Design and performance analysis of a circular(4)

ProcIMechEPartC:

JMechanicalEngineeringScience2016,Vol.230(2)189–205!IMechE2015

Reprintsandpermissions:

sagepub.co.uk/journalsPermissions.navDOI:10.1177/0954406215572435pic.sagepub.com

Designandperformanceanalysisofacircular-arcgearpumpoperatingathighpressureandhighspeed

YangZhou,ShuanghuiHaoandMinghuiHao

Abstract

Thispaperpresentsahigh-pressureandhigh-speedgearpumpforaerospaceapplicationandintroducesacircular-arctoothprofilethathasnotrappingfeatureandwhosegearsareincontinuousone-pointcontactintheplaneofrotation.Basicdimensionsaredeterminedandperformanceparametersofthegearpumpareobtainedbystructuraldesignofthegearpump.Theperformanceparametersarediscussedfordifferenttoothprofiles.Adisctoolandhobaredesignedtogeneratethecircular-arctoothprofile.Acomputer-aideddesign(CAD)systemforthedesignofthecircular-arcgearpumpisdevelopedwithfeaturesoftoothprofiledesign,tooldesign,andnumericalcontrol.ThetestgearswereprocessedtoverifythecorrectnessofCADsystem.Astudyoftheoutletpressureofthecircular-arcgearpumprevealsthatthedevelopedhigh-pressureandhigh-speedgearpumphaslowoutletpressurefluctuations.

Keywords

High-pressureandhigh-speedgearpump,circular-arctoothprofile,performanceparameters,CAD,pressurefluctuations

Datereceived:15September2014;accepted:12January2015

Introduction

Pumpsystemsarepopularmechanicalsystemsforaerospacehydraulicsystems,operatingsystems,andaircraftenginefuelsystems.1Thepistonpumpandgearpumparewidelyusedinaerospaceapplications.Thepistonpumpcanworkinawiderangeofworkpressuresandathighrotationalspeeds.However,apistonpumpexhibitspoorstabilityandissensitivetooilpollutionandeasilybecomesstuckwhenrunathighspeed.Incontrast,agearpumpissimpleandinsensitivetooilpollution.Thedrawbacksofthegearpumprelatetolowrotationalspeedsandlowpressure;aninvolutegearpumpisoftenfoundtohavetrappedoilandoutletflowripples;2–6thebend-ingstrengthofinvolutegearislow.Inresponse,acircular-arctoothprofile7–10whosegearsareincon-tinuousone-pointcontactintheplaneofrotationhasbeendeveloped.Thecitedstudieshavefocusedonthetoothprofiledesignandpressurefluctuationswhenthecirculararcisrunatlowrotationalspeed.Asaerospacestabilityandminiaturizationareurgentrequirements,thedevelopmentofahigh-speedminia-turizedhigh-pressuregearpumpwillbeofimportantscientificsignificanceandapplicationvalue.Thefol-lowingneedtobeconsideredwhenrunningagearpumpathighspeedandhighpressure.(a)Theleakingofagearpumpincreasesunderhigherpressureandathigherrotationalspeeds.11–14Thetraditionalmodelof

agearpumpisnolongerapplicableundersuchcon-ditions.(b)Ariseinthejournalbearingtemperatureundersuchconditionsaffectsthehydraulicoilviscos-ityandrotorstability.15(c)Thestabilityofgearwhenthepumpisrunathigh-speedandhighpressure.16,17(d)Thesystemsealshavechangedundersuchcondi-tions.18–22(e)Processingandmanufacturingismadedifficultbyminiaturizationofthepump.3,23,24Themainpurposeofthisstudyistoprovideabasemodelofhigh-pressureandhighspeedgearpumpforaddressingtheaboveproblems.Acircular-arctoothprofileisintroducedandtheflowrate,radialforce,axialforce,andtorquearestudied.Acomputer-aideddesign(CAD)systemforthedesignofagearpumpisdeveloped,allowingdesignofthetoothprofile,designofthetool,andnumericalcontrolofcircular-arcgearpump.TheinternalflowfieldofthisgearpumpisstudiedemployingFluentsoftware.Thebasemodelprovidedtakesintoaccountthemainphenomenarelatingtoleakages,temperaturerises,androtor

SchoolofMechanicalandElectricalEngineering,HarbinInstituteofTechnology,Harbin,P.R.China

Correspondingauthor:

MinghuiHAO,SchoolofMechanicalandElectricalEngineering,HarbinInstituteofTechnologyNo.92,WestDa-zhiStreet,Harbin150001,P.R.China.

Email:hao_minghui@163.com

Downloaded from pic.sagepub.com at Tsinghua University on April 14, 2016

190

stabilityatrotationalspeedsoftensofthousandsofrevolutionsperminuteandpressuresabove25MPa.

ProcIMechEPartC:JMechanicalEngineeringScience230(2)

TheequationfortheinvoluteBCisx0¼uÁRbÁcosðuÀ󰀂0ÞÀRbÁsinðuÀ󰀂0Þy0¼uÁRbÁsinðuÀ󰀂0ÞþRbÁcosðuÀ󰀂0Þ

'

ð2Þ

Designofcircular-arctoothprofile

ThegearpumpstudiedisdepictedinFigure1.Itisanexternalgearpumpwithtwingears(drivinggearanddrivengear),journalbearing,oilseal,circlip,cover,andbody.Figure2showsthetoothprofile.Thetoothprofileisgeneratedbyacirculararc(ABandCD)andinvolute(BC).TheequationforthecirculararcABisgivenby

x0¼rÁcosðukÞy0¼rÁsinðukÞþR

'

ð1Þ

whereudenotesthefunctionvariablesofinvoluteBCandRbisradiusofbasecircle.Theangle󰀂0isgener-atedbytheyaxis,involuteandcoordinateoriginO.TheCDequationinthe(O,x,y)coordinatesystemisx0¼ÀrÁcosðunÞþRÁsinð’Þy0¼ÀrÁsinðunÞþRÁcosð’Þ

'

ð3Þ

whereukdenotesthefunctionvariablesofcirculararcAB,Ristheradiusofpitchcircle,andristheradiusofcirculararc.

where’¼󰀃=Z,undenotesthefunctionvariablesofcirculararcCD,Risradiusofthepitchcircle,Zistheteethnumberandristheradiusofcirculararc.Sincethetoothprofileissymmetricalwithyaxis,thewholetoothprofileisobtainedbyequations(1)to(3).Figure3showsthetoothprofilewiththenumberofteethZ¼7teeth,normalmodulusofmn¼2mm,ref-erencecirclenormalpressureangle󰀄¼28󰀄,andfacewidthB¼10mm.

PointsBandCarejunctionsofinvoluteandcir-culararc.Thus,thecoordinateandfirst-orderderiva-tiveareequalatBpointforinvoluteandcirculararc.Similarly,thecoordinateandfirst-orderderivativeareequalatpointCforinvoluteandcirculararc.ThecoordinateofpointBatcircular-arcisðx01,y01Þ,andthecoordinateofpointBatinvoluteisðx02,y02Þ.ThecontinuityrequirementsatpointBare&

x01¼x02y01¼y02

ð4Þ

Figure1.Structureofthecircular-arcgearpump.

y001y002

¼x001x002

ð5Þ

Figure2.Coordinatesystemsofthecircular-arctoothprofile.

Figure3.Circular-arctoothprofile.

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Zhouetal.

Likewise,thecoordinateofpointCatinvoluteisðx03,y03Þ,andthecoordinateofpointCatcirculararcisðx04,y04Þ.ThecontinuityrequirementsatpointCare&

x03¼x04y03¼y04

ð6Þ

191

whereðx0,y0,z0ÞiscoordinateoftoothprofileinthecoordinatesystemS0ðOÀx,y,zÞ.LookingalongtheZ-axis,󰀂ispositivewhentherotationisclockwise.Likewise,󰀂isnegativewhentherotationisanticlockwise.

Thehelicalsurfaceequationisobtainedfromequa-tions(1)–(3)and(9)9

x¼x0cosð󰀂ÞÀy0sinð󰀂Þ=y¼x0sinð󰀂Þþy0cosð󰀂Þ

;

z¼ÆpÁ󰀂

y003y004

¼x003x004

ð7Þ

ð10Þ

Thecontinuityrequirementsofthetoothprofileareobtainedbyequations(4)to(7)8

<󰀂0¼tanð󰀄0ÞÀ󰀄0þ2󰀃Zr¼Rbð󰀄0þ󰀂0Àtanð󰀄0ÞÞ:

u󰀄¼󰀄0

ð8Þ

where󰀄0isthepressureangleatpithcircular,Rbistheradiusofbasecircle,u󰀄isthecirculararcparametervalueatpointB,andZistheteethnumber.

Thetransversecontactratioofacircular-arcgearpumpis0.5becausethereisonlyonepairofteethincontactduringtheengagementcycle.Thehelicalcircu-lar-arcgearpumpisdiscussedinthispaper.Thehelicalsurfaceisobtainedbycoordinatetransformationfromtransverse,asisshowninFigure4.ThecoordinatesystemS0ðOÀx,y,zÞisfixedcoordinatesystems.ThecoordinatesystemS1ðO1Àx1,y1,z1ÞisobtainedbythecoordinatesystemsS0ðOÀx,y,zÞrotateandmovealongzðz1Þaxis,andtheangleofrotationis󰀂.Thehelicalsurfaceequationisobtained232xcosð󰀂Þ4y5¼4sinð󰀂Þz0

32323

Àsinð󰀂Þ0x00cosð󰀂Þ054y05þ40501Æp󰀂z0

ð9Þ

where‘‘À’’correspondstothedrivinggearand‘‘þ’’

correspondstothedrivengear,󰀂isaparametervari-H

able.pistheparameterofthehelicalsurfacep¼2󰀃,andHislead.ThegearpumpwithtwingearsofZ¼7teeth,normalmodulusofmn¼2mm,referencecirclenormalpressureangleof󰀄¼28󰀄,andfacewidthofB¼10mmisdiscussedinthispaper.

Theperformanceparametersofgearpump

Theoreticaldisplacementandflowrate

Thedisplacementisobtainedbydifferentialalongaxisdirection(z-axis)ofgear.

22

dV¼ðR2eÀRÀfÞdz

ð11Þ

wherefisthedistancebetweenthemeshingpointandpitchpoint,Reistipradius,andRisradiusofpitchcircle.

󰀄󰀅

󰀃mnrRb󰀃mn2

þÁV¼2RrÁ

sinð󰀅Þpsinð󰀅Þ

ð12Þ󰀄󰀅󰀄󰀅

1Rb2󰀃mn3ÀÁ3psinð󰀅ÞwhereVisthedisplacementperradian,pisthepar-H

ameterofthehelicalsurface¼2󰀃,Histhelead,󰀅isÀ󰀃mnpÁZ

thehelixangle󰀅¼arcsinH,andmnisthenormalmodule.

ThefacewidthisobtainedbyB¼

󰀃mnH

¼sinð󰀅ÞZ

ð13Þ

Theequation(12)isrewrittenasfollowB3rRbB2R2

Àb22p12p22

mnRb󰀃B󰀃RbB

þV¼

2cosð󰀅Þ6Z2V¼2RrBþ

3

m2nZ󰀃BV¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi22224ð1þtan2ð󰀄ÞÞÁB2Àm2n󰀃ÁBÀmn󰀃

ð14Þð15Þ

þ

Figure4.Coordinatetransformationfromtransversetothe

helicalsurface.

23m2n󰀃B

224ðð1þtan2ð󰀄ÞÞÁB2Àm2n󰀃Þ

ð16Þ

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192ProcIMechEPartC:JMechanicalEngineeringScience230(2)

Figure5.Flowrateversusdifferenttoothprofileparameters:(a)flowrateversusnumberofteeth,(b)flowrateversusmodulusand(c)flowrateversusfacewidth.

TheflowrateisobtainedbyQðL=minÞ¼

2󰀃VÁn106ð17Þ

Theinfluenceofnumberofteethonflowrate.ThenumberofteethZisvariable,thefacewidthB¼10mm,thepressureangle󰀄¼28󰀄andthemodulusmn¼2inthissection.Equation(17)canbesimplifiedQðzÞ¼f1zQZþf2zQwheref1zQ¼

2Â106

f3mQ󰀃Bnqffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffi¼.,f2zQf2mQ

f2

2

32mQ

À󰀃2Ám2n

B2

À󰀃2m2n

m2n

wherenistherotationalspeed,andtherotational

speedisn¼10,000rpminthispaper.Theflowrateanddisplacementareconstantwhenthetoothprofileparametersaredefined.Asmentionedearlier,thetoothprofileisdeterminedbythetoothprofilepar-ameters.Here,letusdiscusstherelationshipbetweenthetoothparametersandflowrate.

Theinfluenceofmodulusonflowrate.Themodulusmnisvariable,thefacewidthB¼10mm,thepressureangle󰀄¼28󰀄andthenumberofteethZ¼7inthissection.Equation(17)canbesimplifiedf1mQf3mQ

QðmnÞ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþf2mQ

f2mQB22À󰀃2À󰀃2Ámm22À󰀃nm2n

n

2

3

ð19Þ

À󰀃

Theinfluenceoffacewidthonflowrate.ThefacewidthB

isvariable,themodulusmn¼2,thepressureangle󰀄¼28󰀄andthenumberofteethZ¼7inthissection.Equation(17)canbesimplifiedf1BQðBÞ

QðBÞ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2BQðBÞfðBÞfðBÞ

À3BQÁ1À4BQ

B2B4B2þ󰀉f5BQðBÞÀ

f3BQðBÞ

B3f2BQðBÞBð18Þ

ð20Þ

BZn

,f2mQ¼ð1þtan2ð󰀄ÞÞB2,f3mQ¼wheref1mQ¼󰀃2Â10633

󰀃Bn

.1:2Â107󰀊Downloaded from pic.sagepub.com at Tsinghua University on April 14, 2016

Zhouetal.

wheref1BQðBÞ¼2nÂ106,f2BQð2B3Þ¼1þtan2ð󰀄Þ,

mn󰀃n2

f3BQðBÞ¼f4BQðBÞ¼m2n󰀃,f5BQ¼1:2Â107.

Therelationshipbetweenthetoothprofileparam-etersandflowrateisobtainedbyequations(18)to(20),asisshowninFigure5(a)to(c).Theflowrateincreasesnonlinearlywiththemodulusandfacewidth.However,theflowrateincreaseslinearlywithnumberofteeth.

m2󰀃2Zn

193

respectively,thefirstderivativesofx0andy0.Thetorqueisobtainedbyequation(14)

ZZpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZZ

^ÁdsT¼ðxeyÀyexÞÁpx2þy2ÁdFt¼ð23Þ

Theinfluenceofmodulusontorque.Themodulusmnisvariable,thefacewidthB¼10mm,thepressureangle

󰀄

󰀄¼28andthenumberofteethZ¼7inthissection.Equation(23)canbesimplified

f1mTf3mT

TðmnÞ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþf2mT

2f2mTB22Á22À󰀃À󰀃À󰀃m22nmmn

n

Torqueandradialforce

Mðx,y,zÞisapointonthetoothprofileinthe

ðOÀx,y,zÞcoordinatesystem,asshowninFigure6.Theradialforce,tangentialforceandaxialforcecausedbyliquidoilare8󰀄󰀅>yxffi>ffieyÁp^ÁdsdFr¼pffiffiffiffiffiffiffiffiffiexþpffiffiffiffiffiffiffiffiffi>>22xþyx2þy2><󰀄󰀅

ð21Þyxffipffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi>^ee¼ÀÁpÁdsdFtyx>>x2þy2x2þy2>>:

^Áds8dFz¼ez0Áp

nx¼pðx0sinð󰀂Þþy00cosð󰀂ÞÞ>>>>00>n¼Àpðxcosð󰀂ÞÀyy>00sinð󰀂ÞÞ>>>00>>nx

ex¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið22Þ22nxþn2>yþnz>>ny>>ey¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi>22>nxþn2>yþnz>>nz>:ez¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi222

nxþnyþnz

ð24Þ

ÁpB󰀃Z

wheref¼1mT4000,ÁpB3󰀃2f3mT¼24000.3

f2mT¼ð1þtan2ð󰀄ÞÞB2,

Theinfluenceofnumberofteethontorque.ThenumberofteethZisvariable,thefacewidthB¼10mm,the

󰀄

pressureangle󰀄¼28andthemodulusmn¼2inthissection.Equation(23)canbesimplifiedasTðzÞ¼f1zTZþf2zTwheref1zT¼

ÁpB󰀃f3mTqffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffi,f2zT¼f2mTf2

2mTÀ󰀃2Ám2n

B

À󰀃2m2n

m2n

3

ð25Þ

4000

À󰀃2

^ispressure.Itisrelatedtothesizeoftheworkwherep

areashowninFigure7.Thepressuredistributiononthegears,whichvarieswiththerotationangle,iscom-putedbyarithmeticdsistheinfinitesimalÀprocessing.ÁareaonÁgear.nx,ny,nzisnormalvectorandÀ

ex,ey,ezistheunitnormalvector.x00andy00are,

Theinfluenceoffacewidthonflowrate.ThefacewidthBisvariable,themodulusmn¼2,thepressureangle

󰀄

󰀄¼28,andthenumberofteethZ¼7inthissection.Equation(23)canbesimplifiedf1BTðBÞ

TðBÞ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2BTðBÞðBÞðBÞ

Á1Àf4BTÀf3BT

B2B4B2þ󰀉f5BTðBÞ

f2BTðBÞBð26Þ

À

f3BTðBÞB3󰀊Figure6.3Dmodelofgear.Figure7.Pressuredistribution.

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194

wheref1BTðBÞ¼

󰀃m2n1þtan2ð󰀄Þ2

ProcIMechEPartC:JMechanicalEngineeringScience230(2)

ÁpÁ󰀃m2nZ4000

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2BTðBÞ¼1,f3BTðBÞ¼2

1þtanð󰀄Þ

Áp󰀃2m2

n

f4BTðBÞ¼󰀃2m2n,f5BTðBÞ¼24000Áð1þtan2ð󰀄ÞÞ.Therelationshipbetweenthetoothprofileparam-etersandtorqueisobtainedbyequations(24)to

(26),asisshowninFigure8(a)to(c).Thetorqueincreasesnonlinearlywiththemodulusandfacewidth.Thetorqueincreaseslinearlywithnumberofteeth.

Theradialforcecausedbyliquidisxy

^ÁdsdFr¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexþpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieyÁp

2222xþyxþy

!

󰀃

from0to2Z.Thetransversetoothprofilecoordinateswithdifferent’are ! ! !\"0xcosð’ÞÀsinð’Þx0

¼ð28Þ\"0y0sinð’Þcosð’Þy

whereðx0,y0Þistheinitialtransversecoordinatewithvariable’¼0.

Thetoothprofilecoordinateinhelicalsurfacecoordinatesystemis&

\"0cosð󰀂ÞÀy\"0sinð󰀂Þx¼x

ð29Þ

\"0sinð󰀂Þþy\"0cosð󰀂Þy¼xTheunitnormalvectoris

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8

0022qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

00:ey¼Àpðx22\"0cosð󰀂ÞÀy\"0sinð󰀂ÞÞ=n2xþnyþnz

ð30Þ

ð27Þ

Thehelicalgearisdiscussedinthispaper.The

radialforcesvarywithrotationofrotor.Thevariable’hasbeenintroduced,andtherangeofvariable’is

Figure8.Torqueversusdifferenttoothprofileparameters:(a)torqueversusnumberofteeth,(b)torqueversusmodulus,and(c)torqueversusfacewidth.

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Zhouetal.

Theradialforceoftheinfinitesimalareaonthex-axisandy-axisis

8RRx

pffiffiffiffiffiffiffiffiffiffieÁpds^xþy

195

obtainedfromequations(32)and(33).&

Fx¼FrxþFNx

Fy¼FryÇFNy

ð34Þð35Þ

ð31Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2Fr¼F2xþFy

Theradialforceinthex-axisandy-axisis

obtainedby

8RRx0y00Àx00y0RRx0y00Àx00y0

:Fry¼pÁp^\"0sinð󰀂Þþy\"0cosð󰀂ÞÞdsðxx2þy20

0

ð32Þ

Thenormalforcebetweenmeshinggearsis8TFNx¼FNcosð󰀄1Þ:

FNy¼FNsinð󰀄1Þ

ð33Þ

wheretheangleis󰀄1,󰀄1¼arccosðRb=RÞ.Rbisbasecircle,andRispitchcircle.Theradialforceis

where‘‘À’’correspondstothedrivinggearand‘‘þ’’correspondstothedrivengear.Itishardtoobtainradialforcebytheanalyticalmethodsbecausepres-^andangle’isvariables.Thus,theradialforcesurep

canbeobtainedbynumericalmethods.Theangle’isgivenacertainvalueinthispaper,andthentheradialforcecanbeobtainedbyequations(32),(34)and(35).Theradialforceofthedrivengearisthemaximumradialforce.Figure9(a)to(c)showstherelationshipbetweentoothprofileparametersandthemaximumradialforce(theradialforceofthedrivengear).Theresultsshowthattheradialforceincreasesnonlinearlywiththemodulus,numberofteeth,andfacewidth.Figure10showstheperiodicvariationintheradialforcedistributiononthegears.Gearshavebasic

Figure9.Radialforceversusdifferenttoothprofileparameters:(a)radialforceversusnumberofteeth,(b)radialforceversusmodulusand(c)radialforceversusfacewidth.

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196ProcIMechEPartC:JMechanicalEngineeringScience230(2)

ÁpB󰀃󰀊f1mF

qffiffiffiffiffiffiffiffiffiffi,f2zF¼󰀉.wheref1zF¼qffiffiffiffiffiffiffiffiffiffiffiffif2

2mFÀ󰀃2Á

m2n

B

À󰀃2m2n

23

24

f2mFÀ󰀃2m2n

Theinfluenceoffacewidthonaxialforce.Thefacewidth

Bisvariable,themodulusmn¼2,thepressureangle

󰀄

󰀄¼28andthenumberofteethZ¼7inthissection.Equation(37)canbesimplifiedf1BFðBÞ

FðBÞ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2BFðBÞðBÞðBÞ

Á1Àf4BFÀf3BF

B2B4B2þ󰀉f5BFðBÞ

f2BFðBÞBð40Þ

À

f3BFðBÞB3󰀊Figure10.Radialforceversusengagementangle.

wheref1BFðBÞ¼

󰀃m2n1þtan2ð󰀄Þ2

4B

ÁpÁ󰀃mnpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi,f2BFðBÞ¼1,f3BFðBÞ¼2

1þtanð󰀄Þ

Áp󰀃3m2

22

dimensionsofthenumberofteethZ¼7,modulus

mn¼2mm,facewidthB¼10mm,pressureangle󰀄¼28󰀄.Thevariationintheradialforceis󰀆F¼

maxðFrÞÀminðFrÞ

maxðFrÞþminðFrÞ2n

,f4BFðBÞ¼󰀃2m2n,f5BFðBÞ¼24BZð1þtan2ð󰀄ÞÞ.ð36Þ

Theradialforcefluctuationofthedrivengearis

1.32%andthatofthedrivinggearis0.91%.

Axialforce

Theaxialforceofthegearpumpisobtainedbyequa-tion(21)Fz¼

ZZ

^ez:pds

ð37Þ

Therelationshipsbetweenthetoothparameters

andtheaxialforceareobtainedbyequations(38)to(40),asisshowninFigure11(a)to(c).Theaxialforceincreaseslinearlywiththenumberofteeth,increasesnonlinearlywiththemodulus,anddecreasesnonli-nearlyasthefacewidthincreases.

Theflowrate,axialforceandtorquearerepre-sentedbyanequationwhentheeffectofmodulusisdiscussed,asisshowninequations(18),(24),and(38).Likewise,theflowrate,axialforce,andtorquearerepresentedbyanequationwhentheeffectoffacewidthandnumberofteetharediscussed.Therela-tionshipbetweenmodulusmnandfacewidthBisobtainedfromaboveequationsmn5

B󰀃

ð41Þ

Theinfluenceofmodulusonaxialforce.Themodulusmnisvariable,thefacewidthB¼10mm,thepressure

󰀄

angle󰀄¼28andthenumberofteethZ¼7inthissection.Equation(37)canbesimplifiedasf1mFf3mF

FðmnÞ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþf2mF

f2mFB22À󰀃2À󰀃2Ámm22À󰀃nm2n

n

22

Outletpressureandpressurefluctuations

attheoutlet

Theoutletpressuresandfluctuationsaresimulatedbythefinite-volume-basedcodeANSYFluent13.0.ThecomputationdomainhasbeenshownFigure12(a).Table1showsthetoothprofileparametersandgeo-metricalcharacteristicsofgearpump.Thetimederivativeiscalculatedbyafirst-orderimplicitscheme,andthiswayistheonlymethodforsolutionoftransientformulation.ThegradientiscalculatedbyGreen-Gaussnode-basedmethod.ThepressureinboundariesisobtainedbyPRESTO!whichisusedtosolvehigh-speedrotationflowandhighdistortedareaflow.TheQUICKschemeisusedforspatialdis-cretization.Apressure-basedhasbeenselected,sinceitappliestoincompressiblefluid.ThestandardkÀ\"turbulentmodelhasbeenselected,sincethismodelperformswellinexternalgearpump.25ð38Þ

ÁpB2󰀃324Z.

󰀃22

wheref1mF¼ÁpB4,f2mF¼ð1þtanð󰀄ÞÞB,f3mF¼

Theinfluenceofnumberofteethonaxialforce.ThenumberofteethZisvariable,thefacewidthB¼

󰀄

10mm,thepressureangle󰀄¼28andthemodulusmn¼2inthissection.Equation(37)canbesimplifiedasFðzÞ¼f1zFþ

f2zFZ

ð39Þ

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Zhouetal.197

Figure11.Axialforceversusdifferenttoothprofileparameters:(a)axialforceversusnumberofteeth,(b)axialforceversusmodulus,and(c)axialforceversusfacewidth.

Figure12.Thecomputationdomainandoutletmesh:(a)computationdomainofgearpumpand(b)meshatApointofoutlet.

Themeshisdynamicformedbytriangularcells,26asshowninFigure12(b).Smoothingmeth-odsandLocalremeshingmethodsareemployedfordynamicmeshupdatemethods.Figure13showscellnumberevolutionincomputationdomain,andthesimulationsareperformedwithgridcontaining15,45,820cells.Thenumberofcellsischangedbyremeshingalgorithmineachtimestepuntilequilibrium,andgridsstabilizedaround84,000.

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198

Table1.Geometricalcharacteristicsandtoothprofileparameters.

NumberofteethModulus

Diameterofthegears

Distancebetweencentersofgears

NormalpressureangleofreferencecircleRadiusofthechamberDiameteroftheoutletDiameteroftheinlet

72mm

21.334mm18mm28󰀄

10.697mm15mm9mm

ProcIMechEPartC:JMechanicalEngineeringScience230(2)

TooldesignDisctool

Thissectionpresentsthedisctoolsrequiredtomachinehelicalgearswithacircular-arctoothprofile.Thegearcoordinatesystem(O,x,y,z)anddisc-toolcoordinatesystem(O1,X,Y,Z)areshowninFigure17.PointMðx,y,zÞisonthehelicalsurface.ðex,ey,ezÞisunitnormalvectorofpointM.ThenormallineatcontactpointMthroughthedisctoolaxis(Zaxis).Thedisctoolaxisequationinthecoord-inatesystem(O,x,y,z)is

'

\"¼ax

ð42Þ

\"cotð󰀄Þ\"¼Àyzwhereaistheshortestdistancebetweentheaxisofthe

gearandtheaxisofthedisctool,and󰀄istheinstal-lationangleofthetool.Thenormalequationis\"Àxy\"Àyz\"Àzx

¼¼

nznxny

ð43Þ

Theequationforthecontactlineisobtainedfrom

equations(42)and(43)

ðaÀxÞðnyþnztanð󰀄ÞÞþnxðyþztanð󰀄ÞÞ¼0

ð44Þ

Theequationforthecontactlineindisctoolcoord-inatesystem(O1,X,Y,Z)is

Figure13.Cellnumberevolutioninthecomputationzone.

9

Xh¼xÀa=Yh¼ycosð󰀄Þþzsinð󰀄Þ

;

Zh¼Àysinð󰀄Þþzcosð󰀄Þ

ð45Þ

Thefluidanalysismakesthefollowingimportantassumptions:

1.Theliquidoilisincompressiblefluid.

2.Thefluidanalyzedisafullyturbulentmodel.3.Theliquidoilcharacteristicsaredynamicviscosity¼0.048Pa.s,andthedensity¼960kg/m3at40󰀄C.

4.Themodelissetinletpressure¼0MPa;rotationalspeed¼10,000rpm.Figure14(a)to(d)showsthepressurecontourofrateoutletpressuresat5MPa,15MPa,25MPaand30MPaafter0.005sofoperation.Thepressuredropsfromtheoutlettoinletbyarithmeticprocess-ing,asshowninFigure7.Thereismaximumpressurewherethegearsmesh.

Figure15(a)to(d)showstheoutletpressuredistributionatdifferentrateoutletpressuresafter0.005sofoperation.Thepressuresincreasefromthewalltothecenterofoutlet.ThepeakPiscausedbyvortex.

TheoutletpressurefluctuationsareshowninFigure16(a)to(d).Theoutletpressurefluctuationskeep0.007–0.008MPaunderdifferentratepressures.

Theprofileofthedisctoolisobtainedusingequa-tion(45)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi)

2R¼X2hþYh

Zd¼Zh

ð46Þ

Hobdesign

ThecoordinatesystemsofthehobareshowninFigure18.TheangleÆisdeterminedbytheaxisof

!

thehobzpandtheaxisofthegearZ.Supposethatv1isthevelocityofpointMinthegearcoordinate

!

system(O2,XP,XP,ZP)andv2isthevelocityof

!

pointM(O1,x,y,z)inthehobcoordinatesystem.v12isrelativevelocityandnisanormalvectorofpointMinthegearcoordinatesystem.

Thespatialmeshingequationsare

!

nÁv12¼0

pÁsinðÆÞðx00cosð󰀂þ’2ÞÀy00sinð󰀂þ’2ÞÞÀp1cosðÆÞþpÁi21¼ðx0x00þy0y00Þ

p1

!

ð47Þð48Þ

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Zhouetal.199

Figure14.Pressurecontourofdifferentrateoutletpressures:(a)5MPa,(b)15MPa,(c)25MPaand(d)30MPa.

wherex0,y0arethetransversetoothprofilecoordin-ates,󰀂isaparametervariable,i21istheratiooftrans-missionbetweengearandworm,and’2istherotaryangleofthegear.pisparameterofthehelicalsurface

Hp¼2󰀃,andHislead.

Thehelicalparameterofthehobp1isdefinedasp1¼R0tanðl0Þ

ð49Þ

whereR0isthereferenceradiusofthehobandl0istheleadangle.Thebasewormoftheaxialdirectionprofilehasazerorakeangleandastraightgashrakeface.Theequationfortheengagingsurfaceis

8

y¼ðx0sinð󰀂þ’2Þþy0cosð󰀂þ’2ÞÞcosðÆÞþz2sinðÆÞ:

z¼Àðx0sinð󰀂þ’2Þþy0cosð󰀂þ’2ÞÞsinðÆÞþz2cosðÆÞ

ð50Þ

Therotaryangleoftheworm’01is

’2z’01¼þ

i21p1

Theequationforthebasewormis8

z1¼p1Á󰀂

ð52Þ

ð53Þ

Theprofilehavingapositiverakeangleandstraightgashrakefaceistheintersectinglineoftherakefaceandhelicoid.Therakefaceissomedistanceefromthecentero.Theequationforthepositiverakeangleandstraightgashprofileisobtainedfromequa-tion(53)as

8

z1¼p1Á󰀂

Thetransverseequationofthebasewormis&

x10ðuÞ¼xcosð’01Þþysinð’01Þy10ðuÞ¼Àxsinð’01Þþycosð’01Þ

ð51Þ

ð54Þ

Thedesignofdisctoolandhobisaccordingtodifferentsituationsofprocessingworkshop.

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200ProcIMechEPartC:JMechanicalEngineeringScience230(2)

Figure15.Theoutletpressuredistributionat0.005s:(a)5MPa,(b)15MPa,(c)25MPaand(d)30MPa.

Disctoolisusedformillingmachine,andhobisusedforgearhobbingmachine.Disctoolisnon-instantaneous-poleenvelopeinprincipal.However,hobiscompletelyenvelopeinprincipal.Theadvantagesanddisadvantagesofgeneratingtoolsareasfollow:

(a)Workingefficiency

Disctoolshavesimplestructure,andalsocanbedirectlyprocessedbygrindingwheel.Hobneedstodesignbasicworm,andalsoneedstofinishingmachiningafterroughmachining.Thus,theworkingefficiencyofdisctoolishigherthanHob.(b)Machiningaccuracy

Themachiningaccuracyofhobishigherthandisctool.Thus,Hobshouldbeusedwhenmachiningaccuracyisconsidered.

Designsoftwareforthecircular-arcgearpump

Designsoftwareforthecircular-arcgearpumpisdevelopedinthissection.ThestructureofthedesignsoftwareisshowninFigure19.

ThepumpdesigninterfaceisshowninFigure20.Thetoothparametersandperformanceparametersareobtainedusingtheinterfaceofthepumpdesign.Figure21(a)and(b)showsthetoothprofiledesigninterfaceandtheinterfaceoftwogearsmeshing.ThetooldesigninterfaceisshowninFigure22.Thecutterrotationface,cutterrakeface,andcutterradialcross-sectionalcanbeobtainedwiththedesigninterfaceofthedisctool,whichisshowninFigure22(a).Thetransverseprofileofthebasicwormandrakefaceprofileareobtainedwiththedesigninterfaceofthehob,whichisshowninFigure22(b).

TestgearsweregeneratedwiththeCADsystem.Figure23showstestgearshavingbasicdimensionsofthenumberofteethZ¼7,modulusmn¼2mm,facewidthB¼10mm,pressureangle󰀄¼28󰀄,tipdiameterDe¼21:334mmandrootdiameterDf¼14:667mm.ThetestgearswereprocessedbyagrindingwheelunderBM3000Cbroachgrinding.Thegrindingwheelprofilehasthesameprofileasthedisctool,whichwasobtainedwiththeinterfaceshowninFigure22(a).Anexaminationrevealedthatthesizeandaccuracyofthetestgearsmeettherequirements,andtheCADsystemwasthusdemonstratedtoperformwell.

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Zhouetal.201

Figure16.Pressurefluctuationsattheoutlet(a)5MPa,(b)15MPa,(c)25MPaand(d)30MPa.

Figure17.Coordinatesystemsanddisctool.

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202ProcIMechEPartC:JMechanicalEngineeringScience230(2)

Figure18.Coordinatesystemsofthehobandrakefaceofthehob.(a)Coordinatesystemsofthehoband(b)Rakefaceofthehob.

Figure19.Structureofthedesignsoftwareofthecircular-arcgearpump.

Figure20.Structureofthedesignsoftwareofthecircular-arcgearpump.

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Zhouetal.203

Figure21.Designinterfaceofthetoothprofileandinterfaceofgearsmeshing.(a)Designinterfaceofthetoothprofileand(b)interfaceoftwogearsmeshing.

Figure22.Tooldesigninterfaces.(a)Designinterfaceofthedisctooland(b)designinterfaceofthehob.

Conclusion

Thispaperinvestigatedthetoothprofiledesignandperformanceanalysisofahigh-speedandhigh-pressurecircular-arcgearpump,andpresentedaCADsystemforgearpumpdesignwithfeaturesofthetoothprofiledesign,tooldesign,andnumericalcontrolofacircular-arcgearpump.Themainresultsobtainedinthestudyareasfollow.

1.Theeffectsoftoothprofileparametersonper-formanceparameterswerediscussed.Theper-formanceparameterschangednonlinearlywiththemodulusandfacewidth.Theflowrate,axialforce,andtorquechangedlinearlywithnumberofteeth.

Figure23.Drivinggearanddrivengear.

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204

2.Theflowrate,axialforce,andtorquearerepre-sentedbyanequationwhentheeffectofmodulus,numberofteeth,andfacewidtharediscussed,respectively.TherelationshipbetweenmodulusmnandfacewidthBismn5B󰀃.Thisisausefulguidefordesignofcircular-arcgearpump.

3.Theoutletpressurefluctuationswerestudiedwhenthegearpumpwasrunat10,000r/minwithdifferentoutletpressuresof5,15,25and30MPa.Theoutletpressurefluctuationswere0.0071,0.0072,0.0072and0.0075MPa,respectively.Thegearpumphaslowoutletpressurefluctuations.4.ACADsystemwhosefeaturesincludegearpumpdesign,tooldesign,performanceanalysis,andnumer-icalcontrolwasdevelopedforfurtherstudy.Theper-formanceofthesystemwasdemonstratedinthemanufacturingoftestgears.Abasemodelisprovidedforgearpumpwithhighpressureandhighspeed.DeclarationofConflictingInterests

Theauthor(s)declarednopotentialconflictsofinterestwithrespecttotheresearch,authorship,and/orpublicationofthisarticle.

ProcIMechEPartC:JMechanicalEngineeringScience230(2)

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Funding

Theauthor(s)receivednofinancialsupportfortheresearch,authorship,and/orpublicationofthisarticle.

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Zhouetal.205

Appendix1Notation

aa1

B

ðex,ey,ezÞDeDfHi21

mn

ðnx,ny,nzÞ

rradiusofcirculararc

pparameterofthehelicalsurfacep¼H

shortestdistancebetweentheaxisofthep1helicalparameterofthehob2󰀃gearandtheaxisofthedisctool

RradiusofpitchcircleshortestdistancebetweentheaxisoftheRbradiusofbasecircle

gearandtheaxisofthehobR0referenceradiusofthehobfacewidth

Znumberofteeth

theunitnormalvector󰀄pressureangletipcircleofgear󰀅helixangle

rootcircleofgear󰀂parametervariableofhelicalsurfacelead

l0leadangleofhob

ratiooftransmissionbetweengearandÆinstallationangleofhobtoolworm

’2rotaryangleofthegearnormalmodulus’0rotaryangleofthewormnormalvector

󰀄

1installationangleofdisctool

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