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Lennart Rubbert, R. Bitterli, N. Ferrier, S. Fifanski, I. Vardi, et al..
Isotropic springs
based on parallel flexure stages.
Precision Engineering, Elsevier, 2015, 43, pp.132-145.
ContentslistsavailableatScienceDirect
Precision
Engineering
j ou rn a l h o m e pa g e :w w w . e l s e v i e r . c o m / l o c a t e / p r e c i s i o n
Isotropic
springs
based
on
parallel
flexure
stages
L.
Rubbert
∗,
R.
Bitterli,
N.
Ferrier,
S.
Fifanski,
I.
Vardi,
S.
Henein
Instant-Lab,ÉcolePolytechniqueFédéraledeLausanne,Neuchâtel,Switzerland
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:Received28January2015
Receivedinrevisedform6May2015 Accepted2July2015
Availableonline17July2015 Keywords: Centralspring Flexure Compliantmechanism Stiffness Isotropydefect
a
b
s
t
r
a
c
t
Wedefineisotropicspringstobecentralspringshavingthesamerestoringforceinalldirections.In pre-viouswork,weshowedthatisotropicspringscanbeadvantageouslyappliedtohorologicaltimebases sincetheycanbeusedtoeliminatetheescapementmechanism[7].Thispaperpresentsourdesignsbased onplanarserial2-DOFlinearisotropicsprings.Weproposetwoarchitectures,bothbasedonparallelleaf springs,thenevaluatetheirisotropydefectusingfirstlyananalyticmodel,secondlyfiniteelement anal-ysisandthirdlyexperimentaldatameasuredfromphysicalprototypes.Usingtheseresults,weanalyze theisotropydefectintermsofdisplacement,radialdistance,angularseparation,stiffnessandlinearity. Basedonthisanalysis,weproposeimprovedarchitecturesstackinginparallelorinseriesduplicatecopies oftheoriginalmechanismsrotatedatspecificanglestocancelisotropydefect.Weshowthatusingthe mechanismsinpairsreducesisotropydefectbyonetotwoordersofmagnitude.
©2015TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Thebiggestimprovementintimekeeperaccuracywasdueto
theintroductionoftheoscillatorasatimebase,firstthependulum
byChristiaanHuygensin1656[10],thenthebalancewheel–spiral
springbyHuygensandHookeinabout1675,andthetuningforkby
NiaudetandBreguetin1866[14].Sincethattime,thesehavebeen
theonlymechanicaloscillatorsusedinmechanicalclocksandinall
watches.
In[7],wepresentednewtimebasesformechanicaltimekeepers
which,intheirsimplestform,werebasedonaharmonicoscillator
firstdescribedin1687byIsaacNewtoninPrincipiaMathematica
[13,BookI,PropositionX].Thisoscillatoristheisotropicharmonic
oscillator,whereamassmatpositionrissubjecttoacentrallinear
(Hooke)force.
Sincetheresultingtrajectorieshaveunidirectionalrotation,this
oscillatorhastheadvantageofsolvingtheproblemofinefficiency
oftheescapementbyeliminatingitor,alternatively,simplifyingit
[7].Isochronismisthekeyfeatureofagoodtimebase,andinthis
case,thespringofthespring–masssystemmustbeasisotropicas
possible,meaningthatineverydirection,thespringstiffnessand
massmustremainthesame.Inaddition,itshouldbeplanarinorder
tobeeasytomanufactureatanyscale(notethatNewton’smodel
impliesplanarmotion,bypreservationofangularmomentum).
∗ Correspondingauthor.
E-mailaddress:lennart.rubbert@insa-strasbourg.fr(L.Rubbert).
Inthispaper,wemechanizeNewton’smodelbydesigningnew
planarisotropicsprings.Ourdesignsarebased ontheprinciple
ofcompliantXY-stages[1,3,11,12]whicharemechanismwithtwo
degreesoffreedom(2-DOF)bothofwhicharetranslations.Asthese
mechanismsare composedof compliant joints [9] theyexhibit
planarrestoringforcessocanbeconsideredasplanarsprings.In
theliterature,manyplanarflexibleXY-stageshavebeenproposed
and ifsomemay beimplicitly isotropic,nonehasbeen
explic-itlydeclaredtobeisotropic.Thiscouldbeexplainedbythefact
that,ingeneral,XY-stagesarecontrolledinclosed-loops[17]and
isotropy stiffnessdefectsare thereforenot necessarily amatter
ofconcern.Moreover,weuseaserialarchitectureinsteadofthe
parallelonegenerallyseeninXY-stagesusedactuatorintegration
applications.
SimonHenein[6,p.156,158]proposedtwonon-planar
archi-tectureXY-stagesexhibitingplanarisotropy.Thefirstiscomposed
oftwoserialcompliantfour-barmechanisms,alsocalledparallel
armlinkage,whichproduce,forsmalldisplacements,translations
inXandY(seealso[5]).Thesecondiscomposedoffourparallel
armslinkedbyeightsphericaljointsandabellowconnectingthe
mobileplatformtotheground.
Inthispaperthedesignsoftwocentralspringsbasedonparallel
leafspringsarepresentedafterabriefpresentationofthecontext.
Foreachofbothdesigns,theiranalyticalmodelispresentedand
comparedtotheperformancebasedonfiniteelementanalysisand
onexperimentaldataofphysicallyconstructedprototypes.
Someimagesofthesedesignshaveappearedin[7]andsome
appearinrecentpatentapplications.
http://dx.doi.org/10.1016/j.precisioneng.2015.07.003
0141-6359/©2015TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense( http://creativecommons.org/licenses/by-nc-nd/4.0/).
Fig.1. EllipticalorbitundercentralHookeLaw.
2. Context
Inordertounderstandtheellipticaltrajectoriesofplanets
pre-dicted by Kepler’sLaws, Isaac Newton considered thepossible
centrallawsproducingellipticalorbitsandheshowedthatapart
fromtheinversesquarelaw,alinearHooke’slawwouldalso
pro-duceellipticalorbits,1seeFig.1.
Newton’sresultis veryeasily shown.Consider apoint mass
movingintwodimensionssubjecttoacentralforce
F(r)=−Kr,
whereristhedistanceofthemasstothecenter.ApplyingNewton’s
secondlawF=ma,wheremisthemassoftheparticleandaits
acceleration,givesthegeneralsolution
r=(A1sin(ω0t+ϕ1),A2sin(ω0t+ϕ2)), (1)
forinitialconditionsA1,A2,ϕ1,ϕ2andfrequency
ω0=
K
m.
Thisshowsthatorbitsareelliptical,butalsothattheperiodonly
dependsonthemassmandthestiffnessKofthecentralforce,and
notontheenergyofthesystem,whatisgenerallycalled
isochro-nism.Thislastpropertyisthekeyfeatureofhorologicaltimebases
inwhichtheregulationmustbekeptindependentoftheenergy
source.Itfollowsthatthisoscillatorisagoodcandidatetobeatime
baseforatimekeeper,anobservationfirstmadeinourprevious
article[7].
Inordertoexploitthisoscillatorasamechanicaltimebase,
New-ton’smodelmustbefollowedascloselyaspossible.Inparticular,
themechanism’scentrallinearrestoringforcemustbeasisotropic
aspossible.Expressedquantitatively,theisotropydefectmustbe
minimized.
3. Definitionofisotropydefect
Thefirststepinanalyzingtheisotropydefectofacentralspring
istogiveaprecisedefinitionofwhatismeantbyisotropydefect.
Inparticular,sinceisotropydefectonlyappliestocentralsprings,
theterm“central”willbesuppressedwithoutambiguity.Thebasic
contextofourisotropydefectcomputationsisgiveninFig.2.
3.1. Baselinebehavior
Inordertoevaluateisotropydefect,abaselineisrequiredfor
comparison.WeassumethatourspringhasidealstiffnessK.Aforce
F ofmagnitudeFanddirection isapplied,where willvary
between0◦ and360◦ and themagnitudeFwillbeconstant(i.e.
1Theoccurrenceofellipsesinbothlawsisnowunderstoodtobeduetoarelatively
simplemathematicallyequivalence[4]anditisalsowell-knownthatthesetwo casesaretheonlycentralforcelawsleadingtoclosedorbits[2,15].
Fig.2.Basicmodelofisotropydefect.
independentof).Underthisforce,thepointOonthespringmoves
totheidealpositionP,andbyHooke’sLaw, OPhasmagnitudeF/K
anddirection,seeFig.2.Therefore,asvariesbetween0◦and
360◦,theidealpointPdescribesaperfectcircleofradiusF/K,and
therestoringforceislinearandisotropic.
3.2. Definitions
The previous example illustrates ideal behavior with zero
isotropydefect.Divergencefromthisexamplewillbeusedto
mea-suretheisotropydefect.Thus,whenisotropydefectdoesoccur,the
forceFwillmovethepointOtoapointQgenerallydistinctfrom
P,seeFig.2.Theisotropydefectvectorinthedirectionisdefined
tobePQ.
In ordertoevaluateand comparetheisotropy defectof our
mechanisms, it is more convenient tohave scalar measuresof
isotropy defect. We thereforedefine thesimplerradial isotropy
defectgivenby
= OQ− OP.
Notethatthismeasureofisotropydefectconsidersthediscrepancy
betweenthemagnitudesoftheactualdisplacementandtheideal
displacementforanglewhich isdifferentfromthemagnitude
PQoftheisotropydefectvector,seeFig.2.
Theangularisotropydefectϕisdefinedastheanglebetween
OPandOQ,asmeasuredinthecounterclockwisedirection,see
Fig.2.
Inordertodefinestiffnessisotropydefect,wefirstintroducethe
notionofstiffnessinagivendirectionby
k= F
OP+
= F
OQ
.
Thestiffnessisotropydefectinthedirectionisthen
k=K−k,
whereKistheidealstiffnessdefinedasthemaximumofkforall
angles.Notethatkisnon-negative,bydefinitionofK.
Therelativestiffnessisotropydefectinthedirectionisdefined
as =kk = OP ,
Fig.3. Conceptualdesignofourisotropicsprings.
wheretherighthandexpressionfollowsbyapplyingK=F/ OP.
Wefinallydefinethestiffnessisotropydefecttobemax,the
max-imumofoverall0◦≤<360◦.ThiswillbeusedinSection 7to
evaluateisotropyperformance.
Inaddition,weneedourcentralspringstobelinear,i.e.,the
relationbetweenforceanddisplacementshouldbelinear.Inorder
toevaluatethisquantitatively,foreachmechanismandforeach
forcedirection,asecondorderpolynomialfitisappliedtothe
dis-placementversustheforce.Formally,thismeansthattheforceis
writteninpolarcoordinatesrandasF=(A+Br+Cr2)u
,with
uaunitvectorinthedirection.ForalinearspringA=C=0and
Bdefinestheconstantspringstiffnessinthedirection.Thus,C
quantifiesthestiffnesslinearitydefect.
Remark3.2. Inordertosimplifynotation,thesubscriptinthese
quantitiesmaybesuppressedinthefollowing.
4. Isotropicspringdesign
Formula(1)ofSection2showsthatharmonicisotropic
oscilla-torsolutionscanbeconsideredassumsoftwoindependentlinear
1-DOFoscillators,oneinthexdirectionandtheotherinthey
direc-tion.Forthisreason,wehavechosentoconsiderplanarisotropic
springsfollowingtheconceptualconfigurationshowninFig.3.This
configurationconsistsoftwolinearspringsoflengthL0(inneutral
position)placedorthogonallyinserialconfiguration.Thegeometry
ischosensuchthatbothspringsareintheirneutralpositionswhen
pointsPandOarecoincident.
4.1. The1-DOFparallelspringstage
Thebasicbuildingblockofourdesignisthelinearspring,and
thiscanberefinedbyreplacinglinearspringswith1-DOFparallel
springstages,asshowninFig.4.Thepointisthatparallelspring
Fig.4. 1-DOFparallelspringstage.
stageshavetheadvantageofhavingasimplecompactarchitecture
soareeasytomanufactureanddownscale[16].
Asitconstitutesthemainbuildingblockofourcentralspring
designs,wepresentanextensiveanalysisofthis architecturein
ordertofullydescribeitsstiffnessandkinematicproperties.
Fig.4consistsoftwoparallelleafsprings(blades)attachedto
thegroundandtoamobilerigidbodywhosepositionisuniquely
determinedbytheimageofthepointO=(0,0).NotethatthepointO
liesexactlyhalfwayalongtheverticalleafsprings,thatis,atheight
L/2fromtheground(thissatisfiesoneofthemainconditionsof
Appendix).Asdefinedpreviously,whenaforceFisappliedatpoint
O=(0,0)itmovestoQ=(x,y),asillustratedinFig.4.TheforceFcan
bedecomposedintohorizontalandverticalcomponents
F=(Fx,Fy)=(Fcos,Fsin),
where is theangle betweenthe horizontalx-axis and F. We
assumethatdeformationsaresufficientlysmallsothatbeam
short-eningtheory,asdescribedin[8,Section8.7],canbeapplied.Asa
consequence,andasshowninAppendixA,theeffectivestiffness
kx=Fx/xalongthex-axisisverywellapproximatedbythelinearized
approximationtotheanalyticmodel
kx= 24EI
L3 +
6Fy
5L, (2)
whereI=bh3/12,EistheYoungmodulusofthematerialandh,b,
Larethethickness,thewidthandthelengthofthebeam,
respec-tively.
Remark4.1.1. Strictlyspeaking,formula(2)isanabuseof
nota-tionsinceitreferstoanapproximationofthehorizontalstiffness
kx.InAppendix,wemakethisdifferenceexplicitbyreferringtokx
astheactualstiffness,kcom
x thecompleteanalyticstiffnessandklinx
asthelinearizedanalyticstiffnessgiveninformula(2).
Remark4.1.2. ThenecessaryconditionthattheforceFbeapplied
atthemid-heightofthebeamlengthL/2willholdforallthe
mech-anismarchitecturespresentedinthispaper.
Remark4.1.3. Formula(2)showsthatforthisstructure,the
stiff-nesskxdependsonFywhichmeansthatthisspringissensitivetoa
forceappliedalongthey-axis.Formula(2)alsoshowsthatthe
influ-enceofFyonthehorizontalstiffnesskxcanbereducedbyincreasing
thebeamlengthL.
The other result we will useconcerning the 1-DOF parallel
springstageisthat,asshowninAppendixA.12,thetrajectoryof
therigidblockisparabolicsinceitsdefiningpointQ=(x,y)isvery
wellapproximatedby
x,−3x2 5L . (3)Remark4.1.4. Wenotethatourformula(2)forthehorizontal
stiffnessisderivedusingEuler–Bernoullitheory,soasisstandard
inthistheory,theverticaldisplacementofthemobileblockis
con-siderednegligible and takentobezero.The parabolic trajectory
computationincludedaboveisusedexclusivelyforthecomputation
ofthekinematics.
4.2. Thesimple2-DOFparallelspringstage
Inordertogeta2-DOFcentralspring,twoparallelspringstages
arethereforeplacedorthogonallyandinaserialconfiguration,as
showninFig.5.
ForaradialforceFactinginthex–yplaneandappliedatangle
Fig.5. Simple2-DOFparallelspringstage.
centerofthemechanismisobtainedbysummingthecontribution
ofthe1-DOFparallelspringstages and
x=x1+x2, y=y1+y2,
where,bytheresultsstatedintheprevioussection,
x1= Fx kx, y1= −3x2 1 5L , x2= −3y2 2 5L , y2= Fy ky. Moreover,asbefore kx= 24EI L3 + 6Fy 5L, ky= 24EI L3 + 6Fx 5L,
whereonceagain
F=(Fx,Fy)=(Fcos,Fsin).
Combiningtheseresultsisonceagainastraightforward
computa-tionwhichyieldsthefollowingexpressionforxandyintermsof
theforceangle
x= Fcos 2Ebh3 L3 +6F5Lsin − 3 5L
Fsin 2Ebh3 L3 +6Fcos5L 2 , y= Fsin 2Ebh3 L3 +6F cos 5L − 3 5L Fcos 2Ebh3 L3 +6F sin 5L 2 .4.3. Compound2-DOFparallelspringstage
InSection4.1weshowedthattheparallelleafspringstagehas
parabolicdisplacement,however,aspointedout inSection 4.1,
isotropicharmonicoscillatorsolutionsarebuiltfromorthogonal
linearoscillators.Weremedythissituationbyusingthecompound
parallelspringstageshowninFig.6.Whenforceisappliedalong
thexdirectiononly,thesecondordertermsofeachstageexactly
cancelasFy=0,theirstiffnessalongthex-axisareequalandthe
sec-ondordertermyofeachstagesareofsameamplitude,resultingin
perfectrectilinearmotion.
Thesameholdsintheydirection,sothecompound2-DOF
par-allelstageyieldsafaithfulmechanizationoftheisotropicharmonic
oscillator’sxandycomponents,eachtakenseparately,andis
there-foreagoodcandidatefortheminimizationofisotropydefect.The
compound2-DOFparallelspringstageisillustratedinFig.7.
IfaconstantradialforceFisappliedatangle,thentheposition
ofthecenterOofthecompound2-DOFparallelspringstageis
dis-placedbytheactionofthefourparallelleafspringstages,theseare
Fig.6.Compoundparallelspringstage.
numbered to inFig.7.TheresultingpositionQ=(x,y)satisfies
x=x1+x2+x3+x4, y=y1+y2+y3+y4,
andwritingF=(Fx,Fy)asbeforeyields
x1= Fx kx1, x2= Fx kx2, y3= Fy ky3, y4= Fy ky4, y1= 3x2 1 5L, y2=− 3x2 2 5L, x3=− 3y2 3 5L , x4= 3y2 4 5L, where kx1=24EI L3 − 6Fy 5L, kx2= 24EI L3 + 6Fy 5L, ky3= 24EI L3 + 6Fx 5L, ky4= 24EI L3 − 6Fx 5L.
Notethatthefirstexpressionmakesitclearthatthefourparallel
springscanbeassembledsequentiallyinanyorder.Thereforethe
compoundspringmechanismcanbeobtainedbycombiningtwo
simplespringmechanismswiththecorrectorientation.
4.4. Prototypedesign
Inordertoevaluatethesetwocentralsprings,wemanufactured
twoprototypesillustratedinFigs.8and9.Theprototypesare
com-posedofaluminumframesandspringsteel(X10CrNi18-8)blades
whosegeometricandmaterialYoungmodulusaregiveninTable1.
Thealuminum framesof theprototypesweremachinedwitha
numericallycontrolledmachiningtoolandtheflexiblebladeswere
lasercut.Theblade manufacturingandassemblytolerancesare
Fig. 8. Simple 2-DOF parallel spring stage prototype. Frame dimensions 220×220mm2.
Fig.9. Twostagesassembledforthe2-DOFcompoundplanarspringprototype. Framesdimensions150×150mm2.
Table1
Flexiblebeamgeometricandmaterialparametersforeachprototype.
b[mm] h[mm] L[mm] E[GPa]
Prototype1 20 0.2 50 200
Prototype2 20 0.3 50 200
±7mand±60mforthethicknesshandforthelengthL, respec-tively.Thecompound2-DOFparallelspringstageiscomposedof twolayers,asillustratedinFig.10.EachlayergivesoneDOFinthe
xandydirections,respectively.Thebottomlayerframeisrigidly
Fig.11.Experimentalsetupwithsimple2-DOFspringstageprototype.
fixedtothegroundandthetoplayerframeisthemobileplatform
marked .Thelayersareconnectedthoughthecenterelements
marked ,seeFig.10.
5. Experimentalevaluation
5.1. Experimentalsetup
TheexperimentalsetupisshowninFig.11.Theprototypeis
rigidlyfixedtotheground.Thecenterofthemovingplatformis
connectedtoaforcesensor(Kistler9207)byaflexiblebeamwhich
actsasaconnectingrod.Theforcesensorismountedonalinear
stagedrivenmanuallybyaVerniermicrometer(NewportSM-25)
whichhastheoreticaldisplacementresolutionof1m;itsroleis
toimposeadisplacementontheprototypecenter.
Thissystemisfixedonaturntable,thatistosay,atable
allow-ingthedisplacementdirectiontobevariedby15±0.04◦.Foreach
orientation,anincreasingdisplacementisappliedtotheprototype
centerbyactuatingthemicrometer.Theforcesensorhasan
abso-luteprecisionof0.8598mNwithadecreasingdriftof0.5mN/s.
Thexandypositionsofthemovingplatformaremeasuredthanks
totwodisplacementsensors:KeyencelaserheadLK-H082having
measurementrangeof±18mmandabsoluteprecisionof±7.3m.
Dataisacquiredwithanacquisitioncard(NIPCI-6052E)and
asso-ciatedtothemeasuredforcethroughaLabviewsoftwareprogram,
wherea3.30GHzIntelCorei5-2500computerwith64-bit
archi-tectureisused.
Fig.12.Polarplotsofdisplacement,inmillimeters,ofthecenterofeachmechanism.(a)Centerdisplacement,simple2–DOFspringstage,5N.(b)Centerdisplacement, compound2–DOFspringstage,12N.
5.2. Loadcasedescription
Measurementsfrom1Nto5Nwith1Nstepsweremadeonthe
firstprototype.Measurementsfrom1Nto12Nwith1Nstepswere
madeonthesecondprototype.Bothprototypemeasurementswere
madewithforcedirectionsgoingfrom0◦to345◦in15◦steps.
Themeasurementofstiffnessinasingledirectiontakesabout
10sforeachforcelevel,whichmeansthatformeasurementsupto
5N,theforcesensorexhibitsadrifterrorof2.5mN(0.05%error)
andformeasurementsupto12N,theforcesensorexhibitsadrift
errorof6mN(0.05%error).
6. Results
6.1. Analyticalversusnumericalandexperimentaldata
Three different type of methods were used to create data
describingthemechanism.
6.1.1. Analyticalmodel
TheanalyticalmodelofSection 4wasimplementedusingthe
springparametersgiveninTable1.Theresultingdatawas
com-putedusingMatlab.Eachplotisgivenwithangular1◦steps.
6.1.2. Finiteelementmodel
Finiteelementanalysis(FEA)wasdoneusingComsol,with1N
forcesteps,5◦ orientationstepsanda0.001convergencefactor.
Thesimpleandcompound2-DOFparallelspringstagewere
com-putedupto5Nand12Nrespectively.Thesimulationswerebased
onnon-linearmodels.Totalcomputationtimeforthesimpleand
compound2-DOFparallelspringstagewasabout6daysand9days,
respectively,ona6cores(12threads)Xeonprocessorcadencedat
3.2GHzwith48GBofRAM.
6.1.3. Experimentalmeasurements
The displacement of each prototype’s center as a function
of forcedirectionwasacquired forall forcelevels.The Vernier
micrometerisactuatedmanuallyandthedataisacquired
automat-ically.Totalacquisitiontimeforthesimpleandcompound2-DOF
parallelspringstageswasapproximately1.5and2h,respectively.
6.2. Dataplots
6.2.1. Displacementpolarplots
Displacementpolarplotsdisplayinpolarcoordinatesthe
dis-placedpositionsQofthecenterofeachmechanismforarangeof
forceorientations,whereQreferstothenotationofSection 3.2.
Fig.13.Polarplotsofradialisotropydefect,analyticalmodel,FEMandprototype.(a)Radialisotropydefectinmillimetersforthesimple2-DOFparallelspringstagefor5N radialforce.(b)Radialisotropydefectinmillimetersforthecompound2-DOFparallelspringstagefor12N.
Fig.14.Angulardisplacementasafunctionofforcedirection,analyticalmodel,FEMandprototype.(a)Simple2-DOFparallelspringstageunder5Nload.(b)Compound 2-DOFparallelspringstageunder12Nload.
Fig.12(a)showsthedisplacedpositionsofthecenterofthesimple
2-DOFparallelspringstageaspredictedbytheanalyticalmodel,
computedbyFEAandmeasuredexperimentallywiththe
proto-typeundera5Nload.Fig.12(b)showsthedisplacedpositionsof
thecenterofthecompound2-DOFparallelspringstageaspredicted
bytheanalyticalmodel,computedbyFEAandmeasuredwiththe
prototypeundera12Nload.
Remark6.2.1. Thestiffnesspropertiesoftheprototypesarevery
sensitivetothelengthandthicknesstolerancesofthebladessince
theseparametersappeartothethirdpowerinthestiffness
formu-lation.Thus,manufacturingandassemblytolerancesaresufficient
toexplaindiscrepanciescomparedtothemodel,seeSection4.4.
6.2.2. Radialisotropydefect
RecallfromSection3.2thattheradialisotropydefectwas
quan-tifiedby.Inordertoplottheradialisotropy,foreachgivenforce,
aconstantindependentofissubtractedfromthemeasured
dis-tances,withtheforceanglevaryingfrom0◦to360◦.Thisconstant
valueischosenastheminimumofthemeasureddistancesforthat
givenforce.Thischoiceappearstogivethemostreadableplot.
Theresultingplotsforthesimple2-DOFparallelspringstageare
illustratedinFig.13(a).Here,themaximalradialisotropydefects
fortheanalyticalmodelandtheexperimentalmeasurementsare
65.36% and86.04%, respectively, oftheminimum displacement
distance.
Thesameplotforthecompound2-DOFparallelspringstage,
againwiththedisplacementdistanceoffsetequaltotheminimum
displacementdistance,isplottedinFig.13(b).Here,themaximal
radialisotropydefectfortheanalyticalmodelandtheprototype
areto2.92%and8.87%,respectively,oftheminimumdisplacement
distance.
6.2.3. Angularisotropydefect
RecallfromSection 3.2that theangular isotropydefect was
quantifiedbyϕ.Fig.14(a)and(b)plotstheangularisotropydefect
foreachspring.Recallthatthisquantifiestheangulardisplacement
ofthecenterofthemechanismwhenaradialforceisapplied.
Theseplotsshowthatforthesimple2-DOFparallelspringstage,
a0◦ angularisotropydefectisexpectedat45◦,116◦,164◦,225◦,
287◦and335◦andamaximumangularisotropydefectisexpected
at194◦.
For thecompound 2-DOF parallel springstage, a 0◦ angular
isotropydefectisexpectedat0◦,45◦,90◦,135◦,180◦,225◦,270◦and
315◦,andamaximumisotropydefectisexpectedatthemidpointof
theseangles,thatis,at22.5◦,22.5◦+45◦,22.5◦+90◦,...,22.5◦+315◦.
6.2.4. Stiffnessisotropy
RecallthedefinitionofkinSection3.2whichshowsthatthe
isotropydefectkcanrepresentedbyplottingstiffnesskasa
func-tionofforcedirection.Foraperfectlyisotropicspring,thisshould
beconstantandtheplotahorizontalline.Fig.15(a)and(b)
illus-tratesstiffnessasafunctionofforcedirectionforthesimpleand
compound2-DOFparallelspringstages.
Fig.15.Stiffnessasafunctionofforcedirectionforeachmechanism.(a)Simple2-DOFparallelspringstageupto5Nload.(b)Compound2-DOFparallelspringstageupto 12Nload.
Fig.16.Secondordertermofstiffnesspolynomialfitasafunctionofforcedirection.(a)Simple2-DOFparallelspringstageupto5Nload.(b)Compound2-DOFparallel springstageupto12Nload.
6.2.5. Stiffnesslinearity
Werecallthestiffnesslinearityassumptionsandnotationof
Sec-tion3.2.WefocushereonthesecondordertermCquantifyingthe
stiffnesslinearitydefect.InFig.16,apolynomialfitforCisgivenfor
eachcentralspring.Fig.16(a)showsthatthesimple2-DOFparallel
springstagehasanon-linearitydefecttentimesgreaterthanthe
compound2-DOFparallelspringstageshowninFig.16(b).
More-over,thesimple2-DOFparallelspringstagehasbothpositiveand
negativesecondordertermswhereasthecompound2-DOFparallel
springstagehasonlynegativeorderterms.
Thesimple2-DOFparallelspringstageisalsohighlynon-linear
inthe45◦and225◦directions.Therefore,thesimple2-DOFparallel
springstagehasbothnon-linearityandstiffnessisotropydefects.
7. Improvedmechanismarchitectures
Withthegoalofreducingisotropydefect,onecanimprovethe
performanceofthesimpleandcompoundmechanismsbyplacing
anadditionalcopyinparallel,atthecostofalossofsimplicityand
planarity.Thefollowingresultswereobtainedfromtheanalytical
model.
Forthesimplemechanism,oneplacesinparallelasecond
iden-ticalcopywhichisrotated180◦in-plane(Fig.18(a)).Thisisdonein
ordertocancelouttheminimalandmaximalstiffnessvaluesfound
inFig.15(a)andleadstoanisotropydefectmaxof6.77%instead
of65.36%,withmaxasdefinedinSection3.2.Inordertocompare
stiffnessisotropydefectsoftheoriginalversusthedouble
mecha-nism,thecompoundmechanismhasitsleafspringbeamsstiffness
dividedbytwo(e.g.,beamwidthhalved)andthecomparisonis
showninFig.17(a).Forthesamereason,whentheserial
arrange-mentisconsidered,theleafspringbeamsstiffnessismultipliedby
two(e.g.,beamwidthdoubled).
Aneven greater improvement is possiblefor thecompound
mechanismbyplacingaparallelcopyat45◦(Fig.18(b)).Thiscancels
outtheminimalandmaximalstiffnessvaluesshowninFig.15(b)
andyieldsanisotropydefectmaxofonly0.0262%insteadof2.92%.
Thispairofcompoundmechanismis111timesmoreisotropicthan
thesinglestructure.Onceagain,acomparisonofstiffnessforthe
singleversusthecompoundmechanismispossiblebyhalvingthe
leafspringstiffnessofthecompoundmechanismwiththeresults
showninFig.17(b).
Thisprocesscanbecontinuedbyfurthercombining22.5◦
par-allelin-planerotationsoffourcompoundmechanismstocancel
outminimalandmaximalstiffness.Theisotropydefectcouldthus
be reduced tomax=1.58e−6%by stacking together fourlayers
of compound mechanismswiththeabove orientation.Stacking
fourcompound mechanisms is 16,567times betterthan
stack-ingtwocompoundmechanismsand1,847,200timesbetterthan
Fig.17.Stiffnessofparallelcentralspringswithisotropydefectreduction.(a)Stiffnessof180◦parallelsimple2-DOFparallelspringstagesasafunctionofdirection.(b)
Fig.18.Possiblerealizationsofparallelcentralspringarrangementdecreasingisotropydefect.(a)Explodedviewofrealizationofthe180◦parallelarrangementofsimple
2-DOFparallelspringstages,topview.(b)Explodedviewofrealizationofthe45◦parallelarrangementofcompound2-DOFparallelspringstages,topview.(c)Realization
ofthe180◦parallelarrangementofsimple2-DOFparallelspringstages,topview.(d)Realizationofthe45◦parallelarrangementofcompound2-DOFparallelspringstages,
topview.
Table2
Stiffnessisotropydefectmaxcomparisonbetweenthesingleandmulti-layerparallelarrangementofthesimpleandcompound2-DOFspringstages.
#ofstages Simple Compound
max Defectreductionfactor #globalmax Stageoffset max Defectreductionfactor #globalmax Stageoffset
1 65.36% 1 1 – 2.92% 1 4 –
2 6.77% 9.66 2 180◦ 2.62e−2% 111.53 8 45◦
4 1.87% 35.02 4 90◦ 1.58e−6% 1.85e6 16 22.5◦
8 0.11% 578.26 8 45◦ 1.04e−13% 28e12 32 11.25◦
Table3
Stiffnessisotropydefectmaxcomparisonbetweenthesingleandmulti-layerserialarrangementofthesimpleandcompound2-DOFspringstages.
#ofstages Simple Compound
max Defectreductionfactor #globalmax Stageoffset max Defectreductionfactor #globalmax Stageoffset
1 65.36% 1 1 – 2.92% 1 4 –
2 5.41% 12.08 4 180◦ 5.46e−3% 535.60 8 45◦
4 5.16% 12.66 4 90◦ 3.81e−7% 1.40e9 16 22.5◦
Fig.19.Possiblerealizationsofserialcentralspringarrangementdecreasingisotropydefect.(a)Explodedviewofrealizationofthe180◦parallelarrangementofsimple 2-DOFparallelspringstages,topview.(b)Explodedviewofrealizationofthe45◦parallelarrangementofcompound2-DOFparallelspringstages,topview.(c)Realization ofthe180◦parallelarrangementofsimple2-DOFparallelspringstages,topview.(d)Realizationofthe45◦parallelarrangementofcompound2-DOFparallelspringstages, topview.
the single compound mechanism. For both architectures, the performances of parallel rotated arrangement are presented in
Table2.
Apossiblerealizationforeachcentralspringscombiningtwo
layers,180◦and45◦rotationrespectively,isillustratedinFig.19.
Inordertoavoidcontactbetweentheintermediateparts,shims
areplacedbetweenthetwolayersconnectedrigidlytotheframes
(ground)andmoving platforms (seeFigs.5and7).Notethat
the roles of the moving platform and of the ground can be
inverted.
Furtheriterationsreducetheisotropydefecttoarbitrarilysmall
values.
Remark7. Anothermethodforreducing theisotropydefectis
tostacktwostagesinseriesatspecificshiftangles.Theresulting
improvementisquantifiedinTable3.
8. Conclusion
Thenumber,positionandvalueoflocalmaximumsaspredicted
bytheanalyticalmodelandtheFEMmodelareconsistentandthey
matchtheexperimentalresults.Thisvalidatestheuseofthesimple
analyticalmodelestablishedinSection 4topredicttheisotropy
performanceofsuchstructures.
Tables2and3showthattheisotropydefectoftheoriginal
sim-plespringstagecanbedrasticallyreducedusingtwoapproaches
• Useofcompoundstagesinsteadofsimplestages.
• Parallelorserialstackingofidenticalstagesrotatedbyprecisely
chosenangles.
Fortheparallelarrangements,thequantitativeevaluationofthese
designsshowsthatstackingfoursimplestagesleadstoamaximum
relativestiffnessisotropydefectclosetothatofasinglecompound
stage(valuesshowninboldfaceinthetable).
Forexample,Fig.18showshowpairsofsimpleandcompound
stagehavingmaximumrelativestiffnessisotropydefects65.36%
and2.92%, respectively,canbestacked leadingtoadecreaseof
therelativestiffnessisotropydefectbyafactor9.66and111.53,
respectively, leading to relative residual defects of 6.77% and
AppendixA. Stiffnessandkinematicsofthe1-DOFparallel springstage
InthisAppendix,wegiveaformal,self-containedproofofthe
claimsofSection4.1.Inotherwords,wewillshowthatgiventhe
1-DOFparallelspringstageshowninFig.A.20,horizontalstiffness
kxisverywellapproximatedby
klin
x =24EI
L3 +
6Fy
5L,
andthattherigidblockhasaparabolictrajectorysincethe
coordi-natesofthedistinguishedpointQ=(x,y)isverywellapproximated
by
x,−3x2 5L .Note that, as stated in Remark 4.1.4, our formula for the
hor-izontal stiffness is derived using Euler–Bernoulli theory,sothe
verticaldisplacementofthemobileblockisconsidered
negligi-ble.Theparabolictrajectorycomputationincludedaboveisused
exclusivelyforthecomputationofthekinematics.
A.1. Planofproof
Theproofappliesstandardmethodsofmechanicsofmaterials,
e.g.,asfoundin[8].
WeexaminethesituationillustratedinFig.A.20,inwhicha
hor-izontalforceisappliedtoaparallelspringstagewhenaverticalload
ispresent.Clearlythisverticalloadaffectsthehorizontalstiffness,
andmakingthisexplicitwillbethefocusofourinvestigation.
Thegeneralideaoftheproofistoconsiderasinglebeam,use
Euler–Bernoullitheorytocomputethehorizontaldeformationat
thebeamextremityforagivenhorizontalforceandverticalload,
recoverthehorizontalstiffnessusingHooke’sLaw,thenmultiply
bytwotogetthetotalhorizontalstiffness.
Theverticaltensionandcompressioncasesleadtocompletely
differentanalyticsolutions,despitegivingthesameresultforthe
linearizedhorizontalstiffness,sowewilltakecaretodistinguish
thesecasesinthisAppendix.
OurcomputationsuptoandincludingAppendixA.9are
com-plete analytic expressions with linearization only occurring in
AppendixA.10toAppendixA.12.
WebegintheproofinAppendixA.2withanexplicit
descrip-tionoftheparametersinvolvedandnotethatthreeverticalload
situationsoccur.
InAppendixA.4,webeginthecomputationbyconsideringonly
asinglebeamof ourmodel. Wethen setuptheknown
gover-ningdifferentialequationsofdeformationforeachofthethreeload
situations.
InAppendixA.5,wegivetheknowngeneralanalyticsolutions
tothedifferentialequationsofAppendixA.4.
Fig.A.20.Parallelspringstageunderload.
InAppendixA.6,weapplytheboundaryconditionsinorderto
computethecompleteanalyticsolutionstoourthreeload
situa-tions.
InAppendixA.7,wecompute,forourthreeloadsituations,the
horizontalrestoringforceforagivenhorizontaldisplacementofa
singlebeamextremity(whichisexactlythedisplacementofthe
mobilerigidblock).
InAppendixA.8,weapplyHooke’slawtotheresultofAppendix
A.7toderivethecompleteanalyticexpressionforthehorizontal
stiffnessofasinglebeamatitsextremity,andthisforourthree
loadsituations.
InAppendixA.9,weapplytheresultofAppendixA.8tofindan
completeanalyticsolutiontothehorizontalstiffnessforthe1-DOF
parallelspringstagegivenourthreeverticalloadsituations.This
givesthecompleteanalyticsolutiontoourmainproblem.
InAppendixA.10,wegivethelinearizationoftheformulasof
AppendixA.9completingtheproofofthefirstmainresultofthis Appendix.
InAppendixA.11,wegiveanumericalestimateoftherelative
errorbetweenthecompleteanalyticexpressionandits
lineariza-tionusedinthepaperandshowthatitisboundedby404ppm.
AppendixA.12isindependentofthepreviousanalysisandis
basedonwell-knownresultsofbeamdeformationtheory.These
knownresultsyieldtheverticalandhorizontaldisplacementsof
thebeamextremitiesinourmodel,andapproximatingtosecond
orderyieldstheparabolictrajectory,thesecondmainresultofthis
Appendix.
A.2. Problemdescription
Thetwoverticalbeams inFig.A.20 areidenticalandhave a
momentofinertiaIaroundz,andYoung’smodulusE.The
hori-zontaldisplacementofthemobilerigidblockisgivenbyFandits
verticaldisplacementby.Notethat,byHooke’sLaw,the
horizon-talstiffnessofthemodelissimplykx=Fx/f,whereFx=Fxx.
Ourgoalistofindasimpleapproximationtothehorizontal
stiff-nesskx.OurapproachwillbetoapplyEuler–Bernoullitheoryto
findacompleteanalyticformulakcom
x forthishorizontalstiffness
anduseitslinearizedvalueklin
x asoursimpleapproximation.We
willkeepthesuperscriptsexplicitduringourentirecomputationin
ordertominimizeconfusionregardingthethreedifferentvaluesof
horizontalstiffness:exact,completeEuler–Bernoulliandlinearized
Euler–Bernoulli.
Sincethetechnicalevaluationoftotalhorizontalstiffnessis
car-riedoutforasinglebeam,wedenotethehorizontalstiffnessof
asinglebeamby x,itscompleteanalytic formula xcom andits
linearizedvalue lin
x .
Inthetechnicaldiscussionthatfollows,itisnecessaryto
dis-tinguishthreevertical(axial)loadsituations.WritingFy=Fyy,we
have
1.Noverticalload:Fy=0.
2.Verticaltensileload:Fy>0.
3.Verticalcompressionload:Fy<0.
Inordertodistinguishthesecasesinthecomputationsbelow,
itwillbeconvenienttoworkwiththemagnitudeFabs
y =|Fy|ofFy.
A.3. Reductiontosinglebeam
Thetechnicalpartoftheproofwillbeestablishedbyconsidering
asinglebeamonthestructureofFig.A.20.Thisreductionispossible
sincethehorizontalstiffnessesofthetwobeamsareidenticaland
thetotalhorizontalstiffnessofthestructureisexactlytwicethe
horizontalstiffnessofeachbeam.
Toshowthatthisis true,first notethat thehorizontalforce
isappliedtotherigidblockat verticalheightL/2,and thatthe
oftheloadcarriedbyonebeambecomesFabs y /2.
Itthenfollowsthatthetotalhorizontalstiffnessofthe1-DOF
parallelspringstageunderaverticalloadisrecoveredby
multiply-ingthesinglebeamhorizontalstiffnessbytwo,asclaimedatthe
beginningofthissection.
A.4. Settingupthedifferentialequation
Thegoverningdifferentialequationofthedeformationofan
Euler–Bernoullibeamunderloadanduptothefourthderivativeis
well-known[8,p.464,p.591]
EIx(4)−F2yx=0.
Forourthreeloadsituations,thisgives
1.Noverticalload:
EIx(4)=0.
2.Verticaltensileload:
EIx(4)−F
abs y
2 x
=0.
3.Verticalcompressionload:
EIx(4)+F
abs y
2 x=0.
A.5. Generalbeamdeflectionsolution
Thebeamdeflectionsolutionsforthethreeverticalload
situa-tionsareasfollows,whereω=
Fabsy /(2EI).
1.Noverticalload:
x0=a0+a1y+a2y2+a3y3. (A.1)
2.Verticaltensileload:
xt=b0+b1y+b2cosh(ωy)+b3sinh(ωy).
3.Verticalcompressionload:
xc=c0+c1y+c2cos(ωy)+c3sin(ωy).
A.6. Boundaryconditions
Thebeamextremitiesofourmodelsatisfythefollowing
bound-aryconditions.
x(0)=0, x(0)=0, x(L)=f, x(L)=0,
wherethelastboundaryconditionfollowsfromthefactthatthe
mobilemassdoesnotrotate,asshowninAppendixA.3.
Applyingtheseboundaryconditionsallowsustocomputethe
constantsinthegeneralsolutionsfoundintheprevioussection.
1.Noverticalload: a0=a1=0, a2= 3f L2, a3=− 2f L3. (A.2) c0=− C , c1=− C , c2= f (cos(Lω)− 1) C , c3= fsin(Lω) C ,
whereC=2 (1−cos(Lω)) −Lωsin(Lω).
A.7. Horizontalforceforgivenhorizontaldisplacement
Thedifferentialequationgoverningthehorizontalforceat
ver-ticalheightyis
Fx(y)=−EIx(3)(y)+F2yx(y).
WeareinterestedinthehorizontalforceFx=Fx(L)attheextremity
y=Lofthebeamsinceitisessentiallytherestoringforceappliedto
themobilerigidblock.Sincex(L)=0thisreducesto
Fx=−EIx(3)(L).
Theexplicitanalyticformulafoundintheprevioussectionscan
beappliedtofindthehorizontalforceFx atverticalheighty=L
whichproducesthehorizontaldisplacementf,andthisforthethree
verticalloadsituations.
1.Noverticalload:
Fx=12EI
L3 f.
2.Verticaltensileload:
Fx=EIω
3sinh(ωL)
B f.
3.Verticalcompressionload:
Fx=EIω
3sin(ωL)
C f.
A.8. Completehorizontalstiffnessofasinglebeam
ThecompleteEuler–Bernoullihorizontalstiffness com
x ofa
sin-glebeamisnowcomputedbysimplyapplyingHooke’sLaw
comx =Ffx,
totheformulasoftheprevioussectionforeachofthethreeload
casesandperformingsomeelementarytrigonometric
simplifica-tion.
1.Noverticalload:
comx =12EIL3 . (A.3)
2.Verticaltensileload:
comx = Fabs y /2 L− 2 ωtanh
Lω 2. (A.4)
3.Verticalcompressionload:
comx = Fabs y /2 2 ωtan
Lω 2−L. (A.5)
Weseefromformula(A.5)thatthereexistsaverticalload0
producingzerostiffnessexactlywhenωL= sothat0=
2EI L2 .
Wethen define =Fabs
y /(20)sothat ω= L√ . Thisallows
ustorewritethestiffnessasamultipleof 0,thestiffnessofone
beamwithoutaxialload.Notethatthesuperscript com
0 hasbeen
suppressed,sinceallvaluesofk0willbeidenticalinwhatfollows.
1.Noverticalload:
comx = 0. (A.6)
2.Verticaltensileload:
com x = 0 2 12
1− 2 √ tanh √ 2.
3.Verticalcompressionload:
com x = 0 2 12
2 √ tan √ 2−1
.
A.9. Completehorizontalstiffnessofthe1-DOFparallelspring
stage
AsshowninAppendixA.3,theparallelflexurestageiscomposed
oftwoidenticalbeamsthereforeitscompleteanalyticalhorizontal
stiffnessissimplykcom
x =2 xcom.
Itfollowsthattheverticalcompressionloadinducingzero
hor-izontalstiffnessissimplyN0=20sothat
N0=2
2EI
L2 .
Expressing in termsofN0 gives =Fyabs/(20)=Fyabs/N0.The
totalhorizontalstiffnessforthethreeloadcases,expressedasa
functionofthenoverticalloadcasek0,arefoundbysimply
multi-plyingtheexpressionsoftheprevioussectionbytwo.
1.Noverticalload:
kcomx =k0= 24EI
L3 .
2.Verticaltensileload:
kcom x =k0 2 12
1− 2 √ tanh √ 2. (A.7)
3.Verticalcompressionload:
kcom x =k0 2 12
2 √ tan √ 2−1
. (A.8)
A.10. Linearizedhorizontalstiffnessofthe1-DOFparallelspring
stage
Inthissection,weusetheTaylorseriesexpansionofthe
solu-tions found in the previous section to linearize the complete
analyticstiffnesskcom
x ofthe1-DOFparallelspringstage.
1.Noverticalload:
klinx =k0= 24EI
L3 .
2.Verticaltensileload:
TheTaylorexpansionoftanh (X) uptothirdordergives
tanh
√ 2= √2 − 324 √ + 5240 2√ sothat klinx = k 0 1− 2 10 .
ApplyingtheTaylorexpansionof1+X1 uptosecondordergives
klin x =k0
1+ 2 10 =k0+ 6 5LF abs y . (A.9)3.Verticalcompressionload:
TheTaylorexpansionoftan(X)uptothirdordergives:
tan
√ 2= √ 2 + 3 √ 24 + 5 2√ 240 sothat klin x = k0 1+ 2 10 .
ApplyingtheTaylorexpansionfor 1
1+Xuptosecondordergives
klinx =k0
1− 102 =k0− 6 5LF abs y . (A.10)Finally,onenotesthatinEq.(A.9)onehasFy=Fyabsandthatin
Eq.(A.10)onehasFy=−Fyabs,soapplyingthisandsubstitutingthe
explicitformulafork0foundincase1givesthemainresultofthis
Appendix klin x =24EI L3 + 6Fy 5L, aspromised.
A.11. Errorestimate
Inthissectionweinvestigatethediscrepancybetweenthe
com-pleteanalyticalstiffnesskcom
x andthelinearizedanalyticalstiffness
klin
x asafunctionof =Fyabs/N0.
LetusdefineZcom( )=kcom
x /k0.NotethatZcom( )expressesthe
relativediscrepancyfromthenominalstiffnessk0.Wehave
1.Noverticalload:
Zcom(0)=1.
2.Verticaltensileload:
Zcom( )= 2 12
1− 2 √ tanh √ 2.
3.Verticalcompressionload:
Zcom( )= 2 12
2 √ tan √ 2−1
.
WesimilarlydefineZlin( )=klin
x /k0,whichagainexpressesthe
relativediscrepancyfromthenominalstiffnessk0.
1.Noverticalload:
Zlin(0)=1.
2.Verticaltensileload:
Zlin( )=1+ 2
10 .
3.Verticalcompressionload:
Fig.A.21.Normalizedmodelscomparison.
InFig.A.21thetwonormalizedstiffnessmodelsZcom( )and
Zlin( )areplottedonthesamegraph.Notethat,bydefinition, ≥0.
Wedefinetherelativeerrorε betweenthecompleteand
lin-earizedmodeltobe ε =Z lin( )−Zcom( ) Zlin( ) = klin x −kcomx klin x ,
sothattherelativeerrorε expressestherelativeerrorinZ( )as
wellasinkx.
For thefirstprototype’s beamdimensions(see Table1),the
loadwhichproduces azerostiffnessoftheparallelspringstage
isN0=21.05N.Themaximalappliedverticalloadweconsiderfor
thismechanismisFabs
y =5N,correspondingto =0.2375,which
givesamaximalrelativeerrorε =877ppm.
Forthesecondprototype’sbeamdimensions,seeTable1,the
loadwhichproducesazerostiffnessoftheparallelspringstageis
N0=71.06N.Themaximalappliedverticalloadweconsiderforthis
mechanismisFabs
y =12N,correspondingto =0.1689,whichgives
amaximalrelativeerrorε =404ppm.
Conclusion.The1-DOFparallelspringstagelinearized
analyt-icalhorizontalstiffness modelusedinthis paperisaverygood
approximationtothecompleteanalyticalmodel.
A.12. Parabolictrajectorywithnoverticalload
AsshowninAppendixA.3,themobilerigidblockmoves
trans-lationallysoitspositionisuniquelydescribedbythepositionQ=(x,
y)ofthetranslatedpointO=(0,0).Wenowshowthatthe
trajec-toryiswellapproximatedbyaparabola.Inordertodothis,we
Thisformulaisestablishedusingatruncatedpowerseries
devel-opment,soisvalidonlyforsmallbeamdeflectionangles.
Takingthederivativeofformula(A.1)withrespecttoyandusing
theconstantvaluesofformula(A.2)weget
x(y)=6f L2
y−y2 L , (A.12)Using(A.11)and(A.12),weobtaintheverticaldisplacementasa
functionofhorizontaltranslationf
3f2
5L,
whichistheequationofaparabola.
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