelsaredevelopedinatextbasedenvironmentwhereequationsareusedtodescribethebehaviourandtheyaregiveniconsthatdescribethefunction.Complexcomponentsorthewholesystemarecomposedgraphicallyusingthebeforedefinedbasicmodels.Tobuildthemathematicalmodel,thetoolanalysestheequationsandusesKirchhoff'slawstocouplethecomponents.Thismeansthatthesumofallthroughvariablesatanodeisequaltozero.Forfluidpowersystemthismeansthatthesumofallflowratesisequaltozeroattheconnectionpointoftwoormorecomponents;thereisnostorageelementatanode.Allacrossvariablesatanodehavethesamevalue,i.e.allpressuresareequal. 20.5LibrarySolutionforExample293Fig.20.5.Compositiondiagramofapneumaticcylinder(modifiedscreenshotfromDymola)20.5LibrarySolutionforExampleBuildingtheexamplemodelfromFig.20.1iseasywhenthepneumaticslibraryPneuLibisused.Alltheuserhastodoisstartthetool,hereDy-mola(anon.2006a),openthelibraryPneuLib,pickandplacethecompo-nentsandconnectthem.Finallyhehastoentertherequiredparameters,pressthebutton“simulate”inDymolaandafterlessthan3secondshecanviewthecomputedtimehistoriesofallvariables,whichareidenticaltothoseinFig.20.3. 29420DigitalSimulationFig.20.6.Solutiontoexample(screenshotfromDymola)Figure20.6showsthesimplestcylindermodelavailableinPneuLib.Itdescribesthepressurebuild-upinachamberwithvariablestroke.Itwasusedforthisexamplebecauseitusestheequationsfromsection20.1.Therearemoremodelsinthelibrarythatforinstancealreadyincludethepistonmassoradetailedmodelofthefrictionbetweenbarrelandpiston.Figure20.5givesadetailedmodelofapneumaticcylinderthatincludesfriction,hardstopsattheendandpneumaticstrokecushioning.20.6Multi-DomainModelsTheobject-orientedapproachmakesiteasytomodelsystemsthataregov-ernedbymanydomains.AsimpleexampleistheoilcushioningcylinderfromChap.17.5.Figure20.7showsaDymolascreenshotofthemodel.Afterdrawingthecompositiondiagram,theengineerhastoenterthepa-rameters.Oftenthisisthemosttimeconsumingpartofthewholestudy.Thenhepressesthebutton“simulate”inDymolaandaseriesofcomplextasksisfulfilledbytheprogram.Themathematicalsystemmodelisgener-ated,analysedandifpossiblesimplified.Thisinvolvesalotofadvancednumericalmethodsandsymbolicformulamanipulation(Mattssonetal.2000).FinallyCcodeisgenerated,compiledandlinkedwiththeDymolalibraries.Theuser,however,onlyseesthefollowingmessages: 20.6Multi-DomainModels295Fig.20.7.Modelofanintegratedcushioningcylinder(modifiedscreenshotfromDymola)15constantsfound.65parameterboundvariablesfound.376aliasvariablesfound.410remainingtimedependentvariables.FinishedThisprocesstakesabout4sonaPCwithPentium4processorwith2GHz.Torunthesimulationtakes32sfortheresultsshowninFig.20.8.Initiallythesystemisatrest,thenthepistonretracts.Att=5sthepistonstartstoextractwithmaximumspeedandatt=7.5thevalveisclosedrapidly,resultinginanoilpressuresurge. 29620DigitalSimulation0.4Position0.2s,v0Speed642Air(bar)0200100Oil(bar)0012345678Time(s)Fig.20.8.Simulationresult,notehighpeakpressurewhenclosingvalve 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Indexactuatorcapillary36,37bellows133–137causality287muscle140–141chamberstiffness20piezoelectric94changeofstaterackandpinion145–146isentropic18Scotchyoke147–148isobaric13semi-rotatory145–149isothermal18spring-and-diaphragm271polytropic20vane148–149chokedflow31aircircuitatmospheric8meter-in241–242free2,10meter-out239–241fuse218closedsystem12,22motor151–168coefficientofcompressibility51properties8,10coil82–84standardconditions8Colebrook’sformula38turbine168–170compressedair5ANR10mathematicalmodel6areacompressibilityfactor8effectivecompressibleflow34–35bellows134condensation8ISO635851conductanceprocesscontrolvalve271line58–59effectivecross-sectional52sonic41armature81connectorartificialmuscle140–141losses62–63atmosphericpressure9push-in62push-on62bar10continuityequation27bellows133–137controlbend62–64gain-scheduling264bimorphbender94–95hybrid264Blasius38position247–265bleednozzle195pressure202bucklingofcylinderrod101speed164–168brakechamber139–140statespace256–264stroke-time235–237 320Indexcriticalpressureratio31dewpoint8cushionseal99,102–104diaphragmcylinderchargingvalve212–213duplex127pressureregulator196–197efficiency119processvalve269,271–273friction116–122rolling137–139heattransfer123–126digitalsimulation281–296impact142–143directionalcontrolvalve171–183knocking142–143operation175–181leakage122piloted174modelling112–116poppet173multi-position127–130simulationmodel181–183rodless130–133switchingtime176–178cable132dischargecoefficient32magnetic132–133dither87split-seal130–132droop194rollingdiaphragm137–139dutycylce86singlerod99dwelltime118strokecushioning102–112Dymola289,293–295tandem129triplex127efficiencycylinder119density5,10piezo-electricactuation95–97deadvolume115vanemotor151designelbow63airturbine170electro-mechanicalconverter81–97bellows134energysaving243–244brakechamber139enthalpy16cylinderequationknocking143balance284multi-position128constitutive284rod99ofstate7,8,25rodless130–133expansionratio158rollingdiaphragm138directionalcontrolvalve173–174flowmuscleactuator140–141choked31non-returnvalve186coefficient53oilcushioning246force228pressurecontrolvalvefrictional36diaphragm197function30piston198gain224,227proportionalsolenoid85laminar33,37rack-and-pinion146orifice32–35reliefvalve213subsonic31vaneactuator149turbulent33vanemotor154,155flowcontrolvalve215–219 Index321flowratehysteresis211calculation41restrictions48idealgas6simplified49–50equationofstate7mass10properties6nominal52impactenergyspecificationforvalves50–54impactcylinder142volume10strokecushioning106fluid25incompressibleflow32fluidmechanics25internalenergy16fluidpower1isobaricprocess13fluidics1isochoricprocess11forcebalanceisothermalprocess18inpositioner275isotropicprocess18inregulator196–198forwardflowcharacteristic193laminarflow33,37freeair10Laval31freedischarge28leakagefrequencyresponsecylinder122directcontrolvalve225,226energysavings243line76–79linevanemotor167steady-stateloss55–61frictionfrequencyresponse76–79cylinder116–122mathematicalmodeldynamic118discretized65–69static117frequencydomain76–79steady-state118resonancepeak79factor37–39temperature56–57FRLunit193linearity211fullpressuremotor158Machnumber23gainmassflowrate10dynamicofpositioner274mathematicalanalogy282scheduling264mathematicalmodel282staticofpositioner274McKibbenartificalmuscle140–141gasmeter-in241–242constant8meter-out239–241ideal6microfluidics5law6Modelica288–290gaugepressure9modellingcausal287Hagen37object-oriented287handpiece169motorheattransfer123–126characteristics151–153humidity8process160hydraulics1vane153–168 322Indexmovinggauge9coil93ratio41magnet93regulator193–212static9nominaltotal9diameter52processvalve269flowrate52pulse-widthmodulation86–88nozzlePVdiagramideal28–32constantpressure15model41reversible19nozzel-flapper272–273vanemotor160objectorientedmodelling286–288ratioobserver248,263criticalpressure31oilcushioning245ofspecificheatcapacities19opensystem22relativehumidity5orificereliefvalve212–213dischargecoefficient32repeatability212flow32–35Reynoldsnumber38compressible34critical38incompressible32overlap174Sanville41sealPa10cushioning102–104packing269–270viton101parallelconnection50sensitivity212piezoelectricseriesconnection49–50bender94–95shadingcoil82effect94softstartvalve213–214stack94solenoid81–85plunger81design81–82pneumatics1dynamics92pneumatictimer217proportional85Poiseuille37sonicpolytropicprocess20conductance41positioner270–279velocity23analogueelectro-pneumatic276specificheatcapacitydigital277constantpressure16–17pneumatic275constantvolume11,13pressure9speedabsolute9controlatmospheric9cylinders237–242criticalratio31vanemotors164–168dynamic9ofsoundinair23effective9spool174gain224,228 Index323standardreferenceatmospherepressurerelief212–213meteorological10processcontrol269technical5,8proportionaldirectional221–233startingtorque152quickexhaust191statereconstruction263relief212state-spacemodel256–258shut-off185–192stiffnessofchamber20shuttle189–190strokecushioning102–112soft-start213–214St.Venant30switchingtime176–178subsonicflow31symbols171–172suckingcoil93throttling215Sutherland’sformula26–27twinpressure190–191thermaltimeconstant125vaneactuator148–149thermodynamicprocess11vanemotorconstantpressure13airconsumption165constanttemperature18design155constantvolume11designschemes154general22efficiency151polytropic20mathematicalmodel156-164reversible18speedcontrol166–168throttlingstartingtorque152cylinderspeed239–241throttling164–166vanemotor164–166variableturbomachine168across284turbulentflow33through284VDR84underlap189,191velocityofsound23viscosityvalvedynamic26analogue222kinematic27automaticshut-off218Sutherland’sformula26–27charging212temperaturedependency26check185voicecoil93delay217volumeflowrate10directionalcontrol171non-return185–187Wantzel30non-returnoverride188normallyclosedNC171normallyopenNO171one-wayflowcontrol216pilot172poppet173portnumbering171pressureregulator193–212dome-loaded199