Based on the unique structure of the proposed bridge-type compliant mechanism with double output ports generating bi-motions, a new type of piezoelectrically actuatedtwo-stage four-wayhy
Trang 1j o u r n a l ho me p a g e :w w w e l s e v i e r c o m / l o c a t e / s n a
Mingxiang Linga,∗, Jiulong Wanga, Mengxiang Wua, Lei Caoa, Bo Fub
a Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang, 621900, China
b School of Mechanical Engineering, Sichuan University, Chengdu, 610065, China
a r t i c l e i n f o
Article history:
Received 3 February 2021
Received in revised form 27 February 2021
Accepted 10 March 2021
Available online 15 March 2021
Keywords:
Compliant mechanisms
Piezoelectric actuator
Mechanical displacement amplifier
Flow control valve
Dynamic stiffness matrix
Smart materials
a b s t r a c t
1 Introduction
Advancesinsmartmaterialshaveresultedinarenewed
inter-estindevelopingactuatorswithhighcontrolprecision,compact
structureandlowcostforuseinmicro/nano-manipulations,
ultra-precisionmachiningandmanyotherengineeringfields[1–3].The
main goal ismulti-fold toobtainhigh-frequency response,
suf-ficient stroke,multipledegrees offreedom, fineresolution,and
highreliability.Withthisinmind,manyresearchershavemade
significant strides in developing all kinds of actuators, such as
shape memory alloys, piezoelectric actuators, magnetostrictive
transducers, voice coil motors,and soforth [4,5 Among these
types,piezoelectricallyactuatedcompliantmechanismshavebeen
preferred for someprecision applications due to thefollowing
advantages:
(a)High energy density and simplicity in structure of
piezoelectric actuators with large driving force
(Piezo-∗ Corresponding author.
E-mail address: ling mx@163.com (M Ling).
electric stacks), nano-scale resolution and fast dynamic response
(b)Thebenefitsofmonolithicstructureofflexure-based compli-antmechanismswithoutwear,friction,backlashandreduced requirementofassemblyprocess[6
Strokesofcommerciallyavailablepiezo-stacksareverysmall withthedeformationofabout0.1%to0.15%oftheirlength There-fore, flexure-based amplifying mechanisms are often designed fortheapplicationwithstrokesof severalhundredsof microns andevenmillimeterranges[7 Except forthefunctionalitiesof mechanical amplifyingand motionguiding,another purposeof usingflexure-basedcompliantmechanismsforpiezoelectric appli-cationsisthefactthatpiezo-stackscansupportlittletensileorshear forceloads,andcompliantmechanismscanprovidepreloadand restoringforcethroughelasticdeformation,renderingpiezo-stacks suitableforuseindynamicapplications
Atpresent,leveraged[8–11], bridge-type[12–15]and Scott-Russell [16,17] compliant mechanisms are frequently used as mechanical displacement amplifiers Somederivative or newly developed amplifying mechanisms have also been proposed [18–23].Althoughleveragedcompliantmechanismscanprovide https://doi.org/10.1016/j.sna.2021.112687
0924-4247/© 2021 Elsevier B.V All rights reserved.
Trang 2a desirable displacementamplification ratio, theyrequirelarge
spaces.Bridge-typecompliantmechanismswereoriginatedfrom
theearlyMoonie-type,Cymbal-typeandRainbow-type
amplify-ingmechanisms[24]andarewidelyusedinprecisionengineering
withasatisfyingdisplacementamplificationratioinacompactsize
Moreover,thestructuralsymmetryinthebridge-typeamplifying
mechanismspreventsshearingloadsbeingappliedtopiezo-stacks
One disadvantage of some previous bridge-type compliant
mechanisms,however,liesintheinertialmotionofinternal
actu-ators(e.g.piezo-stacks),limitingthesystem’sdynamicbandwidth
[12–15].Tothisend,weintroduceanimprovedbridge-type
com-pliant mechanism with two outputports in the current study,
whichcangeneratehomodromousbi-motionsinthesame
direc-tionwithonlyasinglegroupofpiezo-stacks.Withthisimproved
design,theinertialmotionofpiezo-stacksisavoided,andthe
fre-quencybandwidthisconsequentlyenhanced
Based on the unique structure of the proposed bridge-type
compliant mechanism with double output ports generating
bi-motions, a new type of piezoelectrically actuatedtwo-stage
four-wayhydraulicservovalveisdesigned,fabricatedandtested
Incomparisontopreviousfour-wayservovalvesactuatedbytwo
orfourpairsofpiezo-stacks[25,26],onlyonegroupofpiezo-stacks
is requiredin thecurrentdesign.Thecontrollercomplexityand
costarereducedtoacertainextent.Inaddition,theissueofoil
contamination and inner leakage in traditional nozzle-flapper
hydraulicservovalvesiseffectivelyovercome,allowingloweroil
cleaninglevelwithhigherreliability
Thispaperprogressesasfollows:Section2describesthe
con-figurationoftheimprovedbridge-typecompliantmechanism.The
kinetostatic anddynamicanalysesbyusingatwo-portdynamic
stiffnessmodelareimplementedinSection3,followedbydesign
andexperimentaltestingforanewtypeofpiezoelectrictwo-stage
servovalveinSection4.Finally,concludingremarksareprovided
inSection5
2 Design of the improved compliant mechanism
Fig 1 compares the schematic configuration of three types
ofpiezoelectricallyactuatedbridge-typecompliantmechanisms
Thesebridge-typecompliantmechanismsconsistofflexurehinges
and rigidlinks Micromotionofpiezo-stacksisamplifiedbased
onthetriangleandflextensionalprinciples.Theleaf-springflexure hingeisshowninFig.1asanexamplebutotherflexurehinges,such
ascircular,corner-filleted,ellipticandhyperbolicprofiles[27,28], arealsocommonlyappliedfordesigning bridge-typecompliant mechanisms
In thetraditional bridge-type compliantmechanisms shown
inFig.1(a),oneoutputportisfixedtoblockwhileanotherport
isservedastheoutputdisplacementport Insucha traditional configuration,exceptforthelongitudinalexpansiondeformation, piezo-stackshaveanextratransverseinertialmovementwithahalf magnitudeofoutputdisplacement.Thisinertialmovement lim-itsthesystem’sdynamicbandwidth Thebridge-typecompliant mechanismshowninFig.1(b)waspresentedbyTian,etal.[29]for useindesigningamicro-gripperbyfurthercombiningaleveraged amplifyingmechanism,whichisnotspecifiedhere.Oneinputport
ofthistypeofmechanismisfixedtoblockwithtwooutputports freetogeneratebi-motionsinanoppositedirection.Ascanbeseen fromthedeformationnephograminFig.1(b),theinertial move-mentofpiezo-stacksiseliminatedbuttheoutputportsstillhave parasiticmotionsalongthelongitudinaldirection
Animprovedbridge-type compliant mechanism isproposed
inthecurrentpapershowninFig.1(c).Theproposedimproved bridge-typecompliantmechanismconsistsofleaf-springhinges, rigidlinks aswellasguiding flexible beamsfixedtotheblock Micro-displacementofpiezoelectricstacksismechanically ampli-fiedwithoutinertial movementnorparasiticmotionby adding fourpairsofguidingflexiblebeams.Inaddition,homodromous bi-motionsinthesamedirectioncanbegeneratedbyre-organizing therelativepositionofflexurehingesincomparisontothe configu-rationinFig.1(a)andFig.1(b),whichwillbeutilizedtodriveanew typeofflowcontrolvalveinthecurrentstudy.Sincethethickness
ofguidingflexiblebeamsismuchsmallerthanthatofotherparts, theoutputdisplacementoftheimprovedbridge-typecompliant mechanismwillnotbeheavilyattenuated
3 Kinetostatic and dynamic analyses
3.1 Two-portdynamicstiffnessmodeling
Inordertosizethegeometricparameters,thetwo-portdynamic stiffness model of theimprovedbridge-type compliant
mecha-Fig 1. Comparison of bridge-type compliant amplifying mechanisms (a) Traditional bridge-type compliant mechanism with one output port (b) Derivative bridge-type compliant mechanism with two output ports generating reversed bi-motions for use in designing a micro-gripper in Ref [ 29 ] (c) Improved bridge-type compliant mechanism with double output ports generating homodromous bi-motions for designing a flow control piezo-servovalve in the current paper.
Trang 3Fig 2.Geometric parameters and discretization of the presented bridge-type
com-pliant mechanism.
Fig 3. Definition of nodal forces and nodal displacements of the ith flexible beam
element with respect to the reference coordinate frame.
nismisformulated,andthegeometricparametersaredefinedin
Fig.2.Themechanismisdiscretizedintoflexurehinges,rigidlinks
and lumpedmass(outputports)withonlykineticenergy
Flex-urehingesandrigidlinksareregardedasflexiblebeamsandare
denotedseriallyfromelements(1)to(20)connectingwithnodes
fromnumber1tonumber16.Alltheclampednodesarenumbered
as0.Theinputforceactuatedbypiezo-stacksisdenotedasfp
AsshowninFig.3,eachflexiblebeamhassixdegreesof
free-dom:axialdisplacementsujanduk,transversedeflectionswjand
wk,rotationsϕjandϕkatthetwonodesjandk.The
frequency-dependentnodalforcesandnodaldisplacementsoftheithflexible
beamcanbecorrelatedbytheirdynamicstiffnessmatrixDi(ω)in
thereferenceframeo-xy[30,31]:
Fi,j(ω)
Fi,k(ω)
=Di(ω)·
xi,j
xi,k
=
ki,1 ki,2
ki,3 ki,4
·
xi,j
xi,k
(1)
where Fi,j(ω)=[Nj; Qj; Mj], Fi,k(ω)=[Nk; Qk; Mk] and xi,j=[uj; wj;
ϕj],xi,k=[uk;wk;ϕk]arethetwonodalforcesandnodal
displace-mentsoftheithflexiblebeaminthereferencecoordinateframe.ω
Fig 4.Condensed configuration of the improved bridge-type compliant mecha-nism.
isthecircularfrequency.Di(ω)=RTi ·De(ω)·Ri,hereDe(ω)isthe dynamicstiffnessmatrixofflexiblebeamsintheirlocalcoordinate frameo-xiyi[30]:
De(ω)=K0−ω2M1−ω4M2−ω6M3 (2) whereK0,M1,M2,M3arethestaticstiffnessmatrixandthefirst three-ordermassmatricesofflexiblebeamelements,respectively, andthedetailedexpressionsarelistedinAppendix
CoordinatetransformationmatrixRiisdeterminedbythe ori-entationioftheithflexiblebeamwithrespecttothereference frameo-xy,asshowninFig.3:
Ri=
⎡
⎢
⎢
⎢
⎢
⎣
−sini cosi 0 0 0 0
0 0 0 cosi sini 0
0 0 0 −sini cosi 0
⎤
⎥
⎥
⎥
⎥
⎦
(3)
Thevalueofeachorientationifortheflexiblebeamelements
inFig.2issummarizedinTable1.Sincethetwonodesoftherigid linksarenotalignedatthecentralaxis,theyareequivalenttobe flexiblebeamswithanorientation=actan(H/l4),asshowninFig.2
Ontheotherhand,thetransfermatrixoftheithflexiblebeam element,whichalsodescribestherelationshipbetweennodalforce andnodaldisplacementofflexiblebeams,canbeeasilyderived basedonEq.(1)as:
xi,k
Fi,k
=Ti·
xi,j
Fi,j
=
ti,1 ti,2
ti,3 ti,4
·
xi,j
Fi,j
=
−k−1i,2·ki,1 k−1i,2
ki,3−ki,4·k−1i,2·ki,1 ki,4·k−1i,2
·
xi,j
Fi,j
Basedontheimprovedtransfermatrixmethodin[31],thetotal transfermatricesoffourbranchlimbsfromtheinputportstothe outputportsinFig.2canbedirectlyderivedas:
⎧
⎪
⎪
⎪
⎪
T1=T5·(ST4)·(ST3)·(SQ2T1)
T2=T10·(ST9)·(ST8)·(SQ7T6)
T3=T15·(ST14)·(ST13)·(SQ12T11)
T4=T20·(ST19)·(ST18)·(SQ17T16)
(5)
Trang 4Table 1
The value of each orientation i for the flexible beam elements (Unit: Degree).
In Eq (5),the specificvalues of Qi (i =2, 7, 12, 17)can be
expressedas[31]:
⎧
⎪
⎪
⎪
⎪
Q2=
I3 O3
k2,4 I3
,Q12=
I3 O3
k12,4 I3
Q7=
I3 O3
k7,4 I3
,Q17=
I3 O3
k17,4 I3
whereki,4(i=2,7,12,17)arethelastthreerowsandthreecolumns
inDi(ω).I3andO3arerespectivelytheunitandzeromatriceswith
thedimensionof3×3
Based ontheabovecondensation, theimprovedbridge-type
compliantmechanism inFig.2 canbefurtherequivalentasthe
two-portmechanicalmodel,asshowninFig.4.Theequivalent
rela-tionshipbetweenthenodalforceandnodaldisplacementofthe
condensedlimbscanbeobtainedagainbasedonEq.(5):
Fi,j
Fi,k
=Di(ω)·
xi,j
xi,k
=
ki,1 ki,2
ki,3 ki,4
·
xi,j
xi,k
=
−t−1
i,2·ti,1 t−1i,2
ti,3−ti,4·t−1i,2·ti,1 ti,4·t−1i,2
·
xi,j
xi,k
wherei=1,2,3,4isthenumberoffourcondensedlimbs.ti,1,ti,2,
ti,3andti,4 aretheblocksub-matricesofthecondensedtransfer matrixTiinEq.(5)
TakingthefourportnodesinFig.4asthestudyobjects,forces imposedoneachnodearethesummationoftheinversenodalforce
ofitsconnectedequivalentlimbs,theinertialforceofthenodeitself
ifitisalumpedmassandexternalforcesaccordingtod’Alembert’s principle.Therefore,thefollowingforceequilibriumequationsets canbeestablished:
⎧
⎪
⎪
⎪
⎪
fi1=F1,j+F4,j
fi2=F2,j+F3,j
fo1=F1,k+F2,k+M·xo1
fo2=F3,k+F4,k+M·xo2
(8)
inwhich, fo1 and fo2 are dummyforcesassumed attheoutput ports.ThedynamicstiffnessmatrixMoftheoutputportscanbe calculatedasfollows[30,31]:
M(ω)=
⎡
⎤
wheremisthemassoftheoutputports
Fig 5.Comparison of the static and dynamic performances with the presented two-port dynamic stiffness model and finite elemental results (Four sets of FEM with H = 0.2
mm, 0.5 mm, 1 mm and 2 mm).
Trang 5Table 2
Key geometric parameters of the improved bridge-type compliant mechanism.
BysubstitutingEq.(7)intoEq.(8),therehas:
⎧
⎪
⎪
⎪
⎪
fi1=(k1,1xi1+k1,2xo1)+(k4,1xi1+k4,2xo2)
fi2=(k2,1xi2+k2,2xo1)+(k3,1xi2+k3,2xo2)
fo1=(k1,3xi1+k1,4xo1)+(k2,3xi2+k2,4xo1)+M·xo1
fo2=(k4,3xi1+k4,4xo2)+(k3,3xi2+k3,4xo2)+M·xo2
(10)
RewritingEq.(10)astheformofmatrix,thetwo-portdynamic
stiffnessmodeloftheimprovedbridge-typecompliantmechanism
canberegularlyexpressedas:
⎧
⎪
⎪
⎪
⎪
fi1
fi2
fo1
fo2
⎫
⎪
⎪
⎪
⎪
=
⎡
⎢
⎢
⎣
k1,1+k4,1 0 k1,2 k4,2
0 k2,1+k3,1 k2,2 k3,2
k1,3 k2,3 k1,4+k2,4+M 0
k4,3 k3,3 0 k4,4+k3,4+M
⎤
⎥
⎥
⎦
·
⎧
⎪
⎪
⎪
⎪
xi1
xi2
xo1
xo2
⎫
⎪
⎪
⎪
⎪
(11)
where fi2=-fi1=[fp; 0;0]and fo1=fo2=0aretheinputandoutput
forces.fpistheinputforceactuatedbypiezo-stacks.xi1,xi2,xo1
andxo2aretheinputandoutputdisplacements
3.2 Parameterinfluenceanalysis
Actuatingforceswiththemagnitudeof100Nandthefrequency
rangefrom0Hzto1000Hzwereexertedontheinputportsinthe
x-directiontocalculatethestaticanddynamicperformances.The
materialwasselectedasspringsteelwithYoung’smodulusof200
GPaanddensityof7850kg/m3.Thegeometricparametersarelisted
inTable2.Foursetsoffiniteelementsimulationwiththe
commer-cialsoftwarepackageANSYSWorkbench15.0wereemployedto
verifythetwo-portdynamicstiffnessmodel.TheSolid186element
waschosentobuildthemodelandtheadvancedsizefunctionof
proximityandcurvaturewasadoptedtorefinetheelement.The
resultswereproven tobeconvergentandaccurateenough.The
static analysisand harmonicanalysisin ANSYSWorkbench15.0
werecarriedouttoevaluatethestaticanddynamicperformances
andthencomparedwiththetheoreticalresults
Whensettingfrequencyω=0rad/s,thedisplacement
amplifi-cationratioR=xo/xiandinputstiffnessKin=fi/xicanbecalculatedby
solvingthelinearequationsetsofEq.(11).Thosetwostaticindexes
wereplottedversusgeometricparameterHshowninFig.5(a)and
(b).Moreover,startingfromtheinitialvalueoffrequencyf=1Hz
(ω=2f)andincrementingstepbystepwithf=1Hz,the
fre-Fig 6. Theoretical prediction of the displacement amplification ratio of the two output ports by offsetting the node position of the input ports.
Fig 7. Picture of the improved bridge-type compliant mechanism actuated by piezo-stacks.
quencyresponseofoutputdisplacementscanbecalculatedandone
iscurvedinFig.5(c)versusfrequencyfwithgeometricparameter
H=0.5mm.Similarly,thefundamentalfrequencyagainst differ-entgeometricparameterHcanbeeasilyobtainedbycheckingthe peaksofthefrequencyresponsecurve,asshowninFig.5(d).One canseethatthetheoreticalmodelwelldescribesthechangetrend
ofthestaticanddynamicperformanceswithrespecttothefinite elementalresults.TheoptimalgeometricparameterH=0.5mm canbestraightforwardlyconfirmedfromthesecurvestofabricate theprototype
Inaddition,itcanbeseenfromFig.5(a)thatthedisplacements
ofthetwooutputportsbecomeinconsistentwiththeincreaseofH duetothetransverseasymmetryoftheimprovedbridge-type com-pliantmechanism.Thisdiscordancecanberelievedbyadjustingthe positionofpiezo-stacksinactualapplications.Inordertosupport thisviewpoint,thedisplacementsofthetwooutputportswere the-oreticallycalculatedwiththesameprocessasthatinFig.5(a)but
byoffsettingthenodepositionoftheinputports.During calcula-tion,thenodepositionofinputportscanberegardedastheforce pointsofpiezo-stacks.FromtheresultsinFig.6,thedisplacements
ofthetwooutputportsbecomeequalinthedomainofgeometric parameterH=0.5mmwhentheoffsetofthenodepositionis0.2 mm
Trang 6Fig 8.Experimental results of the static and dynamic performances of the improved bridge-type compliant mechanism (a) Output displacement under frequency of 1 Hz (b) Frequency response curve.
Fig 9. Schematic diagram and three-dimensional view of the presented piezoelectric two-stage servovalve based on the improved bridge-type compliant amplifying mechanism.
Piezo-stacks(modelNAC5023-H40)withtheaxialstiffnessof
150N/mwasadoptedasthemicromotiongenerator.The
mech-anism wasfabricatedwith theInvarsteel and by thewire cut
electricaldischargemachiningtechnique.Fig.7providesthe
proto-typewiththeassembledpiezo-stacks,whosedimensionis70mm×
64mm×15mm.FromtheexperimentalresultsinFig.8,theoutput
displacementof0.24mmforoneoutputportcanbeobtained,and
thefirst-orderresonancefrequencyoftheimprovedbridge-type
compliantmechanismwasmeasuredas880Hz(including
piezo-stacks).Inaddition,hysteresiserrorofpiezoelectricmaterialcan
beobservedinFig.8(a),andthistypeoferrorcanbefurther
com-pensatedbysomecontrolstrategies[32]butisnottheemphasisof
thecurrentstudy.Ontheotherhand,thetheoreticalpredictionis
ingoodagreementwiththefiniteelementsimulationwithsmall
errorsasshowninFig.5,whilethelargerdiscrepancybetweenthe
theoreticalresultsandexperimentaltestingcomesfromthe
dis-placementattenuationeffectofpiezo-stacksunderaspringload
andprototypingerrors,suchasmismatchingofstructural
param-eters,machiningerrors,misalignmentandassemblingerrors
4 Application for designing a piezo-valve
4.1 Conceptualdesign
A two-stage four-way proportional flow control valve with
thepilotstagebeingtwoslidingspoolsdrivenbytheimproved
piezoelectric bridge-type compliant mechanism is designed, as
schematicallyillustratedinFig.9.Thesecondstageemploysthe
hydraulicamplifierforalargeflowrate,whichhasbeenwidely
adoptedintraditionalnozzle-flapper,jet-deflectorordirect-drive
hydraulicservovalves[33,34]
Twooutputportsoftheimprovedbridge-typecompliant mech-anismaredirectlyconnectedtothetwopilotslidingspools.The inputvoltage generates linear deformation of piezo-stacks and theamplifiedhomodromousmotionmovesthetwoidenticalpilot spoolsslidingbackandforth,openingoneorifice(portaorportb)
tooilsupplyportPwhilesimultaneouslyconnectinganother ori-ficetooilreturnportT.Thisgeneratesapressuredifferenceand forceimbalanceacrossthemainspoolthatcausesittomoveback andforth.Theslidingmovementofthemainspoolopensor over-laystheloadcontrolorificewindowsAandBonsleeveandthus modulatesthefluiddirectionandflowrate
Itshouldbenotedthat theelasticdeformation ofcompliant mechanismprovidesarestoringforcefordynamicmotionofpilot spoolandnoextraspringisneededinthecurrentdesign.Thedesign withslidingspoolsalsopromisestobelessoilcontaminationand lowleakage.Incomparisontopreviousfour-wayservovalves actu-atedby two orfour pairsof piezo-stacks [25,26], onlya group
ofpiezo-stacksisrequiredavoidingthecomplexityofcontroller thankstothebi-motionsoftheimprovedbridge-typecompliant mechanism
4.2 Prototypeandexperimentalsetup Thespoolandsleeveoftheproposedpiezo-valvearemodeof Cr12MoVsteel,andtheprototypeisshowninFig.10.The piezoelec-tricservovalvewasfixedtothevalvetestrigtoevaluateitsstatic anddynamicperformances,andtheexperimentalsetupisshown
inFig.11.Anti-wearhydraulicoil(Mode46)wasusedasthe work-ingmedium.Theflowcharacteristicswererecordedandanalyzed withadatarecorder(YihengInc.,China).Apoweramplifierwith themaximumoutputvoltageof200Vdevelopedinourlabwas
Trang 7Fig 10.Prototype of the presented piezoelectric servovalve.
Fig 11.Experimental setup for evaluating the static and dynamic performances.
utilizedtodrivethepiezo-stacks.Displacementsofthemainspool
wasmeasuredbyaLVDTdisplacementsensor(UM-375,Univo
Sen-sorInc.)andthenfeedbacktoacontrollerfortrackinganopening
orificeof±0.8mm.Alltestswereintendedtobeperformedunder
no-loadconditions
4.3 Measuredresults
Fig.12 showsthestepresponse ofthemainspoolat a
sup-plypressureof30bar.Itisshownthatthemechanicalresponse
Fig 12.Measured results of step response of the main spool.
Fig 13.Measured results of the dynamic bandwidth of the main spool.
time reaching 60 % and 100 % of the maximum opening ori-fice(corresponding toa maximum spoolposition of±0.8mm)
is respectively 5.9 msand 8.5 ms.The fastresponse time in a matterofmicrosecondsiscomparablewithtypicalsmart-material basednozzle-flapperanddirect-driveflowcontrolvalvesin litera-ture[25,26,33,34],butwiththeadvantageofhavinglargerstrokes (largerflowrate)
A second experiment was performed to show the dynamic behavior of the designed two-stage piezo-valve in frequency domain.In this case,sine sweepvoltagesrangingfrom1 Hzto
150Hzwereappliedtothepiezo-stacks.AsshowninFig.13,the dynamicbandwidthwiththesupplypressureof30barcanbe mea-suredasabout120Hzundertheamplitudeattenuationof-3dB
5 Conclusions
Thispaperdescribesthedetaileddesign,modelingandanalysis
ofanimprovedbridge-typecompliantmechanismtoamplifythe microstrokeofpiezoelectricstackswithenhanceddynamic band-width.Thekinetostaticsanddynamicsofthepresentedbridge-type compliantmechanismareformulatedandanalyzedbyusingthe two-portdynamicstiffnessmodel.Theimprovedbridge-type com-pliant mechanismis furtherappliedto designa piezo-actuated two-stagefour-wayhydraulicservovalve.Withtheunique struc-ture of the improved bridge-type compliant mechanism with doubleoutputports,onlyonegroupofpiezo-stacksis required avoidingthecomplexityofcontrollerandreducingthecost Mea-suringresultsshowthestepresponsetimeof5.9msand8.5ms (60%and100%ofthemaximumstroke),aswellasthefrequency bandwidthof120Hz(-3dB)atthesupplypressureof30bar
Trang 8Author statement
Mingxiang Ling: Conceptualization, Modeling, Original draft
preparation,revision
JiulongWang:CAD-Drawing,Designofthecompliant
mecha-nism
MengxiangWu:Designofthepiezo-valve,Experimental
mea-suring
LeiCao:Softwareandnumericalvalidation
BoFu:ReviewingandEditing
Allauthorsreadandcontributedtothemanuscript
Declaration of competing interest
Noconflictofinterestexitsinthesubmissionofthismanuscript,
and themanuscript isapprovedbyallauthorsforpublication.I
would liketodeclareonbehalfofmy co-authorsthatthework
describedwasoriginalresearchthathasnotbeenpublished
pre-viously,andnotunderconsiderationforpublicationelsewhere,in
wholeorinpart
Acknowledgment
The authors acknowledge the research projects funded by
the National Natural Science Foundation of China [grant
num-ber 52075179], the Applied Basic Research Program of Science
andTechnologyDepartmentofSichuanProvinceofChina[grant
number 20YYJC0312], and thePresidentialFoundationof China
AcademyofEngineeringPhysics[grantnumberYZJJLX2019008],
forthefinancialsupportincarryingoutthiswork.Theyarealso
thankfultoallmembersofMicroActuatorandSensorGroupfor
helpfuldiscussionandsupport
Appendix A
For flexiblestraightbeams,theexpressions ofstaticstiffness
matrixandthefirstthree-ordermassmatricesinEq.(2)arelisted
asfollows,whereEistheYoung’smodulus,isthedensity,listhe
length,handdarein-planeandout-of-planethickness,A=h·d,and
I=h3·d/12aretheareaandmomentofinertiaabouttheneutralaxis
ofthecross-section,˛2=l2/Eandˇ4=l4A/EI.Formoredetails
ofhigh-ordermassmatrices,possiblereadersarerecommendedto refertoourpreviousstudy[30]
K0= E
l3
⎡
⎢
⎢
⎢
⎢
⎣
4Il2 0 −6Il 2Il2
4Il2
⎤
⎥
⎥
⎥
⎥
⎦
M1=420Al
⎡
⎢
⎢
⎢
⎢
⎣
4l2 0 13l −3l2
4l2
⎤
⎥
⎥
⎥
⎥
⎦
M2 =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
EA˛ 4
7EA˛ 4
59EIˇ 8
161700l 3
223EIˇ 8
2910600l 2 0 1279EIˇ
8
3880800l 3 − 1681EIˇ8
23284800l 2
71EIˇ 8
4365900l 0
1681EIˇ 8
23284800l 2 −1097EIˇ8
69854400l EA˛ 4
8
161700l 3 − 223EIˇ8
2910600l 2
71EIˇ 8
4365900l
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
M3=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
2EA˛6
31EA˛6
551EIˇ12
794593800l3
3547EIˇ12
12
8475667200l3 − 112631EIˇ12
76810048000l2
127EIˇ12
3972969000l 0
112631EIˇ12
76810048000l2 − 899EIˇ12
28252224000l 2EA˛6
12
794593800l3 − 3547EIˇ12
23837814000l2
127EIˇ12
3972969000l
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Appendix B Supplementary data
Supplementarymaterialrelated tothis article canbe found,
in the online version, at doi:https://doi.org/10.1016/j.sna.2021
112687
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Biographies
Mingxiang Lingreceived the Bachelor degree from Xi’an Jiaotong University, China, in 2009, and the Master degree (First class Hons.) from Harbin Institute of Technology, China, in 2011 From 2011 to 2015, he was a mechan-ical engineer at China Academy of Engineering Physics From 2015 to 2018, he studied as a PhD student at the Department of Mechanical Engineering in Xi’an Jiaotong University and received the PhD degree in 2019 From
2017 to 2018, He was a visiting scholar with the Com-pliant Mechanisms Lab at the Department of Mechanical Engineering, Brigham Young University, USA He is now an associate research fellow at China Academy of Engineer-ing Physics His main research interests include compliant mechanisms, mechanical dynamics, piezoelectric actuator and sensor (acoustic sensing).
Jiulong Wangreceived the Bachelor degree from Sichuan University, China, in 2015 and the Master degree in mechanical engineering from Xi’an Jiaotong University, China, in 2018 He is now an engineer fellow at China Academy of Engineering Physics His research interests include dynamic test and evaluation, nondestructive test-ing and evaluation of mechanical structure.
Mengxiang Wu received the Bachelor degree in the Department of Mechanical Engineering, Three Gorges University, China, in 2015, and the Master degree in the Department of Mechanical Engineering from Shantou Uni-versity, China, in 2019 He is now an assistant engineer at China Academy of Engineering Physics His main research interests include piezoelectric actuator, mechanical opti-mization design.
Lei Caoreceived his Bachelor and Master degrees in the Department of New Energy, from China University of Petroleum(East China), in 2016 and 2020, respectively.
He is now an assistant Engineer at China Academy of Engineering Physics His main research interests include precision control for piezoelectric actuator, mechanical dynamics.
Bo Fureceived the Bachelor degree in 1991 and the Mas-ter degree in 1994 from Sichuan University, Chengdu, China, and the PhD degree in 2005 from Paderborn Univer-sity, Germany, all in mechanical engineering In 1994, he joined Sichuan University, where he is currently a Profes-sor with the School of Mechanical Engineering His current research interests include fluid power transmission and control, piezoelectric systems and ultrasonic technolo-gies, and multi-objective optimization.