NONLINEARITIESANDNOISEINMICROMECHANICALRESONATORS:FROMUNDERSTANDINGTOCHARACTERIZATIONANDDESIGNTOOLSByPavelM.PoluninADISSERTATIONSubmittedtoMichiganStateUniversityinpartialfulÞllmentoftherequirementsforthedegreeofMechanicalEngineering-DoctorofPhilosophyPhysics-DoctorofPhilosophy2016ABSTRACTNONLINEARITIESANDNOISEINMICROMECHANICALRESONATORS:FROMUNDERSTANDINGTOCHARACTERIZATIONANDDESIGNTOOLSByPavelM.PoluninInthisworkweconsiderseveralnonlinearity-basedand/ornoise-relatedphenomenathatwereobservedrecentlyinmicro-electromechanicalresonators.Themaingoalhereistocloselyexaminethesephenomena,understandtheirunderlyingfundamentalphysics,andÞndwaystoemploythemformeasurementpurposesand/ortoimprovetheperformanceofspeciÞcclassesofmicro-electromechanicalsystems(MEMS).Thegeneralperspectiveofthisworkisbasedontheacceptanceofthefactthatnonlinearityandnoiserepresentintegralpartsofthemodelsforthesesystems,andthediscussionisconstructedaboutthecaseswherethesegenerallyÒundesirableÓfeaturescanbeutilizedratherthanavoided.Inthisdissertationweconsiderthreedi!erent,butrelated,topics.Tostart,weanalyzethestationaryprobabilitydistributionofanonlinearresonatordrivenbyPoissonnoiseatanon-zerotemperatureoftheenvironment.WeshowthatPoissonpulseswithlowpulseratescausethepower-lawdivergenceoftheprobabilitydensityattheresonatorequilibriumintherotatingframebothinoverdampedandunderdampedregimes.Wehavealsofoundthattheshapeoftheprobabilitydistributionawayfromtheequilibriumpositionisqualitativelydi!erentforoverdampedandunderdampedcases.Inparticular,intheoverdampedregime,theformofthesecondarysingularityintheprobabilitydistributionstronglydependsonthereferencephaseoftheresonatorresponseaswellasthepulsemodulationphase,whileintheunderdampedregimethereareseveralsingularpeaks,andtheirlocationisdeterminedbytheresonatordecayrateintherotatingframe.Finally,weshowthatevenweakGaussiannoisea!ectstheprobabilitydistributionbysmoothingitinthevicinityofsingularpeaks.Second,wediscussatime-domaintechniqueforcharacterizingparametersformodelsofasinglevibrationalmodeofsymmetricmicromechanicalresonators,includingcoe"cientsofconservativeanddissipativenonlinearitiesaswellasthestrengthsofnoisesourcesactingonthemodeofinterest.Thesenonlinearitiesresultinanamplitude-dependentfrequencyandanon-exponentialdecay,whilenoisesourcescauseßuctuationsintheresonatorampli-tudeandphase.Wecapturethesefeaturesinthemodalringdownresponse.Analysisoftheamplitudeoftheringdownresponseallowsonetoestimatethequalityfactorandthedissipa-tivenonlinearity,andthezero-crossingpointsintheringdownmeasurementcanbeusedforcharacterizationofthelinearnaturalfrequencyandtheDu"ngandquinticnonlinearitiesofthevibrationalmodewhicharisefromacombinationofmechanicalandelectrostatice!ects.Additionally,weshowthatstatisticalanalysisofthezero-crossingpointsintheresonatorresponseallowsustoseparatee!ectsofadditive,multiplicative,andmeasurementnoisesandestimatetheircorrespondingintensities.Finally,weexaminetheproblemofself-inducedparametricampliÞcationinring/diskresonatinggyroscopes.Wemodelthedynamicsofthistypegyroscopesbyconsideringßex-ural(elliptical)vibrationsofafreethinringandshowthattheparametricampliÞcationarisesnaturallyfromanonlinearintermodalcouplingbetweenthedriveandsensemodesofthegyroscope.Analysisshowsthatthismodalcouplingresultsinsubstantialincreaseinthesensitivityofthegyroscopetotheexternalangularrate.Thisimprovementinthegyroscopeperformancestronglydependsbothonthemodalcouplingstrengthandtheoperatingpointofthegyroscope,andwefurtherexplorewaystoenhancethise!ectbychangingtheshapeoftheresonatorbodyandattendantelectrodesandutilizingelectrostatictuning.ACKNOWLEDGMENTSLookingbackatmythreeandahalfyearsofworkingonmyPh.D.dissertationatMichiganStateUniversity,therearemanypeoplewhoprovidedtheirsupport,guidance,motivation,encouragement,andfeedbackandtowhomIamsincerelygratefulforthis.Firstofall,Iwouldliketoexpressmysincerestgratitudetomyacademicadvisers:ProfessorStevenW.ShawandProfessorMarkI.Dykman,fortheirpatience,continuoussupport,andencouragementduringmyPh.D.program.Besidesprovidingtheiracademicinstructionandtechnicalguidance,bothSteveandMarkhavebeenmygreatmentorsinmanyaspectsofmylife(includingsurÞngFloridawavesorÞndingperfectsweetsinEastLansingarea),andtheirmentorshipwillkeepguidingmethroughmyprofessionalcareerandpersonallife.IamtrulyhonoredtohavehadanopportunitytoworkwithProfessorShawandDykmanandlearnfromtheseexperts.IamalsoverygratefultomyGuidanceCommitteeMembers:ProfessorBrianF.Feeny,ProfessorVladimirG.Zelevinsky,andProfessorRanjanMukherjee,fortheirinvaluablefeedbackduringmygraduateprogramandoverseeingthisdissertation.Ihavebeenextremelyfortunatetotakeafewclasseswitheachofthesegentlemen,fromwhomIlearnedmanyinterestingtheoriesandmethodsthathelpedmeinmyworkonthisdissertation.Next,Iwouldliketothankmycollaborators:ProfessorThomasW.Kenny(Stanford),ProfessorHoBunChan(HKUST),andProfessorKimberlyL.Turner(UCSB).Ihavedeeplyenjoyedworkingwiththeseoutstandingscholarsandtheirstudents.SpecialthanksheregoestoYushiYang(Stanford),LilyL.Li(UCSB),andPanpanZhou(HKUST)fornumerousin-depthdiscussions,bothonscienceandlifeingeneral,whichhavebroadenmyhorizonsindi!erentdirections.ivSpecialthanksgoestomyformerandfellowlab-matesforaveryfriendlyandwarmatmospherethatwehavekeptinourlabandthathastriggeredcountlesscoversationsaboutpeculiarscientiÞcproblemsandlifechallenges.TheseincludeNickMiller,ScottStrachan,OrielShoshani,VenkatRamakrishnan,BrendanVidmar,MustafaAcar,GizemAcar,SmrutiPanigrahi,XingXing,KirillMoskovtsev,YaxingZhang,JuanAtalaya,andAyseSapmaz.Ahugethanksgoestomywonderfulfriends:AndreyMaslennikov,OleksiiKarpenko,OlegTarasov,AntonEfremov,BrianZhou,andJenniferKirk,forhelpingmetomaintainahealthywork-and-lifebalance.Ifeelveryluckytohaveyouinmylife.TomyfamilybackinRussia,therearenowordsthatcanexpressmyloveandgratitudetowardsmyfamilymembers.Thankyou,MomandDad,foryourendlesssupport,uncondi-tionallove,andyourencouragementduringdi"cultperiodsofmylife.Lastbutnotleast,thanksgoestomydearandlovelywifeRoshanAngoshtariforhercontinuouslove,support,patience,andunderstandingduringmyon-goinglifeadventure.vTABLEOFCONTENTSLISTOFTABLES.......................................viiiLISTOFFIGURES......................................ixChapter1Introduction...................................11.1WhynonlinearandnoisymodelsforMEMS?....................21.2NonlinearitiesandnoiseinMEMS..........................41.3Summaryofthework.................................10Chapter2SingularprobabilitydistributionofvibrationalsystemsdrivenbyPois-sonandthermalnoises.............................132.1Experimentalsetupandresonatordynamicalmodel.................172.1.1Micromechanicaltorsionalresonator.....................172.1.2Resonatormodelintherotatingframe....................202.2Qualitativepictureofresonatorßuctuations.....................232.3Resonatorstationaryprobabilitydistributionintherotatingframe..........262.3.1Overdampedregime.............................262.3.2Underdampedregime.............................342.3.3Effectsofthermalnoise............................382.4Outlook........................................40Chapter3CharacterizingnonlinearitiesandnoiseinMEMS:ringdown-basedap-proach......................................423.1Deviceunderstudyandmeasurementsetup.....................453.2Model.........................................483.3Characterizationof!0,!1,!2,"and#parameters..................533.3.1Revealingnonlinearfrictionandextracting!1,!2..............553.3.2ExtractingtheresonatorstiffnesscoefÞcients!0,",#............583.3.3SeparatingmechanicalandelectrostaticeffectsinMEMSresonators....603.4Characterizationofthermal,frequencyandmeasurementnoisesources.......663.4.1Measurementnoisetimingjitter$Tµ(i,j,k).................683.4.2Resonatorphaseßuctuationsduringringdown................723.4.3Phasenoisetimingjitter$Tr(i,j,k).....................773.4.4Noisecharacterizationprotocol........................793.5Outlook........................................83Chapter4ImprovingthesensitivityofMEMSring/diskresonatinggyroscopes...854.1Generaldynamicalmodel...............................894.2Non-linearforcedvibrationsofathinspinningring.................924.2.1Gyroscopedynamicswithfully-coupledmodes...............924.2.2DynamicsoftheDriveMode.........................984.2.3DynamicsoftheSenseMode:ParametricAmpliÞcation..........101vi4.2.4Example....................................1044.3Manipulatinggyroscopenonlinearities:geometricandelectrostaticoptimizationmethods........................................1084.3.1Nonlinearelectrostatictuningbyanon-uniformbiasvoltage........1104.3.2Effectofanon-uniformelectrodegapsize..................1134.3.3Shapeoptimizationofthegyroscopebody..................1154.4Outlook........................................118Chapter5Conclusions...................................122APPENDICES.........................................128AppendixAPoissonnoiseintheresonatoreigenfrequency...............129AppendixBMechanicalandelectrostaticeffectsinclamped-clampedbeamresonators131AppendixCDerivationofnonlinearstrain-displacementrelationshipsinafreering..136BIBLIOGRAPHY.......................................139viiLISTOFTABLESTable3.1:Estimatedvaluesofthelinearandnonlineardissipationcoe"cientsandconservativenonlinearityfordi!erentinitialamplitudes.Ring-downmeasurementshavebeenperformedwithVDC=30VandatT=!40"C................................56Table3.2:Estimatedvaluesofthemechanicalnaturalfrequency!0m,theme-chanicalcontributiontotheDu"ngnonlinearity"m,andtheelectro-staticpotentialstrengthCeintheDA-DETFresonator(seeFig.3.1)forthreeindependentringdownmeasurementswiththesamebiasvoltage,VDC=30V...........................65Table3.3:Extractedintensitiesofthemeasurement,frequency,andthermalnoisesources.ThermalnoiseintensityDfisexpressedintermsoftheroot-mean-squareoftheresonatorvibrationamplitudeinthermalequilibrium:Df=4#1!20##a2$th.Noisecharacterizationwasper-formedinÞveindependentringdownmeasurementsatVDC=30VandT=!40"Cwithinitialamplitudea0=265mV.........83viiiLISTOFFIGURESFigure1.1:Schematicrepresentationoftheßexuralvibrationofaclamped-clampedbeam,showninadeformedconÞguration.Theelementdepictedinthebeam,andshowntop-viewintheinset,isusedtoderivethenon-lineare!ectsofmid-linestretching[46]...............5Figure1.2:Frequencyresponseshowingthecombinede!ectsofmechanicalandelectrostaticnonlinearities.Atsmallvibrationamplitudestheme-chanicalcomponentofthenonlinearrestoringforcedominatestheelectrostatice!ects,leadingtohardeningbehavior.Asthevibra-tionamplitudeincreases,theelectrostaticforcesstartstoplayamoredominantrole,resultingintheappearanceoftheso-calledzero-dispersionpointontheresonatorfrequencyresponse,beyondwhichtheresonatorbehaviorissoftening...................7Figure1.3:Analyticalmethods,e!ectivecharacterizationtechniques,andreli-abledesigntoolsarethekeyingredientsinbuildingMEMSresonatorswithoptimalperformancefordi!erentapplications..........10Figure2.1:Micromechanicaltorsionalresonatorunderstudyandexperimentalsetup....................................18Figure2.2:SchematicoftheresonatorresponsetoasinglePoissonpulseexci-tationintherotatingframeinoverdamped(a)andunderdamped(b)regimes.Forpanel(a)wehaveassumed,forsimplicity,thatthesystemisinstronglyoverdampedregime,i.e.|$1|%|$2|,withthecorrespondingeigenvectorscoincidingwiththeXandYdirections.24Figure2.3:Poissonnoise-inducedresonatordynamicsintherotatingframeintheoverdampedregime..........................27Figure2.4:Measuredprobabilitydistribution%(qm)intherotatingframeduetoPoissonpulsesalongtheout-of-phasequadraturewith&=0.94HzandthermalnoiseatT=300Kintheoverdampedregime.IncontrasttothedatashowninFig.2.2a,theresonatoreigendirectionsintheexperimentare(1,±('!$))in(X,Y)spaceandarenotalignedwitheitherthein-phaseorout-of-phaseresonatorcomponents.ThelatterfactallowsustoobservesingularorßatpeakstructuresoftheresonatorprobabilitydistributionataÞnitedistancefromtheoriginbychangingtheobservationdirection..................33ixFigure2.5:Theresonatordynamicsandtheassociatedprobabilitydistributionintherotatingframeintheunderdampedregime...........35Figure3.1:Top:COMSOLmodelofamicromechanicalDA-DETFresonatorshowingthesymmetricvibrationalmodeunderstudy.Theexpected(usingFEManalysis)valuesoftheresonatorlinearparametersintheexperimentareasfollows:e!ectivemodemassmeff&0.2µg,qualityfactorQ&103!104,andnaturalfrequencyf0'1.2MHz.Thesquaredenotesthelocationofthecross-sectionalSEM.Bottom:SEMfroma45"-viewangleoftheresonatorencapsulatedwiththeepi-sealprocess..............................45Figure3.2:Variable-phaseclosed-loopfeedbacksystemwithaddedcapabilityforringdownmeasurements.TheencapsulateddevicesareplacedintoaThermotronS-1.2cenvironmentalchamberfortemperaturestabiliza-tionat!40"C..............................46Figure3.3:Measuredamplitude-frequencyresponsesofthemicromechanicalres-onatorinthefeedbackloopwithVDC=30Vanddi!erentvaluesofVACatT=!40"C.Eachcircledenotesthepositionofthenonlinearresonancewherethesystemhasbeenpreparedforthesubsequentringdownmeasurement..........................47Figure3.4:Ameasuredringdownresponseoftheresonatorunderstudy;VDC=30VandVAC=250mV.Redsolidlineindicatesextractedvibra-tionalenvelopea(t).Insetshowsatime-expandedviewoftheinitialportionofthesignal...........................48Figure3.5:Theroad-mapfortheringdown-basedcharacterizationmethod.Sep-arationofdissipativeandconservativeparametersofthesystemun-derstudyisachievedthroughindependentanalysesofthevibrationenvelopeandtheformoftheresonatorÒbackbone.ÓFurthermore,small-andlarge-amplitudecomponentsofthesystemringdownsig-nalareusedfore!ectivecharacterizationoftheresonatorlinearandnonlinearcoe"cients,whilethejitterintheringdownzero-crossingpointsprovidesinformationaboutnoisesourcespresentinthesystem.54Figure3.6:Post-processingmethodusedforextractingtheresonatorvibrationalenvelopefromtheringdownmeasurement.Initialheterodyningoftheresonatorringdownresponsewithin-phaseandquadraturesignalsat!ssandsubsequentisolationofthesystemslowly-varyingquadraturesallowsonetoreconstructtheresonatorvibrationalamplitudea(t)..56xFigure3.7:MeasuredvibrationalamplitudeoftheDETFresonatorduringitsringdownresponsewithVDC=30V(solidline).Thedashedlinerepresentstheexponentialdecayoftheresonatoramplitudeatlowvibrationamplitudes,extendedthroughouttheamplituderange.Up-perinset:nonlinearfrictioncausestheringdownamplitudeenvelopetodeviatefromexponentialatlargeamplitudes,whichcanbeusedforcharacterizationof#2.Lowerinset:thee!ectofnonlineardis-sipationontheringdownresponsebecomesstrongerastheinitialamplitudeincreases............................57Figure3.8:Vibrationfrequencyoftheresonatorduringtheringdownasafunc-tionofitsamplitudefordi!erentvaluesofinitialamplitude.Duetoamplitude-dependentfrequencypulling,thefrequencyvarieswithamplitude,allowingcharacterizationof!0,"and(fromasinglemeasurement.Discretedotsrepresentextractedvaluesofthevi-brationfrequency!k/2)duringtheringdownresponse(errorbars'1!10Hz,notshown).ThesolidlinesrepresentthecurveÞtsofextractedÒbackbonesÓusingthemodelinEq.(3.12).........59Figure3.9:Schematicrepresentationofthee!ectsofthermal(f(t))andfre-quency(*(t))noisesontheresonatordynamics,andthee!ectofmeasurementnoise(µ(t))onthequalityofthereadoutsignal....66Figure3.10:Measurementnoisecontributiontothejitterinthezero-crossingpointsintheresonatorringdownresponse.Asexpected,theres-onatormotionwithlargeramplitudeislesssusceptibletothee!ectsofmeasurementnoise,whichisthefundamentalnecessityforhighsignal-to-noiseratioforprecisefrequencygeneratorsandclocks....70Figure3.11:TimeintervalsT(1,j,4)calculatedwithinthe1stringdownsegment.Duetothepresenceofmeasurement,thermal,andfrequencynoises,thesetimeintervalsslightlydi!erinlength,which,whenanalyzedstatistically,allowsonetorevealthesenoisesourcesandestimatetheirintensities..............................80Figure3.12:Qualitativebehaviorof##T2(i,j,k)$jasafunctionofkwithintheithringdownsegment.Importantly,themeasurementnoisecontri-butiontothetimingjitterdoesnotvanishask(0,whichisusedforestimatingthemeasurementnoiseintensityM.Incontrast,thecorrespondingcontributionsofthermalandfrequencynoisesourcestothetimingjitterarebothk!dependent,buthavedi!erentmagni-tudesindi!erentringdownregimesduetotheirdi!eringamplitudedependence................................81xiFigure3.13:Characterizationofmeasurement,thermalandfrequencynoisesources.82Figure4.1:Schematicrepresentationofthesystemunderstudy:auniformcir-cularringrotatingataconstantangularrate$aboutthez!axiswithsegmentedelectrodesrepresentingthemeansforelectrostaticactuationandreadout.Segmentationofelectrodesisanessentialfeatureofthedevice,necessaryforproducingspatially-dependentdrivingforcesthroughVAC(+,t),andfortuningthegyroscopicdriveandsensemodesviaanon-uniformdistributionofthebiasvoltagesVDC(+)[27]................................93Figure4.2:Degenerateellipticalmodesoftheuniformcircularringunderstudy.95Figure4.3:Representativesteady-statefrequencyresponsecurvesoftheringdrivemodedescribedbyEq.(4.14)fordi!erentvaluesoftheforcingamplitudeF.Theblue,red,andblackcurvescorrespondtoforcingmagnitudesF0,2F0,4F0.Responsesareobtainedundertheassump-tionthatelectrostaticforcesdominatetheDu"ngnonlinearity,i.e.,|"e|)|"m|.Solidanddashedcurvesrepresentstableandunstableresponseamplitudes...........................100Figure4.4:E!ectofself-inducedparametricampliÞcationonfrequencyresponsesoftheringsensemodedescribedbyEq.(4.16)fordi!erentvaluesofthedispersivemodalcouplingcoe"cientCd,whereweconsiderthecaseCd/!2B)C1,C2,C!,sothattheparametricpumpingcoe"-cient$isessentiallyproportionaltoCd.FrequencyresponsesareobtainedforaR/%=0.2and$/!B=2*10!4.Thedashedcurveisthenon-ampliÞedresponse(Cd=0),whiletheredandbluecurvescorrespondtothesensemodefrequencyresponseswithCd/!2B=!0.5*104,!1.1*104respectively;thesenumbersarecho-sensuchthattheanalyticalresultsofSection4.2.3remainvalid,thatis,sothatthestatedapproximationshold.SignalampliÞcationfromtheintermodalcouplingisevident....................104Figure4.5:Increaseofthesensitivityofthegyroscopering(withoutsuspension)duetononlinearmodalcouplingasafunctionofsystemanddriveparameters.Thesolidredlinerepresentsthea+!!+curvewherethegainGdivergesaccordingtothelinearmodeldescribedbyEq.(4.15);thisistheprimaryArnoldtongueforthesensemode[141].ThemeshedregionisthesetofoperatingconditionswherethesolutionfoundinEq.(4.16)isunstable......................106xiiFigure4.6:DependenceoftheÞgureofmeritinEq.(4.22)onthevalueoftheobjectivefunctionK.QualityfactorsofthegyroscopedriveandsensemodesareassumedtobeQA=QB=1200[19]............110Figure4.7:Behavioroftheelectrostaticcomponentsofthedispersivecouplingstrength(blackdashedcurve),theDu"ngnonlinearity(blacksolidcurve),andtheobjectivefunction(bluesolidcurve)asafunctionofthevariationinthebiasvoltagerDC..................112Figure4.8:Manipulatinggyroscopenonlinearitiesviaanon-uniformelectrodegapsize..................................115Figure4.9:Manipulatingnonlinearparametersofathinspinningringbymodi-fyingtheradialringthicknessh(+)...................117FigureC.1:DeformationofthegyroscopesegmentKLMNintoK1L1M1N1..137xiiiChapter1IntroductionGenerally,micro-electro-mechanicalsystems(MEMS)thatcanbedescribedbylinearmod-elsaremorewidelydesiredforapplications,whilesystemsexhibitingnonlinearbehaviorareavoided.Thereareanumberofreasonsforthis,includingthefactthatthedynamicbehavioroflinearsystemsiswellunderstood,thatthesesystemsaresigniÞcantlysimplerfromcharacterizationanddesignpointsofview,andthattheyallowforrelativelysimpletuningintermsofsystemmodelparameters.Atthesametime,MEMSoperatingintheirlinearregimeshavecertainfundamentallimitationsintermsoftheirperformance,duetotherestrictednatureoftheirinput-outputpropertiesandthefactthatthelinearrangeisoftenrestrictivelysmallinmanyapplications.Furthermore,linearsystemsarenotsuitableforapplicationsthatnecessarilyinvolvenonlinearphenomena,likethelimitcyclesrequiredforfrequencygenerators[1,2],certaintypesofatomicforcemicroscopy[3Ð5],andthesub-andsuper-harmonicsneededforfrequencyconversion[6,7]insignalprocessingapplications.NoisesourcesaregenerallytreatedasunwantedinMEMSsincetheynegativelya!ectdeviceperformancebyreducingtheirprecision.Whilethisisgenerallytrue,systematicapproachesforreducingthee!ectsofnoiseonthesystemdynamicscannotbefullyrealizedwithoutadetailedunderstandingofthenoisepropertiesand,preferably,knowledgeoftheirori-gins[8Ð10].Asaresult,furtherimprovementintheperformanceoftheseMEMSrequiresacomprehensiveunderstandingofvariousnonlinearandnoise-inducedphenomena.InthischapterwedescribethescientiÞcbackgroundthatmotivatedpresentresearch1andoutlinethemaincontributionsdescribedinthisdissertation.InSection1.1webrießyintroducereaderstotheÞeldofmicro-electromechanicalsystemsanddiscusstheimportanceofstudyingnonlinearbehaviorinMEMS.InSection1.2weelicitthephysicaloriginofthenonlinearitiesandnoisesourcesinMEMSmodels,aswellastheire!ectonthesystemdynamicbehavior,anddescriberecentÞndingsthatmotivatedcurrenttheseresearche!orts.InSection1.3wesummarizeourmajorcontributionstothisÞeld.1.1WhynonlinearandnoisymodelsforMEMS?Designforlineardynamicrangeisanonlinearproblem.ÐStevenW.ShawMicro-electromechanicalsystemsisawellestablishedÞeldofsciencethatwasinitiatedbythepioneeringconceptsandideasofFeynman[11],Nathanson[12]andPetersen[13].HerewefocusonMEMSresonators,thatis,vibratorydeviceswithlightdamping,whichhaveachievedsuccessinnumerousapplications,includingstablefrequencygeneration[14Ð17],precisesensing[18,19],andsignalprocessing[20,21],tonamebutafew.Foradetailedoverviewofdi!erenttypesofMEMSoperatingintheirlinearregimewereferthereaderto[22],whileherewefocusonnonlinearbehaviorofMEMSresonatorsandhighlightfeaturesthatmakethisanimportantandinterestingÞeldforresearch.Intime-keepingandsensingapplications,quartzcrystalresonatorsprovidehighlystableandprecisefrequency-selectiveelements,buttheirrelativelylargesize,ascomparedwithMEMS,makesitchallengingtointegratethemwithintegratedcircuittechnologies.Thisisacrucialshortcominginmodernapplications,suchashealthmonitoringandsmartphoneapplications,whicharedrivenbycontinuosminiaturization.MEMSprovideapromising2alternativetoquartzresonatorsduetotheirsmalldimensions,lowpowerconsumption,andexcellenton-chipintegrability[23Ð25].Additionally,theirsmalle!ectivemassmakesMEMSidealcandidatesforanumberofsensingapplications,forexampletheprecisemeasurementofacceleration[26],angularrate[27],mass[18],force[28]andelectronspin[29].Ontheotherhand,theseveryfeaturesofMEMSresonatorsmakethemextremelysensitivetovariousnoisesources,suchasthosearisingfromthermalandelectronicßuctuations,whichdegradetheperformanceofMEMSdevices.SpeciÞcally,thestrongsusceptibilityofMEMSresonatorstonoisesourcesresultsinrelativelylowsignal-to-noiseratios(SNR)whenoperatingintheirlineardynamicrange.ThestandardwaytoincreaseSNRandimprove,forexample,sensormeasurementpreci-sion,istodriveMEMSresonatorstolargervibrationamplitudes.However,astheresonatorvibrationenergyincreases,variousnonlineare!ectscomeintoplay,thusmakinglinearsys-temmodelsnolongervalid.Inthissituation,wehavetwogeneraloptionstochoose:oneoptionistooperatethedeviceattheonsetofnonlinearity;inthiscasethesystemresponseessentiallyremainslinear,butwelimitourselvesbyrelativelylowvibrationamplitude,re-sultinginlowSNRandlimiteddeviceperformanceandprecision.Anotheroptionistodrivetheresonatorinitsnonlinearregime,whichallowsonetoimprovetheSNR,butatthecostoffacinganddealingwithnonlinearity-inducedphenomena,suchasamplitude-dependentvibrationfrequency[30,31]andnonlinearfriction[32Ð35].Asnotedabove,nonlinearsystemmodelsaresigniÞcantlymorecomplicatedtoworkwithwhencomparedtolinearmodelsand,asaresult,themajorityofcontemporaryMEMS-basedsystemsoperateintheirlinearrange[36Ð38].Recently,severalresearchgroupshaveinvesti-gateddi!erentapproachestoincreasethelinearrangeofexistingMEMSresonators[39Ð42].Inthissituation,aspracticeshows,inordertodesignaresonatorwithincreasedlinear3range[43]orreducedphasenoise[1],itisÞrstnecessarytoobtainacomprehensiveunder-standingofthenonlinearitiesand/ornoisesourcesoneistryingtominimizeoreliminate.Inthisthesiswetakeadi!erentapproach;ratherthanavoidingand/orneglectingnonlinearitiesandnoise,wefocuson:(i)methodsforcharacterizingnonlinearityandnoise,(ii)achievingabetterunderstandingoftheire!ectsonsystemdynamicbehavior,and(iii)utilizingthisknowledgetodesignsystemswithimprovedperformance.1.2NonlinearitiesandnoiseinMEMSAnyvibrationalsystemwillexhibitnonlinearbehaviorundersomeoperatingconditions.LinearmodelsserveasÞrst-orderapproximationstothesystemdescription,andincertaincasestheaccuracyprovidedbytheseapproximationsissu"cientforanaccuratedescrip-tionofobservedphenomena,whileinothersituationsonemustnecessarilygobeyondthisassumptionandaccountforhigher-order(nonlinear)e!ects.InMEMSresonatorsnonlinearitiescanbeseparatedintwomajorgroups.TheÞrstgroupofnonlineare!ectsarisefrommechanicalpropertiesandareassociatedwiththeresonatorbodyitself.Theseincludenonlinearitiesduetoresonatorvibrationsatlargeamplitudes,andcanbefurtherclassiÞedintomaterialandkinematice!ects.MaterialnonlinearitiesariseatlargevibrationamplitudesduetothefactthatmaterialpropertiessuchasYoungÕsmodulusactuallydependonthevibrationamplitude[31]andtheorientationofthestrainÞeldwithrespecttocrystallographicprincipalaxes,whichisimportantin,forexample,resonatorsmadeofsinglecrystalsilicon[38]orotheranisotropicmaterials.Inaddition,ithasbeenfoundrecentlythatdopinglevelcana!ectresonatorresponseatlargevibrationamplitudesinanon-trivialway[44].Generally,materialnonlinearitiesa!ecttheresonatorpotential4xuuu+dudxdxA!Figure1.1:Schematicrepresentationoftheßexuralvibrationofaclamped-clampedbeam,showninadeformedconÞguration.Theelementdepictedinthebeam,andshowntop-viewintheinset,isusedtoderivethenon-lineare!ectsofmid-linestretching[46].energywhileleavingthekineticenergyunchanged.Ontheotherhand,kinematicnonlin-earitiesresultfromÞnitedeformationsoftheresonatorbodyduringitsvibration.Inthiscase,theparticularwaynonlineare!ectsrevealthemselvesdependsofthevibrationalmodeshapeandcanresultinnonlinearsti!nessand/ornonlinearinertialterms.Inparticular,forßexuralvibrationsofaclamped-clampedbeamresonator,whichwillbeconsideredinChapter3,themid-linestretchingoftheresonatoratlargevibrationalamplitudesresultsinanonlinearmodalrestoringforce[45];seeFig.1.1.Thismid-linestretchingleadstoanamplitude-dependentvibrationfrequencyandhastobetakenintoaccountatrelativelysmallvibrationamplitudes,speciÞcally,atamplitudesontheorderofthebeamthickness.Whenitcomestoßexuralin-planevibrationsofrings,forexample,fortheellipticvibra-tionalmodesinmicromechanicalgyroscopes,kinematicnonlinearitiesappearnotonlyinthesystempotentialenergy,butalsoalterthekineticenergyoftheresonatorbody,leadingtoinertialnonlinearities,asdiscussedindetailinChapter4.Kinematicnonlinearitiesalsoa!ecttheresonatorkineticenergyincantileverbeamresonators,whicharecommonlyused5insensingapplications[47,48].Asecondmajorgroupofnonlineare!ectsisassociatedwiththeinteractionoftheres-onatorbodywiththeenvironment,forexample,usedforactuationoftheresonatorandfortransductionofsignalsfromtheresonatorresponse.Inthisworkwefocusouratten-tiononelectrostaticallydrivenMEMSresonatorswithcapacitiveactuationandreadoutschemes[22].Theelectrostaticpotentialisahighlynonlinearfunctionoftheresonatordisplacementanddependsinverselyonthedistancebetweentheresonatorbodyandtheattendantelectrodes.Whenthevibrationamplitudeaissmallascomparedwiththeelec-trodegapsized,a%d,thecorrespondingelectrostaticforcescanbeapproximatedaslinearfunctionsoftheresonatormodalamplitude.Ontheotherhand,whena&d,evena&0.1d,thelinearapproximationoftheelectrostaticforcesisnolongervalidandonemustkeephigher-ordertermsintheelectrostaticforcemodel.NotethattheaforementionednonlinearitiesinMEMSresonatorshavedi!erentphysicaloriginsand,asaconsequence,generallycontributetotheresonatordynamicsatdi!erentvibrationamplitudes.Therefore,itcanhappenthatmorethanonesourceofnonlinearityaf-fectstheresonatordynamicbehaviorforagivenvibrationenergy.Inthiscase,nonlinearitiesoriginatingfromdi!erentsources,e.g.mid-linestretchingandtheelectrostaticpotential,es-sentiallyrenormalizeeachotherinthecorrespondingnonlineartermsintheresonatormodalrestoringforce.Thisinterplayofdi!erentnonlineare!ectscanresultinveryinterestingandnon-trivialresonatorresponses,suchasthatshownonFig.1.2.Damping,orfriction,alsooriginatesfromtheinteractionoftheresonatorwithitsenvi-ronment,whichnecessarilyhastemperatureT.Thisinteractioniscapturedbythecouplingoftheresonatortoathermalbath,andthee!ectsweareabouttodescribe,whichincludedampingandnoise,haveathermalorigin.Inausefulframework,theresonatorismod-6Figure1.2:Frequencyresponseshowingthecombinede!ectsofmechanicalandelectrostaticnonlinearities.Atsmallvibrationamplitudesthemechanicalcomponentofthenonlinearrestoringforcedominatestheelectrostatice!ects,leadingtohardeningbehavior.Asthevibrationamplitudeincreases,theelectrostaticforcesstartstoplayamoredominantrole,resultingintheappearanceoftheso-calledzero-dispersionpointontheresonatorfrequencyresponse,beyondwhichtheresonatorbehaviorissoftening.eledasinteractingwithasetofmicroscopicÒoscillatorsÓthatdescribethemodesoftheenvironmentaccordingtothefollowingmodel,¬q+U,(q)=!,Ui(q,{qk}),q,(1.1a)¬qk+!2kqk=!,Ui(q,{qk}),qk,(1.1b)whereU(q)istheresonatorpotential,whichcanbeharmonicoranharmonic,qkdenotesthecoordinateassociatedwithamodeofthebath(k=1,2,...,N,whereN)1),andUi(q,{qk})istheinteractionpotentialbetweentheresonatorofinterestandthemodesofthethermalbath.Inthesimplestcase,thisinteractionpotentialhastheformUi(q,{qk})=q!k-kqk,(1.2)7where-kdescribesthecouplingstrengthoftheresonatorofinterestwiththekthmodeoftheenvironment.ConsideringthissimpleformofUi(q,{qk}),onecanshowthatthisinteractionleadstoirreversiblelossesoftheresonatorvibrationenergy,i.e.,decayofunforcedvibrations,andresultsinarandomforceactingontheresonantmode[49].Whentheresonatorvibrationamplitudeisrelativelysmall,itsdecayrateisessentiallyconstantandtherandomforceisindependentoftheresonatormodalcoordinate,whichismodeledasanadditivenoise[33].Asexpected,whentheresonatorvibrationenergyislarge,thecouplingoftheresonatortotheenvironmentbecomesnonlinear,whichresultsinanamplitude-dependentfrictioncoe"cient(nonlineardamping)andadditionalparametricnoise[50].Thenatureofthenoiseprocessesactingontheresonatoris,however,notlimitedtothermalorigins.Dependingonthesystemofinterestanditsapplication,therecanexistdi!erentmicroscopice!ectsinteractingwiththeresonatordirectlyorthroughtheattendantelectrodesandthusa!ectingthesystemdynamicbehavior.Inamasssensingexperiment,forexample,afrequentlyencounteredsourceofnoiseisarandomprocessassociatedwiththeattachment/detachmentofmoleculestotheresonatorsurface[51].DuetotheÞnitemassofthesemolecules,theresonatore!ectivemasschangesinadiscretemanner,m(m+#m,whichresultsindiscretechangeoftheresonatornaturalfrequency,!+#!="km+#m'!#1!#m2m$,(#!'!!#m2m,(1.3)wherekistheresonatore!ectivelinearsti!nessanditisassumedthat#m%m,i.e.,themassofindividualmoleculesismuchsmallerascomparedwiththeresonatore!ectivemass.Dependingonthetimescaleofthisattachment/detachmentprocess,theassociatedfrequencynoisecantreatedaswhiteGaussian,Poisson(seeChapter2andAppendixA)or8telegraph,and,ingeneral,willhavedi!erente!ectsontheresonatordynamics.Anothertypeofnoisethatisinherentlypresentinvirtuallyallsystemsisso-calledmea-surementordetectornoise,whichappearsduetoimperfectionsofthesensing/detectionscheme.Typically,thisnoisemanifestsitselfinarandomperturbationaddedtothemea-suredsignal,forexample,avoltagecorrespondingtotheresonatormodaldisplacementorvelocity.Whilethisnoiseisassumedtohavenoe!ectontheresonatordynamicsitself,itse!ectonthemeasuredsignalcanbefrequentlyattributedtosomeÒunaccountedÓnoisesactingontheresonator,whichcanobviouslyleadtodiscrepanciesinanalyticalpredictionsand/ormisleadingconclusions.Asaresult,itisimportanttobeabletodistinguishdi!erentnoisesourcespresentinMEMSresonatorsinordertoexplaintheirdynamicbehavior,andtoaccountforthemtoimproveperformance.ThebriefoverviewofnonlinearandnoiseprocessespresentedabovemotivatestheneedforabetterunderstandingofthesephenomenainMEMSresonatorsandthedevelopmentofaccuratecharacterizationtechniquesforquantifyingthesee!ects.Athoroughandcom-prehensiveunderstandingofthesenonlinearandnoise-inducede!ectsallowsnotonlyfore!ectivepredictionoftheresonatorresponseacrossawiderangeofmodelparametersandoperatingconditions,butalsoprovidesinformationthatisessentialfordesigningMEMSresonatorswithdesireddynamicbehavior,seeFig.1.3.Atthesametime,characteriza-tionmethodsalsoplayanimportantroleindevelopingafundamentalunderstandingoftheunderlyingphysics,whilealsoprovidinginformationaboutMEMSperformance.Inthislight,themaingoalofthisdissertationistopushforwardthefrontiersofourfun-damentalunderstandingofdi!erentnonlinearitiesandnoiseprocessesinMEMSresonatorswithafocusoncharacterizationtechniquesanddesignmethods.9Figure1.3:Analyticalmethods,e!ectivecharacterizationtechniques,andreliabledesigntoolsarethekeyingredientsinbuildingMEMSresonatorswithoptimalperformancefordi!erentapplications.1.3SummaryoftheworkTheÞrsttopicinthisdissertationisananalysisofthestationaryprobabilitydistributionofanonlinearparametricallydrivenresonatorundertheactionofPoissonpulsesinthepresenceofthermalnoise.Inthiscase,thePoissonpulsesaremodulatedatonehalfofthedrivingfrequency,wherethesystemsensitivitytotheexternalforceishighest.Weconsiderthedynamicsofamicro-mechanicalresonatormodelintherotatingframeandexamineitsprobabilitydistributionprojectedontotheselecteddirectionasafunctionofthePoisonnoisepulserateanditsmodulationphase.ModulatedPoissonpulsesresultine!ectivejumpsintheresonatorstatesintherotatingframeandweshowthatPoissonnoisewithlowpulseratescausesapower-lawdivergenceoftheprobabilitydensityattheresonatorequilibriuminboththeoverdampedandunderdampedregimes.Additionally,wehavefoundthatintheoverdampedregimetheformofthesecondarysingularityintheprobabilitydistribution,ataÞnitedistancefromtheequilibriumposition,stronglydependsonthePoissonnoisemodulationphaseandthephaseoftheresonatorresponse.Incontrast,thestructureoftheprobabilitydensityfunctionintheunderdampedregimeisessentiallyindependentof10theselectedresonatorphaseandPoissonpulsedirection.Wealsoexaminethee!ectofweakadditiveGaussiannoise,aconsequenceofnon-zerotemperatureoftheenvironment,ontheshapeoftheprobabilitydistribution.Analysisshows,andexperimentalresultsfromcollaboratorsatHongKongUniversityofScienceandTechnologyverify,thatevenweakGaussiannoisea!ectstheprobabilitydistributionbysmoothingitsshapeinthevicinityofPoissonnoise-inducedsingularpeaks.Second,wedescribeatime-domaintechniqueforcharacterizingparametersofasinglevibrationalmodeinsymmetricmicro-mechanicalresonators.Inparticular,weshowthatamodalringdownresponsecanbeusedforcharacterizationofnotonlylinearmodelpa-rametersQ(qualityfactor)and!n(naturalfrequency),butalsoofcoe"cientsrepresentingconservativeanddissipativenonlinearities,aswellastheintensitiesandstatisticalpropertiesofvarioustypesofnoisesourcesactingonthemodeofinterest.Thesenonlinearitiesresultinanamplitude-dependentfrequency(conservative)andanon-exponentialdecay(dissipative),whilethenoisesourcescauseßuctuationsintheresonatoramplitudeandphase.SpeciÞcally,weshowthatthebehaviorofthemodalvibrationamplitudeisindependentofthenonlinear-itiesintheresonatorrestoringforce,whichallowsustoestimatelinearandnonlinearfrictionconstants.Ontheotherhand,themodalvibrationfrequencyisdeterminedsolely(uptonoise-inducedßuctuations)bytheformofthesystempotentialenergy,whichcanbeusedtocharacterizethelinearnaturalfrequencyandcoe"cientsofthecubic(Du"ng)andquintictermsinthemodalrestoringforce.Furthermore,weshowthatinsomecasesitispossibletoindividuallycharacterizemechanicalandelectrostatic(bothlinearandnonlinear)e!ectsthatinßuencethemodedynamics,andweformulatetheconditionsforsuchcharacteriza-tion.Finally,weillustratethatastatisticalanalysisofthezero-crossingpointsinthemodalresponsecanrevealthepresenceofadditive(thermal),multiplicative(frequency),andmea-11surement(readout)noisesources,providingauniquewayforindependentcharacterizationofthenoisestrengths.Finally,weexaminetheproblemofself-inducedparametricampliÞcationinring/diskresonatingMEMSgyroscopes.Weshowthatthedegenerateellipticalmodesofgyroscopeswithring-likegeometriesarecoupleddispersivelythroughbothsti!nessandinertialnon-linearities,andweillustratethee!ectsoftheelectrostaticpotentialonindividualmodalnonlinearitiesandonthestrengthofnonlinearmodalcoupling.Wefurtherstudythee!ectofthedispersivemodalcouplingonthegyroscopeperformanceandsensitivity.Inparticular,weillustratethattheback-actionofthesensemodevibrationsonthedrivemodedynamicscanbeneglectedwhenthestrengthofthemodalcouplingandexternalangularratearesmall.ThisfactallowsustosimplifythegyroscopemodelandtreatthesensemodeasbeingÒdrivenÓbythedrivemodethroughtheCoriolisforce,aswellasparametricallythroughnonlinearmodalcouplinge!ects,whichresultsinasubstantialincreaseinthesensitivityofthegyroscopetotheexternalangularrate.Furthermore,weshowhowthisextragain,duetoparametricampliÞcation,dependsonthedrivemodevibrationamplitudeandthecouplingstrength.Finally,weusethismodeltodemonstratewaysforadditionalimprovementofthegyroscopeperformanceasanangularratesensorbyadjustingthemodalcouplingstrengththroughvariationsinthegeometryofthegyroscopebodyandattendantelectrodes,aswellasutilizingelectrostatictuningmethods.ThesemodelingandoptimizationapproachesareaÞrststepincontrollingandexploitingnonlinearbehaviortoimprovesensordesign.12Chapter2SingularprobabilitydistributionofvibrationalsystemsdrivenbyPoissonandthermalnoisesInthischapterwestudytheprobabilitydistributionofatorsionalmicromechanicalresonatordrivenbymodulatedPoissonpulsesinthepresenceofweakthermalnoise.SpeciÞcally,weaklydampedmicromechanicalresonatorisdrivenparametricallyintotheresonance,andbothPoissonandthermalnoisesourcesperturbtheresonatordynamicsaboutitsdetermin-isticstate.ModulatedPoissonnoiseisasequenceofburstsoftheperiodicsignalwiththefrequencyclosetotheresonatornaturalfrequency.Theseburstsappearrandomlyandtheirtimesofarrivalareindependentfromarrivalsofpreviouspulses.Themeantimebetweenpulsesismuchlongerthantheresonatorcharacteristicdecaytimeandthedurationofeachburstismuchlongerthantheresonatorvibrationperiod.Thisproblemisoffundamentalimportanceasitaddressesthecombinede!ectsoftwonoiseshavingdi!erent,descriptions,namelyÒcontinuousÓanddiscreteintime.IntheabsenceofPoissonnoise,theweakthermalnoise,havingasymmetricprobabilitydistribution,essentiallyÒthermalizesÓtheresonatorarounditsstablestate,inwhichcasetheresonatorprobabilitydistributionintherotatingframeisGaussianalonganyselecteddirection.Incontract,thePoissonnoisehasdrastically13di!erente!ectontheresonatordynamics.Thediscretenessofthenoisepulsesandtheasso-ciatedasymmetryofPoissondistributionareevidentwhentheresonatorparametersaresuchthatitsdynamicsinthevicinityofthestablestateexhibitsoverdampedbehavior.Forex-ample,ifPoissonpulsesallhavethesamesign(theunipolarcase),theresonatorprobabilitydistributionbecomesstronglyasymmetricwithrespecttothestablestateinthedirectionofthePoissonpulses.DependingonthePoissonmeanpulserate,theprobabilitydistributioncandivergeinthelocationofthestablestate(thecaseoflowpulserate)ordecaytozerowithapower-lawofthedistancetothestablestate(thecaseoffrequentpulses).Whenthesystemparametersarechoseninsuchawaythatresonatorperformsdecayingoscilla-tionsaboutitsstablestate(i.e.,theunderdampedregime),thesystemstillcanexhibitasingularityinthedistributionnearthestablestate,buttheprobabilitydistributionbecomesessentiallysymmetricaboutit.InthischapterweshallalsoshowthatPoissonnoiseresultsininterestingandnon-trivialfeaturesoftheresonatorprobabilitydistributionataÞnitedistancefromthestablestate,andillustratethesmoothinge!ectsoftheweakthermalnoiseonthesingularbehavioroftheprobabilitydensityfunctions.Comparedwiththeworksdescribedinsubsequentchapters,themotivationforthisstudyismorefundamentalinnature.Asoneofthemainsourcesofßuctuationsindynamicalsystems,Poissonnoiseattractsgreatinterestfromresearchersindi!erentareasofscience.FluctuationsinducedbyPoissonpulsescanbee!ectivelyusedtoanalyzemesoscopicdevicesprovidingaccesstophysicsofthesesystemsduetonon-vanishinghigh-orderstatistics[52].However,estimationofnon-Gaussiandistributionsisoftenquitechallengingandinvolvessophisticatedtechniquesindirectmeasurementprocesses[53,54].Alternativetheoreticalapproachesproposedformicroresonators[55Ð57]andJosephsonjunctions[58Ð60]relyonthebreakingofthesymmetryinrandomswitchingofthesystembetweenitsstablestates14thatstemsfromtheinherentGaussiannoise.AlreadytheseexamplesclearlyshowthenecessityforbetterunderstandingofthedynamicbehaviorofmesoscopicsystemsdrivenbyPoissonnoise.AsigniÞcantamountofworkhasbeendoneinanalyzingthefeaturesofactivatedescapeduetoPoissonnoise[61Ð64].Additionally,theprobabilitydistributionofalinearresonatordrivenbyPoissonpulsesinastaticpotentialwasinvestigatedindetailbyIchikietal.[9].Inparticular,ithasbeenshownthattheprobabilitydistributionexhibitsasingularpeakintheoriginforarbitrarydamping,aslongasthePoissonmeanpulserateissu"cientlysmall.Thebehaviorofthedistributionawayfromtheorigin,however,dependsonthedynamicregimeofthesysteminthevicinityofitsstablestate(overdampedorunderdamped),therelativestrengthoftheresonatordampingascomparedwithitsnaturalfrequency,andonthemeanfrequencyofthePoissonpulses.InthischapterwetakeonestepfurtherinunderstandingPoissonnoise-induceddynamicsofmesoscopicsystemsbyexaminingthestationaryprobabilitydistribution(intherotatingframe)ofaparametricresonatordrivenbymodulatedPoissonpulsesandweakthermalnoise.Inparticular,weshowthat,upontransformationtotherotatingframe,modulatedPoissonnoisetransformsintoregularPoissonpulseskickingthesystemawayfromitsequilibriumalongthedirectionspeciÞedbythenoisemodulationphase.Forsu"cientlyweakpulses,theresonatorprobabilitydistribution,whenbeingobservedfromdi!erentdirectionsdictatedbythephasebetweentheresonatorin-phaseandquadraturecomponents,resemblesessential(singular)featuresofthelinearsysteminastaticpotentialdrivenbyPoissonnoiseonly,see[9].ThemaincontributionofthepresentstudyliesinourabilitytocontrolboththedirectionofthePoissonpulsesintherotatingframeaswellastheobservationphase,whichallowsustostudyaspectsoftheresonatorprobabilitydensityfunctionthatcannotbe15observedinthesystemanalyzedbyIchikietal.Furthermore,wealsoinvestigatethee!ectsofweakthermalnoiseontheresonatorprobabilitydensityfunction,whichbecomessmoothenedbythermalnoiseinthevicinityofthesingularpeaks.Fromtheapplicationpointofview,themodelconsideredinthisworkcanbedirectlyappliedtotheanalysisofmesoscopicvibrationalsystemswithdirectfrequencynoiseofPoissontype.Inthiscase,however,thedirectionofthePoissonpulsesintherotatingframeispredeterminedbythelocationofthesystemstablestate,buttheresonatornoise-induceddynamicscanstillbeanalyzedfromdi!erentobservationdirections.Theremainingpartofthechapterisorganizedasfollows.InSection2.1wedescribethemicromechanicalresonatorunderstudyinexperimentalworkperformedbyourcollaboratorsinH.B.ChanÕsgroupattheHongKongUniversityofScienceandTechnology,andprovidethecorrespondingresonatormodel.Next,wediscussthequalitativepictureoftheresonatordynamicsintherotatingframeinthepresenceofPoissonandthermalnoisesourcesinSection2.2.InSection2.3.1weanalyzethenoise-inducedstationaryprobabilitydistributionintheoverdampedregimeandshowthat,inadditiontothesingularityatthestablestate,theresonatorprobabilitydensitycanexhibitasecondarysingularpeakataÞnitedistancefromthesystemÞxedpoint.ThecaseofaweaklydampedresonatorisconsideredinSection2.3.2,whereweillustratethatinthiscasetheresonatorprobabilitydensityfunctioncanexhibit,inadditiontothemainpeakatthesystemÞxedpoint,multiplesingularpeaksawayfromthestablestate,whicharisefromtheslowoscillatorynatureoftheresponseintherotatingframe.Thee!ectofweakthermalnoiseontheresonatordynamicsisalsodiscussedforboththestronglyandweaklydampedregimesinSection2.3.3.Finally,weprovidesomeconclusionsinSection2.4.162.1Experimentalsetupandresonatordynamicalmodel2.1.1MicromechanicaltorsionalresonatorInthisworkmeasurementshavebeenperformedwithamicromechanicaltorsionalresonator(Fig.2.1a)consistingofamovable,highlydopedpolysiliconplate(200µm*200µm*3.5µm)suspendedbytwocentrallyplacedtorsionalrods(4µm*36µm*2µm).Beneaththeplate,therearetwoÞxedelectrodes(200µm*100µm)oneachsideofthetorsionalsprings.The2µmgapunderneaththeplateiscreatedbyetchingawayasacriÞcialsiliconoxidelayer.Figure2.1bshowsacross-sectionalschematicofthedevicewithactuationandmeasurementcircuitry.Thee!ectivespringconstantismodulatedelectrostaticallybyaperiodicvoltageV!atfrequency!neartwicethenaturalfrequency(!0/2)&20kHz)ridingontopofamuchlargerbiasvoltageVDC,whichisnecessaryfore"cientreadout.Theresonatormotionisdetectedcapacitively.PeriodiccarriervoltagesVcatfrequency!cof5MHzwithoppositephasesareappliedtothetwobottomelectrodes.Astheplaterotates,thecapacitancebetweentheplateandthebottomelectrodeschanges.Alock-inampliÞerconnectedtotheplatedetectsthechargeßowingoutandampliÞesit.Inthiscase,therotationangle+isproportionaltotheamplitudeofthecarriervoltageat!cattheoutputoftheampliÞer.VibrationoftheplateisdetectedbymeasuringthisoutputwiththeampliÞerreferencedatthesidebandfrequency!c+!/2.Alternatively,thereferenceofthelock-incanbesetat!c,withitsoutputmeasuredbyasecondlock-inreferencedat!/2.BothmethodsyieldthequadraturesXandYoftheoscillationsoftheplaterotationangle+(t)=C[Xcos(!t/2)+Ysin(!t/2)](2.1)17(a)Scanningelectronmicrographofthemi-cromechanicaltorsionaloscillatorusedintheexperiment.(b)Schematicofthedeviceactuationandread-outcircuitry(nottoscale).(c)Representativefrequencyresponsesoftheres-onatordrivendirectly(theredcurve)andparamet-rically(thebluecurve).Thegreenstarindicatesthelocationofthedeviceoperatingpointintheexperi-ment.(d)ModulatedPoissonpulses"lf(t).Thedurationofeachpulseismuchshorterthantheinverseofthemeanpulserate#,whichallowsustoapproximatePoissonpulsesasDiracdelta-functionsontheresonatorde-caytimescale,!!1.Figure2.1:Micromechanicaltorsionalresonatorunderstudyandexperimentalsetup.at!/2frequency.Measurementshavebeenperformedatroomtemperature(T=300K)withpressureoflessthan10!6torr.Thedynamicresponseofthetorsionalresonatorunderstudycanbemodeledbythe18followingequationofmotion,¬++2#ú++(!20+hcos!t)++"+3=.lf(t)+f(t),(2.2)where!0istheresonatoreigenfrequency(includingmechanicalandelectrostaticcompo-nents),#=24.7rad/sistheresonatordecayrateinthelaboratoryreferenceframe,"=!6.30*1011s!2isthecoe"cientofthecubicsti!ness(Du"ng)nonlinearity(ofwhichtheelectrostaticcomponentrelatedtothedependenceofthedevicecapacitanceontheplateangulardisplacementisdominantinthisdevice),hrepresentsthemodulationamplitudeofthee!ectivespringconstant,.lf(t)isthemodulatedPoissonnoise(subscriptÒlfÓstandsforthelaboratoryframe)andf(t)isaweakthermalnoisewiththefollowingstatistics#f(t)$=0,#f(t)f(t,)$=D#(t!t,).(2.3)TheintensityDofthermalnoisef(t)isrelatedtotheambienttemperaturebytheßuctuation-dissipationtheoremD=4kBT#/I,whereIistheresonatormomentofinertiaaboutitsaxisofrotation.Withnonoise,whenthespringmodulationamplitudeisbelowthecriticalthreshold,onlythezero-amplituderesonatorresponseisstable.Whenthemodulationisincreasedbeyondthethresholdvalue[65],twostableoscillationstatesemerge,whichhavethesameoscillationfrequency(!/2)andamplitudebutarephaseshiftedby)radians,whichcorrespondstoaclassicalparametricresonance,seeFig.2.1c.TheformofthemodulatedPoissonpulses.lf(t)isshowninFig.2.1d.InordertogeneratesuchanoisesignalweconnecttheoutputofaGaussiannoisevoltagegeneratortothetriggerinputofapulsegenerator.Wheneverthenoisevoltage,inararelargeoutburst,exceedsa19thresholdvalue,thegeneratorproducesasquarepulse.Thethresholdischosenmuchlargerthanthemean-squarenoiseamplitude.Respectively,thedurationofthepulse(tp=2ms)ismuchsmallerthanthereciprocalpulserate,&!1,whichallowsustoapproximatethepulsesasdeltafunctionsintherotatingframeinouranalysis,i.e..(t)=g!j#(t!tj),(2.4)wheregisthepulsearearepresentingthestrengthofPoissonnoise.ThestatisticsofthepulsesisPoissonianinthiscase.Toexploitthehighsensitivityoftheresonatoratfrequenciescloseto!0,wemodulatethePoissonpulses.(t)athalfofthespringmodulationfrequencytoobtain.lf(t)=.(t)cos(!t/2+/p),(2.5)where/prepresentsthephaseofthepulsemodulation.Aswewillshowbelow,/pplaysanimportantroleinstudyingtheresonatornoise-inducesdynamicsasitdeterminesthedirectionofPoissonkicksintheresonatorrotatingframe.2.1.2ResonatormodelintherotatingframeTheresonatordynamicscanbewellcharacterizedbytwoslowdimensionlessvariables,thescaledin-phaseandquadraturecomponentsXandYoftheresonatorrotationangle,deÞnedasfollows,+=C(Xcos(!t/2)+Ysin(!t/2)),ú+=!C!2(Xcos(!t/2)+Ysin(!t/2)),(2.6)20whereC=%2h/(3|"|)isthescalingfactor.Byscalingthetimeas0=#tinEq.(2.2),weobtainthefollowingequationofmotionforXandYcoordinates,úq=G(q)+lp.(0)+f(0),(2.7)whereq=(X,Y)isthevectorconsistingoftheresonatorin-phaseandout-of-phasequadra-tures,respectively.DeterministicnonlinearvectorÞeldG(q)readsG(q)=&'(!X!%$Y!'Y(2!(X2+Y2)sgn("))!Y+%$X!'X(2+(X2+Y2)sgn(")))*+,(2.8)where%$=!!2!02#,(2.9a)'=h4!#,(2.9b)sgn(")="|"|.(2.9c)InEq.(2.7),.(0)representsthedemodulatedandscaledPoissonpulseskickingthesystemintherotatingframealongthedirectionlp=(sin/p,cos/p)determinedbythemodulationphase/p.Themeanpulserateandpulseareaintherotatingframeare&,=&#,(2.10a)g,=g"3|"|2h!2,(2.10b)21respectively.Finally,f(0)=(fX(0),fY(0))istheGaussiannoisevectoractingontheresonatorquadratures.Asdetailedin[49],thecomponentsoff(0)aretwoindependentdelta-correlatedwhitenoisesofequalintensityD,=3|"|Dh!2#.(2.11)SincethemaingoalofthisworkistoanalyzetheresonatorprobabilitydistributionduetothepresenceofmodulatedPoissonpulses,weshallassumefornowthatthermalnoiseisnotpresentinthesystem,i.e.,f(0)=0.However,wewillreturntothediscussionofthethermalnoisee!ectontheprobabilitydensityfunctioninSection2.3.3.Inthislight,intheabsenceofthePoissonnoise,.(0)=0,thestationarysolutionq0toEq.(2.7)satisÞesG(q0)=0.IfthePoissonpulsesareweak,theninordertocapturetheire!ectontheresonatordynamicsinthevicinityofitsstablestate,wecanlinearizeG(q)aboutq0andwritethenoise-inducedresonatorresponseintermsofthefundamentalmatrix!(0),ananalogoftheintegratingfactorforsystemsofdi!erentialequations[66],asfollowsq(0)=!(0),$!-d0,!!1(0,)lp.(0,).(2.12)ThejointprobabilitydistributionoftheresonatorquadraturesXandYcanbedescribedusingthestandardkineticequation[64],t%(q,t)=!!á-G(q)%(q,t).+&,-%(q!lpg,,t)!%(q,t).,(2.13)whichdescribesthetemporalevolutionofthesystemprobabilitydensityandisusefulforstudyingtheescapedynamicsofthesystemoutofitsmetastablestates.Inthiswork,22however,westudythePoissonnoiseinduceddynamicsoftheperiodicallydrivenresonatorbyanalyzingtheprojectionofthestationaryprobabilitydistribution%(q)ontoaselectedobservationdirection,whichisdescribedfromaqualitativepointofviewinSection2.2.WethenproceedtotheformalderivationoftheresonatorprobabilitydistributionalongthechosenresonatorphaseinSection2.3,wherewefollowcloselywiththemethodutilizedbyIchikietal.[9],butoutlineimportantdi!erencesofourapproachalongthediscussion.2.2QualitativepictureofresonatorßuctuationsThebehavioroftheprobabilitydistributioncanbeunderstoodusingaqualitativepictureoftheresonatorresponsetoanisolatedPoissonpulseintherotatingframe.Undertrans-formationfromthelaboratorycoordinatestotherotatingframe,themodulatedPoissonpulsesareconvertedintoregularpulsesthatkickthesystemoutofitsequilibriumpositioninthedirectiondictatedbythemodulationphase/p,asdepictedinFigs.2.2aand2.2b.Intheexperiment,theresonatormotionisverylightlydampedinthelaboratoryreferenceframe,!0)#,andthesystemqualityfactorQ&2500.However,itsdynamicsintherotatingframeofXandYquadraturescanbeeitheroverdampedorunderdamped,depend-ingonthemodulationfrequencyandamplitude.Intheoverdamped(underdamped)case,thenoise-freeimpulseresponseisnon-oscillatory(oscillatory)inthequadraturespace;seeFigs.2.2aand2.2b,respectively.Theoverdampedregimeoccurswhentheeigenvalues$1,2oftheJacobian,G/,q|q0inEq.(2.7)arereal;otherwise,thesystemisunderdamped.LetusdeÞneqm(0)asthesystemresponseq(0)projectedontothedirectionlm=23(a)ResonatorrelaxationintherotatingframeintheoverdampedregimeduetoasinglePoissonpulse.(b)ResonatorrelaxationintherotatingframeintheunderdampedregimeduetoasinglePoissonpulse.Figure2.2:SchematicoftheresonatorresponsetoasinglePoissonpulseexcitationintherotatingframeinoverdamped(a)andunderdamped(b)regimes.Forpanel(a)wehaveassumed,forsimplicity,thatthesystemisinstronglyoverdampedregime,i.e.|$1|%|$2|,withthecorrespondingeigenvectorscoincidingwiththeXandYdirections.(cos/m,sin/m)atangle/mrelativetotheresonatorin-phasecomponentX,qm(0)=lTmq(0)=X(0)cos/m+Y(0)sin/m.(2.14)Whenthepulserate&,issmallcomparedtotheresonatordecayrateintherotatingframe,determinedbymin(|$1|,|$2|),thesystemrelaxesbacktonearlyitsequilibriumpositionq0beforethenextpulsearrives,leadingtoanaccumulationoftheprobabilitydistributionatqm=0.Intheoverdampedregime,thereisnoovershootasthesystemrelaxestowardstheequilibriumpoint,seeFig.2.2a,andthesystemdoesnotaccessthequadraturespaceindirectionsoppositetothepulsedirection,i.e.,thedirectionswherelTmlp<0.Asaresult,%(qm)=0forqm<0(intheabsenceofthermalnoise)andthepeakintheprobabilitydistributionattheequilibriumpointisstronglyasymmetric.24Awayfromthemainpeakattheequilibriumpoint,theshapeoftheprobabilitydis-tributionstronglydependsonthemodulationphase/pandthephaseofthemeasurementdirection/m.Dependingonthevalueof/m,thesystemrelaxationtoitsstationarystatecanbegenerallyviewedasmonotonic(non-monotonic),see/m,1(/m,2)inFig.2.2a,re-spectively.Inthelattercase,thesystemrelaxationalonglmhasaturningpointTwhereúqm=0andtheresonatormotion,asviewedfromlm,slowsdown.Whiletheresonatorre-laxationslowsdown,thesystemnecessarilyspendsmoretimeinthevicinityoftheturningpoint,which,inturn,leadstoanaccumulationoftheprobabilitydensityandresultsinanadditionalsharppeakinthedistributionataÞniteqm.Incontrast,whentheresonatorrelaxationalonglmismonotonic,theprobabilitydistributionisalsomonotonicandsmoothawayfromtheresonatorequilibrium.Note,however,thatiflpcoincideswitheitherofthesystemeigenvectors,theresonatorrelaxesbacktoq0alongthesamedirection.Inthiscase,independentlyoftheobservationdirectionwechoose,theresonatortrajectorytowardsitsequilibriumwillbeseenasmonotonicrelaxationandthesecondarypeakwillnotemerge.Intheunderdampedregime,therelaxationpathtowardstheequilibriumpointinthequadraturespaceisaspiral,asshowninFig.2.2b.Inthiscase,forsu"cientlyrarepulses,thesystemhasaccesstobothpositiveandnegativevaluesofqm,regardlessofthechoiceoflm,andthepeakattheequilibriumpointisalmostsymmetricinstronglyunderdampedregime.Additionally,asthesystemrelaxesbacktowardsq0,itpassesthroughmultipleturningpointsTi,atwhichúqm=0,withpositionsthatdependonthemutualorientationoflpandlm.Asintheoverdampedregime,thesystemspendsmoretimenearthesepointsandadditionalpeaksintheprobabilitydistributionoccur.Importantly,ifthePoissonpulsesaresu"cientlyweaksothatthelinearizationofEq.(2.7)aboutq0holds,thesesecondarypeaksresultinaself-similarstructureoftheresonatorprobabilitydistribution.If,however,25thePoissonpulsesaresu"cientlystrongsothatfullsysteminEq.(2.7)mustbeconsidered,thestructureoftheprobabilitydistributionbeyondthelinearlimitisnolongerself-similar,duetoanamplitude-dependentfrequencyshiftalongthespiraltowardsq0.2.3Resonatorstationaryprobabilitydistributionintherotatingframe2.3.1OverdampedregimeWestudytheprobabilitydistributionintherotatingframeinthestronglyoverdampedregimebyapplyingaperiodicmodulationslightlybelowthethresholdvalue,i.e.,'!1,sothattheonlystablestateisthezero-amplitudesolution.Inthiscasewehave,withoutlossofgenerality,-=|$1|%|$2|'2,-%1.(2.15)ForthePoissonnoiseapplied,thephaseofmodulation/pwasinitiallychosensothateachpulseproducesatranslationinthequadraturespacealongthesloweigendirection;inmeasuringthevibration,/mwasalsochosentoliealongthisdirection.Thischoiceof/pand/misveryconvenientforstudyingtheresonatorprobabilitydistributioninthevicinityofq0becausethesystemmotionintherotatingplaneisessentiallyrestrictedalongthesloweigendirection.Figure2.3ashowsatypicalrecordofqmasafunctionoftimeinthiscase.Astheresonatorrelaxestowardstheequilibriumpoint,thesubsequentpulsesarriveatrandominstantsoftimetoproducefurtherjumps.Toobtaintheprobabilitydistribution%(qm),theqmaxisisdividedintobinsandthetimespentbytheresonatorineachbinismeasured.Inordertoderivetheresonatorstationaryprobabilitydistributionalonglm,westart26(a)Sampleofthereal-timeresponseoftheresonatorquadraturealongthesystemsloweigendirectionintheoverdampedregime.ThePoissonmeanpulserateis#=0.52Hz.(b)Probabilitydistribution%(qm)oftheresonatorquadraturealongthesystemsloweigendirection.(c)Probabilitydistribution%(qm)oftheresonatorquadraturealongthesystemsloweigendirectionshownonalogarith-micscale.Thedashedlinerepresentstheleast-squaresÞtofthepower-lawbehavioroftheprobabilitydistributioninthevicinityofqm=0.(d)Power-lawexponentof%(qm)intheoverdampedregimefordi"erentvaluesofthePoissonmeanpulserate#.Thesolidlineandtrianglesrepresentthetheo-reticalpredictionandexperimentallyextractedvaluesofpod,respectively.Figure2.3:Poissonnoise-inducedresonatordynamicsintherotatingframeintheover-dampedregime.withitsconventionalform%(qm)=##(qm!qm(0))$,(2.16)27see[67].FromEqs.(2.12)and(2.14)wehaveqm(0)=,$!-d0,*"(0!0,).(0,),(2.17a)*"(0!0,)=lTm!(0)!!1(0,)lp,(2.17b)where*"(0)istheresonatorsusceptibilityalonglmtoPoissonpulsesalonglpinthequadra-turespace.Thisformofqm(0)furtheryieldsqm(0)=,0!-d0,*"(!0,).(0,)=,-0ds*"(s).(!s),(2.18)wherewehaveintroducednewtimevariables=!0,andßippedthelimitsofintegration.Inthislight,theresonatorprobabilitydistributions%(qm)becomes%(qm)=##(qm!qm(0))$=,-!-dk2)e!ikqm/eikqm(0)0",(2.19)where#$"denotesstatisticalaveragingoverdi!erentPoissonnoiserealizations.Giventhat.(0,)=g,1j#(0,!0,j)and0,j<0inEq.(2.18),wehave/eikqm(0)0"=/exp2ikg,!j,-0ds*"(s)#(s!sj)30",(2.20)wheresj=!0,j>0.Theexpressionontheright-handsideinEq.(2.20)representsthecharacteristicfunctionalfortheseriesofPoissonpulses[68].Accordingly,wederivethe28generalformoftheresonatorprobabilitydensityfunction%(qm),whichreads%(qm)=,-!-dk2)e!ikqmexp2!&,1(k)3,1(k)=,-0d021!eikg,&"($)3,(2.21)Intheoverdampedregime,theresonatorsusceptibilitycanbeexpressedinthefollowingform*"(0)=*(od)"(0)=c1exp[!-0]+c2exp[!20],(2.22)whereciareconstantsthatdependon/p,/m,andtheorientationofthesystemeigendi-rections.Aswementionedabove,inthiscasethereisnoovershootintherelaxation,whichyields%(qm)=0forqm<0.Sinceinstronglyoverdampedregime|$1|%|$2|,thesecondterminEq.(2.22)decaysmuchfasterthantheÞrstone,.exp[!-0],andtheresonatorsusceptibilitycanbeapproximatedas*(od)"(0)'c1exp[!-0],for0>1.(2.23)Asaresult,theleading-ordercontributionto1od(k)inEq.(2.21)comesfromtheinterval0>1.Bydenotingthiscontributionas1(0)od(k),wehave1(0)od(k)=,-1d021!eikg,c1exp[!'$]3=,-1d021!cos#c1kg,e!'$$!isin#c1kg,e!'$$3.(2.24)Inordertocomputethelastintegral,weintroduceanewvariablez=c1kg,exp[!-0].After29somealgebra,10(k)becomes1(0)od(k)=-!1,c1kg,0dz1!cosz!isinzz=-!1-ln(c1kg,)!Ci(c1kg,)!iSi(c1kg,)+"E.,(2.25)whereCi()andSi()arethecosineandsineintegralfunctions,see[69],and"EistheEuler-Mascheroniconstant.Theresonatorprobabilitydistribution%(qm)inthevicinityofthestablestateisprimarilydeterminedbythelarge!kbehaviorof1(0)od(k)inEq.(2.21),whilethesmall!kpieceof1(0)od(k)describesthesmoothpartof%(qm).Inparticular,forlargekcosingandsineintegralfunctionscanbeapproximatedbytheirasymptoticexpansionsandwehave1(0)od(k)'-!12lnc1kg,+i(c1kg,)!1eic1kg,!i)2+"E3.(2.26)BysubstitutingthisexpressioninEq.(2.21),weobtainthat1(0)od(k)leadstothepower-lawbehaviorof%(qm)neartheorigin,%(qm)',-!-dk2)e!ikqm(c1kg,)!#,/'exp2!&,-#"E!i)2$3.q!podm,pod=1!&,-,(2.27)wherepodisthepower-lawexponentoftheresonatorprobabilitydistributioninthevicinityofthestablestate.Thispower-lawexponentisdeterminedbytheratioof&,tothedecayrate-alongtheslowdirectionoftheresponseintherotatingframe.Forsu"cientlylowpulserates,&,<-,theexponentispositiveandtheprobabilitydistributiondivergesattheorigin.AsweshalldiscussinSection2.3.3,thepresenceofthermalnoiseregularizesthissingulardistribution.30Figure2.3bshowsthemeasuredprobabilitydistributionintheoverdampedregime,whichisinqualitativeagreementwiththetheoreticalanalysis.Figure2.3cshowsthesamedataplottedonasemi-logarithmicscale,withthelinearÞtyieldingthemeasuredpower-lawexponentoftheprobabilitydistributioninthevicinityoftheorigin.Figures2.3band2.3carederivedfromtherecordofqmover30minutes.Figure2.3dshowsthemeasuredexponent(triangles)asafunctionofthePoissonpulserate,alongwiththetheoreticalprediction(solidline)whichusesnoÞttingparameters,demonstratingexcellentagreementbetweenmeasurementandtheory.Inadditiontothebehaviorclosetotheorigindescribedabove,thereareadditionalinterestingfeaturesoftheresonatorprobabilitydensityfunctionforqm/=0.Thesefeaturesareduetothesmallcontributionto1od(k)comingfromthetimeinterval00[0,1].Inthiscase,theintegralinEq.(2.21)canbeapproximatedforlargekbyemployingthemethodofstationaryphase.Dependingonthephase/mofthemeasurementquadratureintherotatingframe,*"(0)canbemonotonic,c1c2>0,ornon-monotonic,c1c2<0.When*(0)isamonotonicfunctionof0,thereisnostationarypoint,i.e.,ú*(0)/=0forall0,andwehave1(1)od(k)=,10d021!eikg,&"($)3&1!exp[ikg,*"(0)]ikg,*,"(0)444410'1!iexp[ic1kg,]-c1kg,+iexp[i(c1+c2)kg,]kg,(-c1+2c2),(2.28)wherethesecondtermcancelstheexponentialtermin1(0)od(k).AsitfollowsfromEqs.(2.26)and(2.28),1(1)od(k)isO(-)ascomparedwith1(0)od(k),whichallowsuswritetheresonator31probabilitydistributionandcorrespondingcorrectiondueto1(1)od(k)as%(qm)',-!-dk2)e!ikqmexp2!&,1(0)od(k)3#1!&,!&,iexp[i(c1+c2)kg,]kg,(-c1+2c2)$,(2.29a)#%(qm)'!&,,-!-dk2)e!ikqmexp2!&,1(0)od(k)3iexp[i(c1+c2)kg,]kg,(-c1+2c2).(2.29b)Asaresult,thecorrectionto1od(k),beingtakenintoaccountinEq.(2.21),yieldsanadditionalpeakin%(qm)withtheshapedescribedby#%(qm).|qm!q2|!pod+1,q2=(c1+c2)g,(2.30)FromEq.(2.29b)itfollowsthatthispeakisasymmetric,sincetheprefactordependsonthesignofqm!q2.Remarkably,%(qm)doesnotdivergeatqm=q2sincepod<1forallvaluesofthepulserate,eventhoughthedivergencecanstillbeseenin,qm%forsu"cientlysmall&.Figure2.4ashowsatypicalformoftheresonatorprobabilitydistributionwhentheobservationdirectionischosensuchthattheresonatorsusceptibility*"(0)ismonotonic.If,however,wechoosetheobservationdirectionsuchthat*"(0)isnon-monotonic,thereisastationarypoint0Tintheresonatorsusceptibilitywhereú*"(0T)=0.Physically,thisstationarypointcorrespondstotheturningpointintheresonatorrelaxationpathasseenfromtheobservationdirection,seeFig.2.2a.Inthiscasetheresultfor1(1)od(k)issimilartothatobtainedin[9]andreads1(1)od(k)&1!")4-c1kg,exp2ic1kg,!sgn(c1))43,(2.31)wheresgn()denotesthesignofthequantityinthebrackets.Followingthesamelineof32qm, arb.00.10.2!(qm)020406080100qm, arb.00.050.1!(qm)0510(a)Reducedpeakduetomonotonicresponsefunction,forc1=0.99,c2=0.1.qm, arb.00.050.1!(qm)050100qm, arb.00.050.1!(qm)0510(b)Sharppeakduetonon-monotonicresponsefunction,forc1=13/2,c2=!1/2.Figure2.4:Measuredprobabilitydistribution%(qm)intherotatingframeduetoPoissonpulsesalongtheout-of-phasequadraturewith&=0.94HzandthermalnoiseatT=300Kintheoverdampedregime.IncontrasttothedatashowninFig.2.2a,theresonatoreigendirectionsintheexperimentare(1,±('!$))in(X,Y)spaceandarenotalignedwitheitherthein-phaseorout-of-phaseresonatorcomponents.ThelatterfactallowsustoobservesingularorßatpeakstructuresoftheresonatorprobabilitydistributionataÞnitedistancefromtheoriginbychangingtheobservationdirection.thoughtsthatledustoEq.(2.29b),itbecomesclearthatthecorrectiontotheprobabilitydistributiondueto1(1)od(k)deÞnedinEq.(2.31)resultsinanadditionalpeakintheresonatorprobabilitydistribution.Inparticular,wehave#%(qm)'&,,-!-dk2)e!ikqmexp2!&,1(0)od(k)3")4-c1kg,eic1kg,!sgn(c1)(/4.|qm!c1kg,|!pod+1/2.(2.32)wherepodisdeÞnedinEq.(2.27).Inthissituation,theprobabilitydistributiondisplayssingularbehaviorawayfromtheoriginforpulseratesforwhichpod>1/2;otherwise,theprobabilitydistributionhasßatpeak,while,q%stillcandisplaydivergence.Figure2.4billustratesthesingularbehavioroftheresonatorprobabilitydistributioninthecasewhentheresonatorsusceptibility*"(0)isnon-monotonicfunctionoftime.332.3.2UnderdampedregimeItturnsoutthatwecanalsoderivethegeneralformoftheresonatorprobabilitydistributionwhentheresonatorisdrivenfarfromthesub-criticalbifurcationandwherethesystembehaviorintherotatingframeisstronglyunderdamped.Theanalysiscorrespondingtothiscaseoftheresonatordynamicsisessentiallyidenticaltothecaseoftheunderdampedlinearresonatorinastaticpotential[9],soherewehighlightthemainresults.Inparticular,intheunderdampedregimetheresonatoreigenvaluesarecomplex,$1,2=!$r±i$i,$i)$r>0,(2.33)where$rdictatestheresonatorrelaxationrateintherotatingframe,while$ideterminesthesystemangularfrequencyofitsspiralmotiontowardsequilibrium.Asaresult,theresonatorsusceptibilityalongchosenobservationdirectionhasthefollowingform,*(ud)"(0)=c3e!)r$sin($i0+(),(2.34)wherec3and(areconstantsthatdepend,asbefore,onthemodulationphaseofthePoissonpulses/p,observationdirectionlm,andtheresonatorparameters.Figure2.5ashowsatypicaltrajectoryofthesysteminthequadraturespaceinresponsetoasinglePoissonpulse.Unlikeintheoverdampedcase,thesystemspiralstowardtheequilibriumpointandspendstimeintheqm<0regionregardlessofourchoiceof/m.Asaresult,intheunderdampedregime,%(qm)/=0forqm<0andtheshapeoftheprobabilitydistributionisnearlysymmetricwithrespecttothestationarystate.First,westudytheresonator34(a)Sampleofthereal-timeresponseoftheresonatortoasinglePoissonpulseintheunderdampedregimeintherotatingframe.(b)Measuredprobabilitydistribution%(qm)pro-jectedontolpintheunderdampedregimeduetoPoissonpulseswith#=4.92HzandthermalnoiseatT=300K.Inset:multiplepeaksoccuratposi-tionspredictedbyqn=(!1)nc3g"e!!r"n.Figure2.5:Theresonatordynamicsandtheassociatedprobabilitydistributionintherotat-ingframeintheunderdampedregime.distributioninthevicinityofstablestate.Inserting*(ud)"(0)intoEq.(2.21)yields,1ud(k)'1(0)ud(k)=$!1r,c3kg,0dz1!J0(z)z,(2.35)wherewehaveintroducedanewvariable,z=c3kg,e!)r$,(2.36)andusedJacobi-Angerexpansion[69]ofthetermexp[izsin$i0]totheleadingorderbecausez(0as0(-.InEq.(2.37)J0(z)isthezero-orderBesselfunctionoftheÞrstkind.Aswementionedabove,integrationovertherangeoflargekgivessingularcontributiontothe35distribution%(qm).Forlargek,wecanapproximate1(0)ud(k)as1(0)ud(k)'$!1r#lnc3kg,2+"E$,(2.37)fromwhereitfollowsthattheprobabilitydistribution%(qm)alsoexhibitsapower-lawbe-haviorinthevicinityofqm=0intheunderdampedcase.Inparticular,wehave%(qm)',-!-dk2)e!ikqm#c3kg,2$!#,/)rexp2!&,$r"E3.|qm|!pud,pud=1!&,$r.(2.38)Thepower-lawexponentpudisdeÞned,asbefore,bytheratioofthemeanpulseratetotherelaxationrateofthesystem,$r,butthepeakattheoriginisnowsymmetricduetotheunderdampednatureoftheresonatordynamicsintherotatingframe.Inadditiontothesingularityattheorigin,theprobabilitydistributionhasmultiplepeaksatÞnitevaluesofqmduetotheexistenceofmultipleturningpointsTn,aswediscussedinSection2.2.Theseturningpointscorrespondtothestationarypointsof*"(0),i.e.,whereú*"=0.Inthestronglyunderdampedcase,$r%$i,thesepointsarefoundtosatisfy$i0n+(=)#n+12$,n=0,1,2,....(2.39)Inthiscase,bypartitioningtheintegrationrangeinEq.(2.21)intomultiplebins,eachofwhichcontainingone0n,wecanobtainthecorrection1(1)ud(k)followingthemethodof36stationaryphase[70],1(1)ud(k)'$!1r#1!$r$i")2c3kg,N!n=0e)r$n/2ei*n(k)$,/n(k)=(!1)n#c3kg,e!)r$n!)4$,(2.40)wherenumberNischosensuchthatc3kg,exp[!$r0N]=zn)1.Inthiscase,asitfollowsfromEq.(2.40),correctionsto1ud(k)duetothepresenceofturningpointsTn.$r/$iandsmallinthestronglyunderdampedregime.Asaresult,theassociatedcorrectionstotheresonatorprobabilitydensityfunction%(qm)become#%n(qm)'!&,$ie)r$n/2,-!-dk2)e!ikqmexp2!&,1(0)od(k)3ei*n(k).|qm!qn|!pud+1/2,(2.41)wherelocationsqnaredeÞnedasqn=(!1)nc3g,e!)r$n.(2.42)Asitfollowsfromthefromof#%n(qm),theresonatorprobabilitydistributionhasmultiplepeaksawayfromthesystemÞxedpoint;thesepeaksformageometricprogressionandex-hibitthesamepower-lawbehaviordictatedbytheratioofthePoissonmeanpulseratetotheresonatorrelaxationrate$rintherotatingframe.Importantly,whenpud>1/2,secondarypeaksintheprobabilitydistributionsaredivergent;intheoppositescenariothedivergencecanonlybeseeninthederivative,qm%.Figure2.5bshowsthemeasuredprobabilitydis-tribution%(qm),inwhichthepositionsofthepeaks,asmarkedbythearrows,areingoodagreementwiththeoreticalpredictionsobtainedbyusingEq.(2.42).372.3.3E"ectsofthermalnoiseThepresenceofweakthermalnoisef(t)a!ectstheresonatorprobabilitydistributioninducedbythemodulatedPoissonpulses.Inparticular,f(0)contributestotheresonatormotionq(0)intherotatingframeasfollows,q(0)=q"(0)+!(0),$!-d0,!!1(0,)f(0,),(2.43)whereq"(0)isPoissonnoise-inducedresonatorresponsedeÞnedisEq.(2.12).Consequently,wecanrewritetheresonatorresponsealongchosenobservationdirectionqm(0)asqm(0)=qm"(0)+,-0ds*X(s)fX(!s)+,-0ds*Y(s)fY(!s),(2.44)wherefXandfYarethecomponentsofthermalnoisevectorf(0)and*Xand*YarethecorrespondingsusceptibilitiesofqmtofXandfY,respectively.AccountingforthestatisticalpropertiesoffXandfY,namely#fX(t1)fX(t2)$=#fY(t1)fY(t2)$=D,#(t1!t2),#fX(t1)fY(t2)$=0,(2.45)see[49],wecanshow,followingthesamelineofthoughtsasinSection2.3.1,thatthepresenceofthermalnoiseresultsinanadditionalexponentialfactor,[68],inEq.(2.21),%",f(qm)=,dk2)e!ikqme!#,+(k)exp2!k2D,A23,A=,-0d0(*2X(0)+*2Y(0)),(2.46)38where%",f(qm)istheresonatorprobabilitydistributioninthepresenceofbothPoissonpulsesandthermalnoise.FromEq.(2.46)itfollowsthatthermalnoise,nomatterhowsmall,dominatesthedynamicsforlargek,sincek2)1(k)&lnkfork(-,andthusweexpecttoseeGaussianbehavioroftheprobabilitydistributioninthevicinityofsingularpeakslocatedattheresonatorstablestateqm=0andattheturningpointsqm=qTalongthesystemrelaxationtrajectory,bothintheoverdampedandunderdampedregimes.Figures2.4a,2.4band2.5bshowthatthermalnoiseregularizessingularbehavioroftheprobabilitydensity,and%(qm)hasnormaldistributioninthevicinityofitsmaxima.Ofcourse,thermalnoisea!ectstheresonatordynamicsalongthewholerelaxationpath.However,itse!ectontheresonatordynamicsbecomesmorenoticeableinthevicinityofpointswheretheresonatorrelativevelocityissmall,andf(0)essentiallyÒthermalizesÓtheresonatorinthecloseproximitytosingular(forrarePoissonpulses)peaks.Forweakthermalnoise,%(qm)quicklydeviatesfromthenormaldistributionawayfromÒsingularÓpeaks.AswementionedinSection2.3.1,thermalnoisealsoleadstoanon-vanishingcontributionto%(qm)forqm<0whentheresonatorisintheoverdampedregime.Physically,theonlywaytheresonatorcanenterthisregionisduetothermalnoisewhenqm"1D,and,thus,theprobabilitydistributionisnormalinqm<0region.Intheunderdampedregime,asitfollowsfromEq.(2.46),f(0)alsodistortsthesingularbehavioroftheprobabilitydensityandmakes%(qm)Þniteforall&overtheentirerangeofqm,whichwasconÞrmedintheexperiment,seeFig.2.5b.TheanalyticalapproachdevelopedinthischapterandexperimentalmethodspresentedaboveallowonetostudythedynamicsofmesoscopicvibrationalsystemswitheigenfrequencyßuctuationshavingPoissonstatistics.Inthiscase,theanalysisisquitesimilartotheonedevelopedinSection2.1,seeAppendixA.However,themaindi!erencebetweenthemodelconsideredinthischapterandsystemswithmultiplicativenoiseofPoissontypeisinthefact39that,inthelattercase,themodulationphaseofPoissonpulsesisessentiallypredeterminedbytheresonatorÞxedpointintherotatingframeand,thus,ishardertomanipulateintheexperiment.However,onestillcanvarytheobservationdirectionand,thus,investigatesingularfeaturesoftheresonatorprobabilitydistributionfromdi!erentangles.Finally,itisworthnotingthatourmethodsarenotlimitedtoparametricallymodulatedresonatorsandcanbesimilarlyappliedtodirectlyexcitedresonantsystems.2.4OutlookAsmentionedatthebeginningofthischapter,thisworkwasmotivatedbythenecessityofabetterfundamentalunderstandingofthedynamicbehaviorofmesoscopicsystemsinthepresenceofPoissonnoise.Inthislight,wehavestudiedtheprobabilitydistributionofthenonlineartorsionalparametricresonatorintherotatingframe.Inadditiontoparametricforcing,theresonatorwasdrivenbymodulatedPoissonpulsesandweakthermalnoise.WiththeabilitytotunethedirectionofPoissonpulsesandchoosethemeasurementquadratureintherotatingframe,wehaveshownthatforsu"cientlysmallpulseratestheprobabilitydistributionexhibitsapower-lawsingularityneartheresonatorequilibriumbothintheoverdampedandunderdampedregimes.Additionally,wehavedescribedthedependenceofthecorrespondingexponentonthePoissonmeanpulserateandthesystemdecayrateintherotatingframe.Wealsofoundadditionalpeak(s)inthedistributionawayfromtheoriginandspeciÞedtheirpositionsandtheconditionsfortheirappearance.Inparticular,wedemonstratedthattheprobabilitydistributionisstronglyasymmetricintheoverdampedregime,whileintheunderdampedregimeithasaself-similarstructure.Weakthermalnoisea!ectsthesystembysmootheningsingularpeaksintheprobabilitydistributionandmaking40thedistributionGaussianinthevicinityofthesepeaks.Ouranalyticalresultsforpower-lawexponentsnearthepeaksandforthepositionsofpeaksintheprobabilitydistributionareinexcellentagreementwithexperimentalobservationsfromthemicromechanicalresonator.41Chapter3CharacterizingnonlinearitiesandnoiseinMEMS:ringdown-basedapproachInthischapterwediscussatime-domaintechniqueforcompletecharacterizationofnonlinearsymmetricMEMSresonatorswhoseresponseisgovernedbyasinglevibrationalmode.Thecharacterizationmethodisbasedontheanalysisofthetransientresponseoftheresonatorwhenexternalforcingisabsent,thatis,onitsso-calledringdownresponse.OurprimarypurposeistoÞndawaytoextractmodelparametersassociatedwithbothdeterministicandrandompropertiesoftheresonatorandthereadoutofitsresponse.Thedeterministicparametersareassociatedwithlinearandnonlinearsti!ness,linearandnonlineardamping,andtheparametersfortherandompartofthemodelaretheintensitiesofadditive(ther-mal),multiplicative(frequency)andmeasurementnoisesources.Thefactthatthedynamicbehavioroftheresonatorvibrationamplitudeduringtheringdownisindependentoftheshapeoftheresonatorpotentialallowsustoinvestigateitsdecayrateasafunctionoftheinstantaneousamplitude,thus,providingawaytoestimatetheresonatorqualityfactorandtheassociatednonlinearityinthedampingforce.Atthesametime,theresonatorvibrationfrequencyintheringdownfollowstheÒbackboneÓcurve,whichentirelydependsontheform42ofthesystempotentialenergy.Inthisworkweutilizethesequenceofzero-crossingpointsinordertoextracttheresonatorvibrationfrequencyasafunctionoftime,or,equivalently,amplitude,andestimatenotonlythelinearnaturalfrequencyofthesystem,butalsochar-acterizethecoe"cientsofhigher-orderterms(Du"ngandquintic)intheresonatorrestoringforce,thusallowingonetodescribehardening,softening,andmixedtypesofconservativenonlinearities.Furthermore,astatisticalanalysisofthesequenceofzero-crossingpointsallowsustoseparatethee!ectsofmeasurementnoiseonthemeasuredresonatorresponsefromresonator-relatedadditiveandmultiplicativenoisesources,andtoestimatetheirrel-ativestrengths.Wehavetestedthistechniqueusingsimulateddata,aswellasringdownresponsesofdouble-anchoreddouble-ended-tuning-fork(DA-DETF)resonatorsthathavebeenmeasuredbyourcollaboratorsinTomKennyÕsgroupatStanfordUniversity.Theprimarymotivationforthisstudyisthefactthatparameterestimationinvibrationalsystemsisachallengingproblemarisinginsystemsofdi!erentsizescales[45,56,71Ð73].Itisimportantsinceitallowsonetodescribethedynamicsofsystemsofinterestusingstan-dardmodels[74Ð76],tounderstandthefundamentalphysicalmechanismsresponsibleforcertainobservede!ects[77,78],andtodesignsystemswithdesiredperformancecharacter-istics[79,80].WhileseveralmethodshavebeendevelopedfornonlinearsystemidentiÞca-tion[81],acommonapproachfordeterminingthemodelparametersofMEMSresonatorsisbasedontheresonantresponseofavibrationalmodetoaperiodicforce.Inthiscase,theresonatoramplitudeandphasearemeasuredasafunctionofthefrequencyoftheexternaldrivingÞeldforaÞxedlevelofthedriveamplitude.Forsystemsoperatinginthelinearregimethisspectralmethodprovidesestimatesforthelinearresonantfrequencyandthequalityfactor[82]fromafrequencyresponse.Whentheresonatorisdrivenintoitsnon-linearregime,theshapeofthefrequencyresponseisdeterminedbybothconservative[31]43anddissipativenonlinearities[32Ð34,83].Asaresult,itisnecessarytoperformseveralmeasurementsatdi!erentforcingamplitudesinordertocompletelycharacterizetheparam-etersofthevibrationalmode[31,82],andtheprecisionislimited,particularlywhereseveralnonlinearmechanismsareinvolved.Estimationofstatisticalparametersofthenoisesourcesinnonlinearmicromechanicalresonatorsisalsoanimportantandchallengingtaskinapplicationsinvolvingparametricsensing[10]andmeasurementoftimeandfrequency[84].Duetotheirdi!erentnature,noisesourcescana!ectresonatoramplitudeand/orfrequency[8],andseveralmethodshavebeendevelopedfornoisecharacterizationindrivensystems.Forexample,inordertoanalyzethefrequencynoiseinself-sustainedoscillators,onecanmeasuretheßuctuationsinthemodalamplitudeandphase[2].Inmorerecentwork[78],byanalyzingtheshapeofthespectrumofadrivennanomechanicalresonator,theauthorsdeterminethepresenceofthedirectfrequencyßuctuationsandestimateitsintensitycomparedwiththethermalnoise.Inbothmethods,however,estimationofthenoisestatisticswillnecessarilyhaveadditionalerrorsduetoinstrumentaluncertaintiesinthedrivingelectronicsand/orelementsofthefeedbackloop[85,86].Webeginourdiscussion,inSection3.1,bydescribingthedeviceunderstudyandtheexperimentalsetupusedformeasurementoftheresonatorringdownresponse.Further,inSection3.2,weprovideamodelforDA-DETFresonator,describeitsvalidityandlimitationsandshowhowtheresonatoramplitudeandfrequency,orphase,behaviordependsonmodelparameters.InSection3.3wediscussthecharacterizationtechniqueitselfandillustratethepost-processingprocedurethatleadstoestimationofdeterministicmodalparameters.Wethenproceedtocharacterizationofnoisesourcesa!ectingtheresonatordynamicsandmeasuredresponseinSection3.4.Finally,weprovideconcludingremarksinSection3.5.443.1DeviceunderstudyandmeasurementsetupInthisworkwecarryouttheringdown-basedcharacterizationfortheDA-DETFresonatorshowninFig.3.1.TheresonatorwasfabricatedusinganepitaxialpolysiliconencapsulationFigure3.1:Top:COMSOLmodelofamicromechanicalDA-DETFresonatorshowingthesymmetricvibrationalmodeunderstudy.Theexpected(usingFEManalysis)valuesoftheresonatorlinearparametersintheexperimentareasfollows:e!ectivemodemassmeff&0.2µg,qualityfactorQ&103!104,andnaturalfrequencyf0'1.2MHz.Thesquaredenotesthelocationofthecross-sectionalSEM.Bottom:SEMfroma45"-viewangleoftheresonatorencapsulatedwiththeepi-sealprocess.process(epi-seal)[87]anditconsistsoftwomicromechanicalbeams200µmlong,6µmwideand40µmthickthatareconnectedonbothendstoperforatedmasses,whicharefurther45anchoredtothebase.Theperforationinthecouplingmassservesasrelease-etchholesanddoesnota!ectthedeviceperformance.Theencapsulationprocessresultsinapressureof<1Painthecavitycontainingtheresonator.Topreparethesystemfortheringdownmeasurement,weÞrstforcetheresonatortooscillateinthenonlinearregimeusingfeedbackloop.Previousresearchhasdemonstratedstableoscillationofthisdevicebeyondthecriticalbifurcationlimitbycontrollingtheoper-atingphaseoftheresonatorwhenthelatterisdrivenintheclosed-loopconÞguration[30].Physically,thefeedbackloopcompensatesthelossesintheresonatorduetodampingandprovidesanadditionalshiftintheresonatorphaseensuringthatBarkhausenstabilitycrite-rionismet.Inthiswork,aZurichHF2LIlock-inampliÞerisusedtocontrolandmaintainavariable-phasefeedbackloop,asshowninFig.3.2.Theoutputofthelock-inampliÞerFigure3.2:Variable-phaseclosed-loopfeedbacksystemwithaddedcapabilityforringdownmeasurements.TheencapsulateddevicesareplacedintoaThermotronS-1.2cenvironmentalchamberfortemperaturestabilizationat!40"C.maintainstheresonatormotionbysupplyingaperiodicsignal(VAC=200!300mV)totwoÒDriveÓelectrodes.Bytuningthephaseshiftinthefeedbackloop,weachievethefrequency46ofself-sustainedoscillationstobeclosetothenonlinearresonance;seeFig.3.3.Toachieveastrongoutputsignal,weapplyaDCvoltage(VDC=30V)totheresonatorbody.Addi-tionally,wemaintainbothdrivingandsensingelectrodesattheÒgroundÓvoltagepotential,thusensuringthesymmetryofthesystempotentialenergy.Figure3.3:Measuredamplitude-frequencyresponsesofthemicromechanicalresonatorinthefeedbackloopwithVDC=30Vanddi!erentvaluesofVACatT=!40"C.Eachcircledenotesthepositionofthenonlinearresonancewherethesystemhasbeenpreparedforthesubsequentringdownmeasurement.TheresonatorresponseisdetectedbytheÒSenseÓelectrodeintheformofcurrentthatiselectrostaticallytransducedduetotheresonatorvibration.ThisoutputcurrentisthenconvertedtoavoltagesignalandampliÞedviaatransimpedanceampliÞer(TIA).Wefurtherpassthesignalthroughaband-passÞlterwithcornerfrequencies1.2MHzand7MHzinordertoremovelow-andhigh-frequencymeasurementnoise,andthensplitthesignalwitha0"powersplitter.Oneoftheoutputsisfedbacktothelock-inampliÞer,wheretheresonantfrequencyandamplitudecanbetracked.ThesecondsignalcomponentgoesintoanAlazarDSOATS9360digitizerforrecordingoftheringdownresponse.Avoltage-controlledRFswitch,placedbetweenthelock-inoutputandtheresonator,actsasthemechanism47forcuttingtheresonatordriving.WhenthetriggervoltageissetÒHighÓ,theconnectionbetweenthelock-inampliÞerandresonatorisclosed,andweobservestableoscillatorysignalviatheAlazarDSO.Oncetheresonatorvibrationsreachsteady-state,thetriggervoltageisswitchedtoÒLowÓ.Inthiscasethefallingedgecutstheinputtotheresonatorcomingfromthelock-inampliÞerandtriggersthedigitizer,allowingustocapturethefullringdownresponse,seeFig.3.4.Thecollecteddataisthenpost-processedforcharacterizationoftheFigure3.4:Ameasuredringdownresponseoftheresonatorunderstudy;VDC=30VandVAC=250mV.Redsolidlineindicatesextractedvibrationalenvelopea(t).Insetshowsatime-expandedviewoftheinitialportionofthesignal.parametersofthevibrationalmodeviatheproceduredescribedbelow.3.2ModelThedynamicsofamicromechanicalresonatorwithcapacitivesensingdependsonbothme-chanicalforcesarisingintheresonatorbodyandtheelectrostatice!ectsduetothebias48voltage[45].Inthiswork,theresonatorßexuraldisplacementy(x,t),wherexisthespatialcoordinatealongthebeam,ismuchsmallerthantheresonatorwidth,y(x,t)%h,whichallowsustoapproximatethemechanicalrestoringforceofthesymmetricvibrationalmodeunderthestudybya3rd-orderpolynomial!20mq+"mq3,whereqisthemodaldisplace-mentcoordinate,!0misthemechanicallinearvibrationfrequencyand"misthemechanicalDu"ngnonlinearitywhichispositiveforaclamped-clamped(CC)beam.Thesemodalparameterscanbeobtainedby(1)approximatingtheresonatordeformationfunctionasy(x,t)=q(t)+(x),(3.1)where/(x)istheidealmodeshapeofaCCbeam,whichweapproximatefurtherbyanassumedmode,givenbyasimplepolynomialthatsatisÞestheCCboundaryconditions,namely,+(x)=16x2(1!x)2,(3.2)and(2)usingtheGalerkinprojectionoftheoriginalequationofmotionforthebeamonthespeciÞcvibrationalmode[45,88].Further,sincetheresonatorisbiasedsymmetricallywemodeltheelectrostaticforceactingontheresonatorduringitsringdownasFel=2[(d!y(x,t))!2!(d+y(x,t))!2],(3.3)wheredisthenominalelectrodegapsizeand2isthestrengthoftheelectrostaticforce,whichdependsontheresonatordimensionsandthebiasvoltage.Inordertoobtaintheexpressionforanequivalentelectrostaticforceactingonthevibrationalmode,onewouldhavetoprojectFelonthismode,whichisgenerallyachallengingtask,seeAppendixB.49However,noticingthat,bydeÞnition,y(x,t)/d<1,wecanexpandFelinaTaylorseriesaboutq=0.Sinced%h,wecankeepinthisexpansionhigher-orderterms.Thesetermscanbecomecomparabletothenonlinearterm.q3wheretheexpansionsofthebothmechanicalandelectrostaticforcesapply.Wewillkeeptermsupto5thorderinq/dandthenperformtheGalerkinprojection.Itisimportanttonotethatthemechanicalandelectrostaticforcesarebothsymmetric.Sincethetermsofdi!erentpowersinqcanbecomecomparableinthesetwoforces,di!erente!ectscancomeintoplaydependingontheamplitude.Themechanicalnonlinearityishardeningandtheelectrostaticnonlinearityissoftening.Thenaturalfrequency(fromthelinearterm)includesbothe!ects,andforthepresentdeviceandbiasvoltagethecubictermisdominatedbymechanicale!ectsandishardening,whilethequinticnonlinearityisdominatedbytheelectrostatice!ectsandissoftening.ThisleadstotheinßectionpointontheamplitudedependenceofthevibrationfrequencyseeninFig.3.3.Aftercombiningmechanicalandelectrostatice!ectstogether,thedynamicsofavibra-tionalmodeofasymmetricmicromechanicalresonatorcanbedescribedformoderatemodalamplitudesbythefollowingphenomenologicalmodel¬q+2(#1+#2q2)úq+q(!20+2!0*(t))+"q3+(q5=f(t),(3.4)whereqisagainthemodaldisplacementcoordinate,!0isthenaturalfrequencyofthemode,#1and#2arethecoe"cientsoflinearandnonlinearfriction,and"and(arethecoe"cientsoftheconservativeDu"ngandquinticnonlinearitiesrespectively.Thelineardampingconstant#1determinestheresonatordecayatsmallvibrationamplitudesandisrelatedtotheresonatorqualityfactorasQ=!0/2#1.Notethat!0isprimarilydeÞnedby50!0m,butisslightlyreducedbythepresenceoftheelectrostaticactuation/sensingscheme(electrostaticfrequencytuninge!ect).Tocompletethemodel,wealsoincludeadditive,f(t),andmultiplicative,*(t),noisesources,whichcanbeofthermalornon-thermalorigin.Qualitatively,thenonlinearandnoisetermsinEq.(3.4)havethefollowinge!ects,totheÞrstorder:thesti!nessnonlinearities"and(causeanamplitude-dependentfrequencyshift,thenonlineardamping#2producesanamplitude-dependentdamping(andanon-exponentialdecay),whilethenoiseprocessesmakeboththeamplitudeandfrequencyßuctuateaboutthedeterministicresponseoftheresonator.ThedecayoftheoscillationamplitudeisdeterminedbythetermsofEq.(3.4)proportionalto#1and#2,andalsobyf(t)and*(t).Thus,inastandardspectralmeasurement,",(and#2(andthenoiseterms[78])leadtoadeviationofthespectralcontourfromtheLorentzian,anditisusuallyimpossibletoaccuratelyex-tracttheseparametersfromasinglefrequencysweep.Incontrast,asweshow,aringdownmeasurementisverysensitivetothesenonlinearitiesandnoisesources.IntheabsenceofthenoisetermsinEq.(3.4),thedynamicsoftheresonatorringdownresponsecanbestudiedintermsofslowly-varying(onthetimescale&!!10)resonatoramplitudea(t)andphase/(t)q(t)=a(t)cos(!0t+/(t)),(3.5a)úq(t)=!!0a(t)sin(!0t+/(t)).(3.5b)SubstitutingthischangeofvariablesintoequationEq.(3.4),applyingthemethodofaver-agingandneglectingfast-oscillatingterms[89],weobtainthefollowingequationsofmotion51forthemodalamplitudeandphaseúa=!##1+14#2a2$a,(3.6a)ú/=3"8!0a2+5(16!0a4.(3.6b)FromEq.(3.6a)itisclearthattheamplitudedynamicsareuna!ectedbytheconservativenonlinearities,whilethephasedependsontheamplitudethroughboth"and(,asexpected.Infact,itcanbeshownthattheamplitudedecayisindependentof"and(eveninthepresenceofnoise[49].Thesolutionfortheresonatorvibrationalenvelopecanbeobtainedinclosedform,a(t)=a0e!!1t/%g(t),g(t)=1+14#2#1a20(1!e!2!1t)(3.7)wherea0istheinitialvalueofthemodalamplitudeintheringdownresponse;seeFig.3.4.Usingthissolutionintheexpressionforú/inEq.(3.6b),weobtainthesolutionfortheresonatorphase/(t)=34!0#2#"!10(#13#2$lng(t)+5(a208!0#2g(t)!e!2!1tg(t),(3.8)whereweomittheinitialresonatorphasesinceitisdeterminedbyanarbitrarychoiceoft=0.Theexistenceofclosed-formsolutionsfortheresonatoramplitudeandphaseallowsustodeveloparingdown-basedtechniqueforestimatingtheresonatorparameters,includingconservativeanddissipativenonlinearcoe"cients,seeSection3.3.ItisworthmentioningapossibleoriginofthenonlineardissipationinMEMSresonators.52Accordingtothemicroscopictheoryofdissipationdiscussedin[49],nonlinearfrictionisanessentialconsequenceofthenonlinearinteractionoftheprimaryresonantmodewithphonons,asisalsothecaseforlinearfriction[49].Forhigh-Qresonators,theadequatedescriptionofnonlinearfrictionisinfactgivenbyEq.(3a);inthephenomenologicalpicture,theterm.#2cancomeeitherfromthefrictionforceoftheformofq2úqorúq3,orfromtheircombination.Ifthephononsthatleadtotherelaxationareinthermalequilibrium,thereisaninterrelationbetweenthenonlinearfrictioncoe"cient#2andtheintensityofthenoise*(t)withthespectrumaround2!0,see[33],similartothefamiliarinterrelationbetween#1andtheintensityoftheadditivenoisef(t).3.3Characterizationof!0,!1,!2,"and(parametersAswementionedintheintroductorypartofthechapterandshowedformallyinSection3.2,theresonatorsti!nessanddissipationparametershavequalitativelydi!erente!ectsonthesystemringdownresponse.Inparticular,theresonatordampingcoe"cients#1and#2de-terminetheresonatorvibrationenvelope,asitfollowsdirectlyfromEq.(3.7).Atthesametime,theresonatorinstantaneousvibrationfrequencyisdictatedbytheformofthesystempotentialenergy,which,inturn,isdeterminedbyacombinede!ectofsti!nessparameters!0,"and(,seeEq.(3.6b).Inthislight,byseparatingtheresonatorvibrationenvelopeandthefrequency-amplituderelationshipinthesystemringdownresponse,wecane!ec-tivelyseparatee!ectsoftheresonatorconservativeanddissipativeparameters,seeFig.3.5.Furthermore,weexpectthatastheresonatorvibrationamplitudedecreases,thesystemreturnstoitslinearregime,meaningthattheresonatorvibrationfrequencyapproachesitsamplitude-independentvalue!0andtheenergydecayisessentiallyexponentialasbeing53Figure3.5:Theroad-mapfortheringdown-basedcharacterizationmethod.SeparationofdissipativeandconservativeparametersofthesystemunderstudyisachievedthroughindependentanalysesofthevibrationenvelopeandtheformoftheresonatorÒbackbone.ÓFurthermore,small-andlarge-amplitudecomponentsofthesystemringdownsignalareusedfore!ectivecharacterizationoftheresonatorlinearandnonlinearcoe"cients,whilethejitterintheringdownzero-crossingpointsprovidesinformationaboutnoisesourcespresentinthesystem.dominatedby#1.Asaresult,thesmall-amplitudepartoftheresonatorringdownmea-surementcanbeusedforimmediatecharacterizationofthesystemlinearparameters:thenaturalfrequency!0andlineardampingconstant#1.Incontrast,nonlineartermsintheresonatordissipativeandrestoringforcesa!ectthelarge-amplitudepartoftheresonatorringdown,whichallowsustoestimatethevaluesof",(and#2coe"cientsfromasingleringdownmeasurement.InordertoextracttheresonatorvibrationenvelopeandtheÒbackboneÓcurve(thedependenceoftheinstantaneousoscillationfrequencyontheresonatoramplitude)fromtheringdownresponse,onecanutilizedi!erentpost-processingmethods,suchasfastFouriertransform(FFT)andHilbertdecomposition,see[90Ð92].InSection3.3.1,wehaveemployed54theheterodyningtechniqueforextractingtheresonatorvibrationenvelopesincethismethodalsoallowsustostudythedynamicbehavioroftheresonatorslowly-varyingquadraturesduringtheringdown[83].Toobtaintherelationshipbetweentheresonatorinstantaneousfrequencyandamplitude,wehaveutilizedthemethodofzero-crossingpointsinSection3.3.2.Whilethismethodisveryintuitiveandsimpletouse,themainadvantageofusingzero-crossingpointsisthatthistechniqueprovidesmeansforstudyingthermal,frequencyandmeasurementnoisesources,whichresultinßuctuations,orjitter,ofthelocationsofzero-crossingpointsinthesystemresponse[84].Analysisofthisjitter,asdiscussedinSection3.4,allowsonetorevealallmentionednoisesourcesand,moreimportantly,toestimatenoiseintensities.3.3.1Revealingnonlinearfrictionandextracting#1,#2Inthisworkwehavestartedtheresonatorcharacterizationprocessbyestimatingthesys-temdampingcoe"cientsfromtheresonatorvibrationenvelope.Inordertoextracttheresonatorvibrationalamplitudea(t),therecordedringdowndatashowninFig.3.4hasbeenheterodynedwiththein-phaseandquadraturecomponentsatthefrequencyofself-sustainedoscillationspriortotheringdownmeasurement,!ss=!0+%!(a0),(3.9)seeFig.3.6.Next,wehavepassedmixedsignalsthroughalow-passÞlterinordertoremovethesecondandotherhigher-orderharmonicsandisolateslowly-varying(onthetimescale&!!10)resonatorquadratures,qx(t)andqy(t).Finally,wehavereconstructedtheresonatorvibrationalenvelopeusingthewell-knownrelationshipbetweentheCartesianandpolar55Figure3.6:Post-processingmethodusedforextractingtheresonatorvibrationalenvelopefromtheringdownmeasurement.Initialheterodyningoftheresonatorringdownresponsewithin-phaseandquadraturesignalsat!ssandsubsequentisolationofthesystemslowly-varyingquadraturesallowsonetoreconstructtheresonatorvibrationalamplitudea(t).coordinatesasa(t)=5q2x(t)+q2y(t).(3.10)Inordertorevealthepresenceofnonlineardissipationintheresonantsystemofinterest,itisconvenienttoplottheresonatorvibrationamplitudea(t)onlogarithmicscale,seeFig.3.7.Asexpected,whentheresonatorringsdown,itsamplitudedecreasesandthee!ectof#2onthevibrationalenvelopebecomessmaller.IntheÞnalpartoftheringdownresponse,theresonatormotionisessentiallyindependentof#2andtheresonatorenergydecaysexponentiallywiththespeeddeterminedby#1andproportionaltotheslopeofthecurveinFig.3.7.ByÞndingthisslopeonecanestimatetheresonatorlineardecayrate#1,seeTable3.1.Asexpected,theresonatordecaypatterndeviatesfromasimpleexponentialTable3.1:Estimatedvaluesofthelinearandnonlineardissipationcoe"cientsandcon-servativenonlinearityfordi!erentinitialamplitudes.RingdownmeasurementshavebeenperformedwithVDC=30VandatT=!40"C.a0(mV)#1(s!1)#2(s!1V!2)"(s!2V!2)((s!2V!4)130116.878931.12*1012!2.2*1013175122.484790.93*1012!2.64*101326511960111.05*1012!2.07*101356Figure3.7:MeasuredvibrationalamplitudeoftheDETFresonatorduringitsringdownresponsewithVDC=30V(solidline).Thedashedlinerepresentstheexponentialdecayoftheresonatoramplitudeatlowvibrationamplitudes,extendedthroughouttheamplituderange.Upperinset:nonlinearfrictioncausestheringdownamplitudeenvelopetodeviatefromexponentialatlargeamplitudes,whichcanbeusedforcharacterizationof#2.Lowerinset:thee!ectofnonlineardissipationontheringdownresponsebecomesstrongerastheinitialamplitudeincreases.formatlarge-to-moderatevibrationamplitudesduetothepresenceofnonlineardamping,seetheupperinsetinFig.3.7.Importantly,thise!ectbecomesstrongerastheinitialamplitudeincreases.AnalysisofequationEq.(3.7)showsthatthemaximumofthisdeviation(onlogarithmicscale)reads%=limt(-2ln#a0a(t)exp(#1t)$3=12ln#1+#2a204#1$,(3.11)whichcanbeusedtoestimatethemagnitudeof#2,seeTable3.1.Alternatively,onecanÞtextractedvibrationalamplitudea(t)withthemodeldescribedinequationEq.(3.7)using,forexample,theleast-squaremethodandestimatebothdampingcoe"cients#1and#2.573.3.2Extractingtheresonatorsti"nesscoe#cients!0,",(Aftercharacterizingtheresonatordissipativeparameters,wecanproceedwithestimatingthesystemnaturalfrequency!0aswellasDu"ngandquinticnonlinearities"and(.AccordingtoEq.(3.6b),theresonatorinstantaneousfrequencydependsontheamplitudea(t)asfollows,!(t)=!0+3"8!0a2(t)+5(16!0a4(t).(3.12)ThisbehaviorofthevibrationfrequencycorrespondstodecayalongtheresonatorÒback-boneÓcurveintheamplitude-frequencyspace.Aswediscussedabove,thee!ectsof(and"areexpectedtodiminishastheresonatorentersitslinearregime,wherethemodalfrequencyapproaches!0.Inordertoextracttheresonatorinstantaneousfrequencyandestimate!0,"and(fromasingleringdownresponse,weanalyzethesequenceofthezero-crossingtimes{0k}intheresonatorresponse,i.e.,thepointsthatsatisfyq(0k)=0,asoutlinebelow.Duringtheresonatorringdown,thevibrationamplitudeandfrequencyarenotconstant,butchangesmoothlyintime(ignoringthee!ectsofnoise).Basedonthis,wepartitiontheringdownresponseintoNsegmentsoflength2)/!0%%t%#!11.WeassumethatthevibrationamplitudeandfrequencyremainessentiallyÞxedwithineachsegment,butchangeinadiscretemannerfromonesegmenttothenext.Inthiscaseourprocedurecorrespondstoadiscretizationofsmoothamplitudeandfrequencyfunctions.Inthisspirit,wedeÞnethevibrationperiodassociatedwithithtimesegmentasTi=2ti,ni!ti,1ni!1=2)!i,(3.13)whereti,jisthejthzero-crossingpointandniisthenumberofzero-crossingpointswithin58theithinterval.ExtractedvaluesofthevibrationperiodTiarethenusedtocomputetheresonatorquasi-instantaneousvibrationfrequencyshowninFig.3.8forN=50.NotethatFigure3.8:Vibrationfrequencyoftheresonatorduringtheringdownasafunctionofitsamplitudefordi!erentvaluesofinitialamplitude.Duetoamplitude-dependentfrequencypulling,thefrequencyvarieswithamplitude,allowingcharacterizationof!0,"and(fromasinglemeasurement.Discretedotsrepresentextractedvaluesofthevibrationfrequency!k/2)duringtheringdownresponse(errorbars'1!10Hz,notshown).ThesolidlinesrepresentthecurveÞtsofextractedÒbackbonesÓusingthemodelinEq.(3.12).partitioningoftheresonatorringdownresultsinthevibrationfrequencybeingaveragedover%t,andmentionedabovejitterinzero-crossingpointsisessentiallyaveragedout.Asaresult,bychoosingspeciÞcvalueofNonewouldalwayscompromisebetweentheamountofdatapointsintheresonatorÒbackboneÓandtheamountofjitterthatisbeingaveraged/retained.Theformoftheresonatorfrequency-amplitudedependenceexhibitsseveralfeaturesofinterest.First,asexpected,thevalueofthevibrationfrequencyatthebeginningoftheringdown,!1,dependsontheinitialvibrationamplitudea0andonsti!nessnonlinearities"and(,duetotheamplitude-dependentfrequencypulling.Astheresonatormotiondecays,thevibrationfrequencychangesinamonotonic(for"(>0oriftheinitialamplitudeisbelowtheturningpoint)ornon-monotonic(for"(<0andtheinitialamplitudeabovetheturning59point)mannerandgraduallyapproaches!0,fromwhereweestimatedthelinearresonantfrequencytobef0'1.2172MHz.Afterobtainingthevibrationperiod(andfrequency)asafunctionoftime,wecanestimatetheresonatorDu"ngandquinticnonlinearitiesbyÞttingtheamplitude-dependentfrequencyshift%!(a)=!(a)!!0(3.14)tothemodeldescribedinEq.(3.12)usingtheleast-squareÞttingmethod,seeTable3.1.Itisworthmentioningthatthezero-crossing-basedmethodpresentedherecanbeeasilyextendedandusedtocapturetheresonatorsti!nessnonlinearitiesofordershigherthan5.Thesehigher-ordernonlinearitieswillresultinadditionaltermsinEqs.(3.6b)and(3.12)thatdictatethebehaviorof%!(a).Clearly,thismethodofzero-crossingpointsisaccurateandverysimplefromacomputationalpointofview,asitallowsonetoextract!0,"and(directlyfromtherawdatawithoutinvolvingtheFouriertransformofasignalthathasanon-stationaryand,generally,non-monotonicvibrationfrequency.3.3.3Separatingmechanicalandelectrostatice"ectsinMEMSresonatorsThedynamicbehaviorofMEMSresonatorswithelectrostatictransductionisdeterminedbybothmechanicalrestoringforcesarisingfromelasticdeformationsintheresonatorbodyandtheelectrostaticinteractionoftheresonatorwithattendantelectrodes,whicharisesfromaÞnitebiasvoltage[45].Knowledgeofbothmechanicalandelectrostaticcontributionstotheresonatoramplitude-dependentfrequencypullinge!ectbecomescrucialforaccuratetuningoftheresonatorparameters,forexample,formaximizingtheresonatorlineardynamic60rangeusingsystematicshapeoptimizationoftheresonatorbody[43].Whiledecouplingofmechanical/kinematicandelectrostatice!ectscanbeachievedinelectrostaticMEMSbyusing,forexample,apiezoelectricstackactuator[93]andopticalsensingmethod[94]toachieveapurelymechanicalresponse,thistechniquerequiresconsiderablechangeoftheresonatordriving/sensingapparatusandadditionalmeasurementtools.Moreover,suchamethodisnotapplicableforresonantsystemsthatareencapsulated[87].Inthislight,thereisaneedforasimpleandreliablecharacterizationmethodthatallowsonetoestimatebothmechanicalandelectrostaticcontributionstothelinearandnonlinearresonatorsti!nessparameters,whileemployinganelectrostaticdrive/sensesetup.Asweshowbelow,itispossibletoadapttheringdown-basedcharacterizationmethoddescribedaboveforseparatingmechanicalandelectrostatice!ectsinMEMSresonatorswithelectrostaticactuationand/orsensingschemes.Mechanicalforcesinßexural-moderesonatorsvibratinginasingleplaneoriginatefromtwoqualitativelydi!erentmechanisms:resonatorbending(ßexure)andmid-linestretching(extension).Inparticular,restoringforcesduetoresonatorbendingnormallydominatethesystemdynamicsatvibrationalamplitudesthataremuchsmallerthantheresonatorthicknesshinthedirectionofvibration.Thee!ectofmid-linestretching,ontheotherhand,isnonlinearintheresonatordisplacementandrevealsitselfathighervibrationamplitudes[95],anditresultsinanamplitude-dependentshiftoftheresonatorvibrationalfrequency[96].Physically,thismanifestsitselfinane!ectivehardeningoftheresonatorsti!ness,resultinginanincreaseinthevibrationfrequencyastheamplitudeincreases.Here,theresonatorthicknesshservesasanimportantcharacteristiclengthscalethatdictatestheappearanceofnonlineare!ectsintheresonatorsti!nessthatariseduetonormalstressesoccurringintheresonatorbodyduringitsvibratoryresponse.61Unlikemechanicalrestoringforces,bothlinearandnonlinearelectrostatice!ectsorigi-natefromthesamephysicalprocessofelectrostaticinteractionoftheresonatorbodywiththeattendantelectrodes.Thee!ectofthisinteractionisexpressedbyasingleelectrostaticpotentialfunction,whichdivergeswhentheresonatoramplitudeapproachestheelectrodegapsized.Sincetheresonatordisplacementy(x,t)cannotphysicallyexceedd1,onecanexpandtheelectrostaticpotentialabouttheresonatorequilibriumpositionandobtaincorre-spondingelectrostaticcontributionstobothlinearandnonlinearresonatorrestoringforces.Importantly,thelargertheresonatordisplacement,themoretermsonehastokeepintheexpansionoftheelectrostaticpotentialforaccuratemodelingofthesystemdynamics.Inthislight,theelectrodegapsizedisanotherimportantlengthscalethatdeterminestheappearanceofcertainelectrostatice!ectsinthepotential,or,equivalently,intheexpansionoftheattendantrestoringforce.Typically,theelectrodegapsizeisthesmalleroftheselengthscalesinMEMSresonatorswithelectrostatictransductionandhighfrequency,sothatd%hoftenholds.Consequently,dependingontherelativemagnitudesofhandd,onemayhavetoretainmechanicaland/orelectrostaticnonlineartermsofdi!erentordersintheresonatordisplacement,aswasdoneinSection3.2.Physically,theelectrostaticforcespulltheresonatorawayfromitsequilibriumpositionandthereforereducetheresonatorsti!nessandcausesofteningoftheresonatorfrequencyresponse[99].Asaresult,theresonatordynamicbehaviorisdeterminedbythecompetitionofmechanical(hardening)andelectrostatic(softening)e!ects,andthisinterplayofdi!erentrestoringmechanismscanresultinpeculiarformsoftheresonatoramplitude-frequencyresponse,suchastheoneshowninFig.3.3.1Infact,y(x,t)"0.7disfrequentlysatisÞedinordertoavoidthepull-inphenomenon[97,98]and/orpreventthedevicefromelectricalshorting.62Byaccountingfortheaforementionedmechanicalandelectrostaticforcesactingontheresonator,wecanobtainthefollowingmodelfortheresonatornaturalfrequencyandamplitude-dependentfrequencyshift(seeAppendixBfordetails),!0="!20m!3Ced,(3.15a)%!(a)=%!m(a)+%!e(a),(3.15b)where!0mistheresonatormechanicalnaturalfrequencyandCe=2-0V2b3%hd2(3.16)isthestrengthofelectrostaticpotentialthatdependsonthebeammassdensity%,beamin-planethicknessh,electrodegapsized,andtheappliedbiasvoltageVb.InEq.(3.15b),%!m,e(a)arethecorrectionstothemodalvibrationfrequencyduetononlinearmechan-icalandelectrostatice!ects,respectively.Closed-formexpressionsfor%!m,e(a)arequitecumbersomeandarerelegatedtoAppendixB.Inpractice,however,modalamplitudesinhigh-frequencyßexural-moderesonatorswithaÒclamped-clampedbeamÓconÞgurationfrequentlysatisfya/h%a/d<1,whichallowsustoapproximatethemagnitudesofthemechanicalandelectrostaticnonlinearfrequencypulling,asfollows,%!m(a)'3"m8!0a2,"m=)4E3%L4,(3.17a)%!e(a)'3"e8!0a2+5(e16!0a4,"e=!35Ce8d3,(e=!693Ce128d5.(3.17b)Notethatwhileweapproximate%!m(a)inEq.(3.17a)onlybytheÞrstnon-vanishing63term.a2,whichcomesfromthecubic(Du"ng)termintheresonatorrestoringforce,weretaincontributionsfrombothcubicandquinticelectrostaticnonlinearitiesin%!e(a).ThevalidityofthisapproximationhasbeenveriÞedintheanalysis;infact,evenwhena'd,thecontributionofthemechanicalquinticterm(duetothemid-linestretching)totheresonatorfrequencyshift,whichis.a4in%!m(a),isseveralordersofmagnitudesmallerthanthecorrespondingcontributionofthemechanicalDu"ngterm,i.e.,5|(m|16!0d4%3"m8!0d2.Incontrast,theelectrostaticpotentialdivergesasa(dandtheelectrostaticquintictermdominatesthesofteningbehavioroftheresonatorfrequencyathighervibrationamplitudes.Inthislight,thecoe"cientoftheresonatorquinticsti!nesstermisapproximatedas('(e,whichwekeepintheanalysisinordertoadequatelydescribethehardening-to-softeningbehavioroftheresonatorfrequencyresponseshowninFig.3.3.Thesequalitativeandquantitativedi!erencesbetweennonlinearmechanicalandelectro-staticforcesallowustocharacterizebothe!ectsusingasingleresonatorringdownresponse.Inparticular,ourabilitytocharacterizetheresonatorlumpedsti!nessparameters:!0,"and(allowsustoestimatethestrengthoftheelectrostaticpotentialCeusingthelastexpres-sioninEq.(3.17b).KnowingCe,wecanfurthercalculatethecorrespondingelectrostaticcontributionstotheresonatornaturalfrequencyandDu"ngnonlinearity"e.Finally,byaccountingfortheseelectrostaticcontributions,wecandeterminethemechanicalnaturalfrequency,!0m="!20+3Ced,(3.18)64Table3.2:Estimatedvaluesofthemechanicalnaturalfrequency!0m,themechanicalcon-tributiontotheDu"ngnonlinearity"m,andtheelectrostaticpotentialstrengthCeintheDA-DETFresonator(seeFig.3.1)forthreeindependentringdownmeasurementswiththesamebiasvoltage,VDC=30V.a0(mV)!0m/2)(Hz)"m(m!2s!2)Ce(m/s2)1301.241*1061.237*10246.51*1051751.238*1061.403*10247.81*1052651.242*1061.155*10246.1*105andtheDu"ngnonlinearity,"m="+35Ce8d3.(3.19)Table3.2showssomedataobtainedfromameasuredringdownresponse.Itisworthmentioningthatthismethodforseparatingmechanicalandelectrostatice!ectsisnotlimitedtoresonantsystemswithßexuralvibrationalmodesandshouldbeapplicabletoothertypesofMEMSresonatorsincluding,forexample,bulk-moderesonators.Whilethephysicaloriginsofmechanicalnonlinearitiescanincludebothgeometricandmateriale!ects,theirappearanceintheresonatormodelisdeterminedbyrelativelylargeresonatordimen-sions,suchastheresonatorthickness(forßexuralmodes)orwidth(forbulkmodes).Incontrast,electrostaticforcesstilloriginatefromtheelectrostaticpotentialwiththecharac-teristiclengthequaltotheelectrodegapsize,whichremainsthesmallestgeometricfeatureofMEMSresonatorswithelectrostatictransduction.Asaresult,theresonatorquinticsti!nesstermshouldstillbedominatedbytheelectrostaticpotentialand,bydrivingtheresonatortosu"cientlylargevibrationamplitude,oneshouldbeabletocharacterizethestrengthsoftheelectrostaticpotentialandmechanicalcontributionstotheresonatorsti!nessparameters.653.4Characterizationofthermal,frequencyandmea-surementnoisesourcesTothispointwehavediscussedhowonecancharacterizethedeterministicresonatorpa-rameters,bothdissipativeandconservative,usingasingleringdownresponsemeasurement.Ofcourse,theresonatordynamicsarenotdeterministicbutareinßuencedbyinteractionsoftheresonatorbodywithanambientenvironment.Theseinteractions,aswedescribedinSection1.2,havedi!erentphysicalorigins;however,thecollectivee!ectoftheseinterac-tionsontheresonatordynamicbehaviorcanbemodeledintheformofrandomexcitationsornoises.1Dependingonanoisenatureandorigin,wecanmodelthemasadditive,orthermal,noise,i.e.,beingindependentoftheresonatorinstantaneousdisplacementandvelocity,or,asmultiplicative,orfrequency,noise.Also,whenonemeasurestheresonatorresponse,theimperfectionsofthedetectingdevicetypicallyresultinanadditionalnoisethatperturbsthemeasuredsignal,whichwemodelasmeasurementnoise.Thee!ectsofthesenoisesourcesontheresonatordynamicsandthereadoutsignalareillustratedschematicallyinFig.3.9,whereweassumethatmeasurementnoisedoesnota!ecttheresonatordynamicsitself,butmerelycorruptsthereadoutsignal.InthissectionweshowhowwecanusetheresonatorFigure3.9:Schematicrepresentationofthee!ectsofthermal(f(t))andfrequency(*(t))noisesontheresonatordynamics,andthee!ectofmeasurementnoise(µ(t))onthequalityofthereadoutsignal.1Thesesameinteractionsarealsothesourceofthedampinge"ectsalreadystudied,wheredissipationisthemeancomponentoftheserandome"ectsandtheßuctuatingcomponentsaremodeledasnoise[49].66ringdownresponseforcharacterizationofmeasurement,additive,andmultiplicativenoisesources.Inparticular,wedemonstratethatthesenoisescausejitterinthezero-crossingpointsintheresonatorringdownand,moreimportantly,theyhavedi!erentsignaturesintheresonatortimingjitter,whichallowsustorevealandseparatetheire!ectsandestimatetheirindividualnoiseintensities.Intermsofnotation,weuse(á)dtodenotethedeterministicvalueofthequantityintheparenthesis,suchastheresonatoramplitude,a,orfrequency,!,while(á)rand(á)µwilldenotevariationsofthisquantityinducedbyresonator-relatedandmeasurementnoisesources,respectively.Thejitterinthezero-crossingpointsmanifestsitselfinthefactthatthelocationofanarbitraryzero-crossingpoint0kdeviatesfromitsdeterministicvalue,0dk,obtainedbysolvingEq.(3.8)for/(0dk)=)k,as0k=0dk+#0rk+#0µk,(3.20)where#0rkand#0µkarerandomshiftsofthelocationof0kduetoresonator-relatednoises(thermalandfrequency)andmeasurementnoise,respectively.Inordertoanalyzethejitterinzero-crossingpointsduringtheresonatorringdown,wehavetoaccountforthenon-stationarynatureoftheresonatorresponseitself,whichultimatelyleadstonon-stationarityoftheresonatorfrequencyßuctuations.Inthislight,similartotheapproachusedinSection3.3.2,wepartitiontheringdownmeasurementintoNsegmentsofwidth!!10%%t%#!11,andassignindexi,unlessspeciÞedotherwise,todenotethecountingnumberoftheseringdownsegments.Next,wecalculatethefollowingtimeintervalswithintheithringdownsegmentT(i,j,k)=0i,j+k!0i,j=Td(i,j,k)+#T(i,j,k),(3.21)67#T(i,j,k)=#Tr(i,j,k)+#Tµ(i,j,k),(3.22)#Tr(i,j,k)=#0ri,j+k!#0ri,j,(3.23)#Tµ(i,j,k)=#0µi,j+k!#0µi,j,(3.24)where0i,jisthejthzero-crossingpointwithintheithringdownsegment.Physically,T(i,j,k)correspondstothetimethatittakesfortheresonatortoaccumulate)kradiansofphasestartingfrom0i,jonthetimeline.Inwhatfollows,weshallassumethatmeasurementnoiseisindependentanduncorrelatedwiththeresonator-relatednoisesourcesf(t)and*(t),whichresultsinthefactthat#0rkand#0µkarealsouncorrelated,i.e.,##0rk#0µk$k20.(3.25)Asaresult,wehavefor#T(i,j,k)##T2(i,j,k)$j=##T2r(i,j,k)$j+##T2µ(i,j.k)$j,(3.26)where#á$jdenotesaveragingoverthejindex,which,inourcase,isassumedtobeequiv-alenttoensembleaveragingfordi!erenttimeintervalsT(i,j,k)withinthesameringdownsegment.3.4.1Measurementnoisetimingjitter#Tµ(i,j,k)Themeasuredresonatorringdownresponseisadigitialrepresentationoftheresonatordis-placementthatisestimated,usinganappropriatedetectingdevice,atspeciÞedinstancesoftimetp.Inthiswork,weassumethatmeasurementnoisedoesnota!ecttheresonator68internaldynamicsandthusmodelthee!ectsofadetectionschemeonthereadoutsignalinthefollowingway,qm(tp)=qr(tp)+µ(tp),(3.27)whereqr(tp)=qrpisthesignalthatdescribestheresonatordynamicsa!ectedbyadditiveandmultiplicativenoisesources,whichwewoulddetectgivenanidealmeasurementappa-ratus,µ(tp)=µpisarandomcomponentaddedtoqrpduetothepresenceofmeasurementnoise,andqm(tp)=qmpisthereadoutoftheresonatordisplacement.Inwhatfollows,weassumeµptobearandomvariablehavingasymmetricprobabilitydistributionwiththefollowingstatistics,#µp$p=0,#µpµs$ps=M#ps,(3.28)where#psisKroneckerdeltasymbol.Asweshallshowlater,thismodel,despiteitsrelativesimplicity,allowsonetoexplaincertainnoise-inducedfeaturesoftheringdownresponsesobtainedfromtheDA-DETFresonator.GiventhesignalintheformofEq.(3.27),asthemeasuredresonatorringdown,weconsiderthesequenceofmeasuredzero-crossingpoints{0mi,j}satisfyingqm(0mi,j)=0.Wedenotethesamplingrateatwhichmeasurementsareperformedasfs,andassumethatitismuchhigherthantheresonatorvibrationfrequency,i.e.,f0%fs,sothatthezero-crossingpoints0mi,jcanbeobtainedfromthemeasureddatawithsu"cientaccuracyusinglinearinterpolationoftheresonatorresponseinthevicinityofqmi,j=0.Thus,foranarbitraryzero-crossingpoint,consideringFig.3.10,weseethat0mi,j=qm2t1!qm1t2qm2!qm1=(qr2+µ2)t1!(qr1+µ1)t2qr2!qr1+µ2!µ1'qr2t1!qr1t2qr2!qr121!µ2!µ1qr2!qr13+µ2t1!µ1t2qr2!qr1,(3.29)69Figure3.10:Measurementnoisecontributiontothejitterinthezero-crossingpointsintheresonatorringdownresponse.Asexpected,theresonatormotionwithlargeramplitudeislesssusceptibletothee!ectsofmeasurementnoise,whichisthefundamentalnecessityforhighsignal-to-noiseratioforprecisefrequencygeneratorsandclocks.where,withoutlossofgenerality,qm1>0andqm2<0arethemeasuredringdowndis-placementsattimest1andt2,respectively,andwehaveassumedthatthemeasurementnoiseisweakascomparedwiththeoriginalsignalduetotheresonatorvibrations,i.e.,|µ2!µ1|%|qr2!qr1|.Byintroducing0ri,j=qr2t1!qr1t2qr2!qr1(3.30)asthezero-crossingpointintheresonatorresponsebeforeameasurementistaken,Eq.(3.29)canberewrittenas0mi,j=0ri,j+#0µi,j,#0µi,j=µ2t1!0ri,jqr2!qr1!µ1t2!0ri,jqr2!qr1,(3.31)where#0µi,jisthejitterinducedbythemeasurementnoisetothelocationofthejthzero-crossingpointontheithringdownsegment.Assumingthepropertiesofµpdescribedin70Eq.(3.28),thestatisticalpropertiesof#0µi,jare##0µi,j$j=0,(3.32a)##0µi,j#0µi,k$jk=M#jk2/(t1!0ri,j)2(qr2!qr1)20j+/(t2!0ri,j)2(qr2!qr1)20j3.(3.32b)InordertocalculatetheexpressioninthesquarebracketsinEq.(3.32b),weassumethattheresonator-relatednoisesonlyslightlyperturbtheresonatormotion,whichallowsustoapproximatethedenominatoras|qr2!qr1|'adi!di(t2!t1),(3.33)whereadiand!diaretheresonatordeterministic(averaged)amplitudeandvibrationfre-quencyduringtheithringdownsegment.DuetotheÞxedsamplingtimeandthefactthattheresonatorfrequency,evenitsdeterministiccomponent(duetofrequencypulling),isafunctionoftime,weassumethatthelocationof0ri,jontheinterval[t1,t2]isarandomvariablewiththeuniformdistributionfordi!erentjÕs,i.e.,prob(0ri,j0[t,t+dt])=dtt2!t1,fort0[t1,t2].(3.34)Thelatterresultsin#(t1!0ri,j)2$j=#(t2!0ri,j)2$j=13(t2!t1)2,(3.35)71andwecanrewriteEq.(3.32b)as##02µi,j$j'23Ma2di!2di.(3.36)Oncewehaveestablishedthestatisticalpropertiesof#0µi,j,wecandothesamefor#Tµ(i,j,k).Inparticular,fromEqs.(3.21),(3.32a)and(3.36)wehave##Tµ(i,j,k)$j=0,(3.37a)##T2µ(i,j,k)$j'43Ma2di!2di.(3.37b)Notethatthecontributionofmeasurementnoisetothejitterinthezero-crossingpointsisindependentofktoleadingorder.AswewillshowinSections3.4.2and3.4.3,thermalandfrequencynoisesresultinaqualitativelydi!erente!ect;inparticular,theirleading-ordercontributiontothetimingjitterisproportionaltok.Thisimportantdi!erenceallowsustoseparatemeasurementnoisefromresonator-relatednoisesources,aswewillconcludeinSection3.4.4.3.4.2ResonatorphaseßuctuationsduringringdownItiswell-knownthatbothadditiveandmultiplicativenoisesourcesa!ecttheresonatordynamicsbyinducingßuctuationsintheresonatoramplitudeandphase[49,50].Atthesametime,theresonatorphaseisintrinsicallyrelatedtotimethroughthesystemvibrationfrequency,whichservesasane!ectivescalingfactorbetweenthesetwoquantities.Asaresult,ifweusetheresonatorphase,or,moreprecisely,thephasedi!erence,asatooltomeasuretime,wewillmeasureitwithacertainerrorduetotheresonator/oscillatorphase72noise.Inthislight,beforeweproceedtodeÞningthetimingjitterduetoresonatorphaseßuctuations,weshallbrießyrevisithowadditiveandmultiplicativenoisesourcescontributetotheresonatorfrequencyand,consequently,tophasenoise.Inordertoillustratethesecontributions,wesimplifytheoriginalresonatormodel,describedinEq.(3.4),asfollows,¬q+2#1úq+q(!20+2!0*(t)+"q2)=f(t),(3.38)wheref(t)and*(t)areadditive(thermal)andmultiplicative(frequency)noisesourcesactingonthesystem.ThemainadvantageofthemodelinEq.(3.38)overtheoneinEq.(3.4)isthelinearityofthedissipativeforce,whichgreatlysimpliÞestheforegoinganalysis.However,weshallreturntothediscussionoflinearvs.nonlineardampinginSection3.4.4,wherewediscussthenoisecharacterizationprotocol.NotethatinEq.(3.38)wehavealsoomittedthequintictermintheresonatorrestoringforcesincethistermdoesnotchangethequalitativepictureoftheresonatoramplitudeandphaseßuctuations;ifnecessary,itcanbeincludedinastraightforwardmanner.Inordertoanalyzetheresonatorphase(andamplitude)ßuctuations,weemployclassicalvanderPoltransformationtotheslowly-varyingcomplexamplitudeu(t),q(t)=u(t)ei!0t+u+(t)e!i!0t,úq(t)=i!0[u(t)ei!0t!u+(t)e!i!0t].(3.39)ByinsertingEq.(3.39)inEq.(3.38)andneglectingfast-oscillatingterms,weobtainthefollowingequationofmotionforu(t),[49],úu=!#1u+3i"2!0|u|2u+iu*0(t)!i2!0f!0(t),(3.40)73where*0(t)representstheslowly-varyingcomponentof*(t)andf!0(t)istheslowly-varyingcomponentoff(t)exp(!i!0t).Byexpressingtheslowly-varyingcomplexamplitudeinthepolarform,u=12aei*,(3.41)wecanseparatetheresonatoramplitudea(t)andphase/(t)asúa=!#1a+1!0#f(i)!0(t)cos/!f(r)!0(t)sin/$,(3.42a)ú/=3"8!0a2+*0(t)!1a!0#f(r)!0(t)cos/+f(i)!0(t)sin/$,(3.42b)wherewehaveexpressedthermalnoiseintherotatingframeas,f!0(t)=f(r)!0(t)+if(i)!0(t).(3.43)Intheabsenceofadditiveandmultiplicativenoisesourcesthesolutiontotheresonatoramplitudeandphasebecomesad(t)=a0e!!1t,(3.44a)/d(t)=3"a2016#1!0#1!e!2!1t$,(3.44b)which,ofcourse,canbeobtainedfromEqs.(3.7)and(3.8)byletting(=0andtakingthelimit#2(0.Next,weassumethattheadditiveandmultiplicativenoisesourcesweaklyperturbtheresonatordynamicsaboutitsdeterministicdecayandrepresenttheire!ectontheresonator74dynamicsasfollows,a(t)=ad(t)+#a(t),(3.45a)/(t)=/d(t)+#/(t),(3.45b)where#a(t)and#/(t)aresmall(|#a(t)|%ad(t)and|#/(t)|%/d(t))perturbationstothedeterministicresonatorsolutioninEqs.(3.44a)and(3.44b).ThisassumptionaboutÒsmallnessÓofamplitudeandphaseßuctuationsdoesnot,ofcourse,holdwhentheresonatoramplitudebecomescomparablewith5#a2th$,theroot-mean-squareofthesystemthermalvibrationsinducedbyf(t).Thus,theanalysispresentedinthissectionisapplicabletotheinitialpartoftheringdown,wheretheresonatorisfarfromthermalequilibrium.Inthiscase,wecanlinearizeEqs.(3.42a)and(3.42b)aboutthenoise-freesolution,whichyieldsthefollowingequationsofmotionfor#a(t)and#/(t),#úa=!#1#a+1!0#f(i)!0(t)cos/d!f(r)!0(t)sin/d$,(3.46a)#ú/=3"4!0ad#a+*0(t)!1ad!0#f(r)!0(t)cos/d+f(i)!0(t)sin/d$.(3.46b)InEqs.(3.46a)and(3.46b),weassumethatthermalandfrequencynoisesourcesarezero-meanrandomprocessesthatareindependentanduncorrelated,i.e.,#*0(t1)f(t2)$20.Additionally,weassumethatthecorrelationtimeofeachoftheserandomprocessestobemuchsmallerthan#!11,whichallowsustoapproximatethesenoisesourcesasdelta-correlated,i.e.,#*0(t1)*0(t2)$=D&#(t1!t2),#f(t1)f(t2)$=Df#(t1!t2).75Usingtheseassumptions,onecanshowthatthethermalnoisecomponentsf(r)!0(t)andf(i)!0(t)possessthefollowingstatisticalproperties[49],#f(r)!0(t)$=#f(i)!0(t)$=0,#f(r)!0(t1)f(i)!0(t2)$=0,#f(r)!0(t1)f(r)!0(t2)$=#f(i)!0(t1)f(i)!0(t2)$=Df2#(t1!t2).(3.47)FromEq.(3.46b)itfollowsthatinordertocalculatestatisticsoftheresonatorphaseßuctuations,i.e.,##/(t)$and##/2(t)$(inthisworkwedonotneedthetwo-timecorrelationfunction##/(t1)#/(t2)$),wemustÞrstcalculatethestatisticalpropertiesoftheresonatoramplitudeßuctuations.Fortunately,Eq.(3.46a)canbesolvedinastraightforwardwayandyields#a(t)=1!0,t0dt,e!!1(t!t,)#f(i)!0(t,)cos/d(t,)!f(r)!0(t,)sin/d(t,)$,(3.48)which,aftersomealgebra,leadstothefollowingexpressionfortheresonatoramplitudecorrelationfunction##a(t1)#a(t2)$=Df4#1!202e!!1|t1!t2|!e!!1(t1+t2)3.(3.49)Wecannowdeterminethestatisticalpropertiesoftheresonatorphase.FromEqs.(3.46b)and(3.47),wehave##/(t)$=0,(3.50a)##/2(t)$=,t0dt1,t0dt2##ú/(t1)#ú/(t2)$=D&t+Df4#1!202e2!1t!1a20+9"2a2064#21!20#1!4#1te!2!1t!e!4!1t$3.(3.50b)76Equation(3.50b)showsthee!ectsofboththeadditiveandmultiplicativenoisesourcesonthevarianceoftheresonatorphase.Asexpected,theresonatorfrequencynoise*0(t)resultsinasimpledi!usionoftheresonatorphase.Incontrast,thethermalnoisecontributionismorecomplicated,duetothenon-stationarynatureoftheresonatorringdownresponse.Inthiswork,however,weareinterestedintheresonatorphaseßuctuationsonthetimescale%t%#!11,inwhichcase##/2(t)$canbeapproximatedasfollows,##/2(t)$'t#D&+Df2!20a20$+Df#12!20a20t2+t3#3Df"2a2032!40+Df#213!20a20$,(3.51)wherewehaveexpandedalltermsinEq.(3.50b)to3rdorderint,whichisnecessarytoillustratetheleadingordere!ectoftheresonatornonlinearity"ontheresonatorphaseßuctuations.FromEq.(3.51)twoimportantconclusionscanbemade.First,theresonatorphasevariancegoesto0ast(0,whichfollowsfromthesimplefactthattheresonatorphaseistheintegraloftheresonatorinstantaneousfrequency.Second,thephasedi!usionconstant,deÞnedasthecoe"cientofthelineartermintinEq.(3.51),hastwocontributionscorrespondingtoadditiveandmultiplicativenoisesources.Importantly,whilethelatterisindependentoftheresonatorinitialamplitude,thecontributionofthethermalnoisegrowsastheinitialamplitudedecreases.Weshallreturntothisimportantdi!erencebetweenthermalandfrequencynoisesourceswhenwediscussthenoisecharacterizationprotocolinSection3.4.4.3.4.3Phasenoisetimingjitter#Tr(i,j,k)Intheprevioussectionwehaveshownhowresonator-relatednoisesourcescontributetoßuctuationsoftheresonatorphase.Hereweshowthatthephasenoise#/(t),inturn,causes77additionaljitter,denoted#Tr(i,j,k),inthezero-crossingpoints,whichisinadditiontothatcausedbymeasurementnoise,#Tµ(i,j,k).Toshowthis,considerthefollowingphasedi!erence,)k=/(0ri,j+k)!/(0ri,j)=,$ri,j+k$ri,jdt!i(t)=,$ri,j+k$ri,jdt!di(t)+,$ri,j+k$ri,jdt#ú/i(t),(3.52)where#ú/i(t)representsthecumulativee!ectofadditiveandmultiplicativenoisesources;seeEq.(3.46b).Byrearrangingtermswecanexpresstheresonatorphaseßuctuationsinthefollowingway,#/(i,j,k)=,$ri,j+k$ri,jdt#ú/i(t)=)k!,$ri,j+k$ri,jdt!di(t)')k!!di(Td(i,j,k)+#Tr(i,j,k)),(3.53)wherewehaveassumedasu"cientlyÞnepartitioningoftheresonatorringdownsignal,sothat!di(t)'!di=const.ThesameassumptionalsoleadstoTd(i,j,k)'Tdik2()k'!diTd(i,j,k),#/(i,j,k)'!!di#Tr(i,j,k).(3.54)Asaresult,thestatisticalpropertiesof#Tr(i,j,k)become##Tr(i,j,k)$j=!!!1di##/(i,j,k)$j=0,##T2r(i,j,k)$j=!!2di##/2(i,j,k)$j.(3.55)78Finally,usingEqs.(3.26),(3.51)and(3.55),wecanexpressthestatisticalpropertiesofthetotaltimingjitterinthemeasuredresonatorringdownresponseas##T2(i,j,k)$j'43Ma2di!2di+##/2(i,j,k)$j!2di'43Ma2di!2di+)k!3di#D&+Df2!2dia2di$,for)k!di%#!11,(3.56)wherewehave,inthespiritoftheringdownpartitioning,discretizedtimetinEq.(3.51)as(k/!diandretainedonlythelowest-order(ink)termdescribingthetemporalevolutionofthevarianceofthemeasuredtimeintervalsT(i,j,k),whichisduetothepresenceofnoisesµ(t),f(t)and*(t).3.4.4NoisecharacterizationprotocolInthissectionwesummarizeourunderstandingofhowthermal,frequency,andmeasurementnoisesourcesa!ecttheresonatorringdownresponse,obtainedinSections3.4.1to3.4.3,andformulatethenoisecharacterizationprotocolusingEq.(3.56)asthecentralresult.First,asfollowsfromEq.(3.56),themeasurementandadditivenoisecontributionstothetimingjitterdependontheresonatorvibrationamplitude.Thus,bypartitioningtheresonatorringdownresponseintoNsegmentsoftimelength%t,wee!ectivelyobtainseveralringdownsub-measurementsthathavedi!erentinitialamplitudesandvibrationfrequencies,whereeachisassumedtobeessentiallyconstantovereachsegment.Next,withintheithsegmentoftheringdownresponse(i=1:N),wecalculatethenumberoftimeintervalsT(i,j,k)thatittakesfortheresonatortoaccumulate)kradiansofphaseor,equivalently,toperformkhalf-cycles;seeFig.3.11.Thesecollectedtimeintervalsarethenanalyzedstatisticallyandthethevariance##T2(i,j,k)$j79Figure3.11:TimeintervalsT(1,j,4)calculatedwithinthe1stringdownsegment.Duetothepresenceofmeasurement,thermal,andfrequencynoises,thesetimeintervalsslightlydi!erinlength,which,whenanalyzedstatistically,allowsonetorevealthesenoisesourcesandestimatetheirintensities.iscalculated.Hereitisimportanttonotethattheringdownpartitioningshouldbedoneinawaythatensuresstatisticalsu"ciencyforT(i,j,k)withdi!erentvaluesofk.Forinstance,theDA-DETFresonatorusedinthisstudyhasaqualityfactorQ'104.Par-titioningtheringdownintoN=50segmentsresultsineachsegmenthaving'200fullcycles,or'400zero-crossingpoints.Fromastatisticalpointofview,thismeansthatwehaveenoughzero-crossingpointstocalculatetimeintervalsT(i,j,k)fork=1:40andreconstructthe##T2(i,j,k)$jfunction.Thisfunction,asfollowsfromEq.(3.56),hasaconstantcontributionduetothemeasurementnoise,andak!dependentpartthatisduetothethermalandfrequencynoises;seeFig.3.12.Thenon-vanishingcontributiontothejitterinthezero-crossingpointsisthesignatureofmeasurementnoiseandcanthereforebeusedforseparatingoutthee!ectsofmeasurementnoiseandestimatingitsintensityM.Inparticular,themeasurementnoiseintensitycanbeestimatedbyplottingtheconstant80Figure3.12:Qualitativebehaviorof##T2(i,j,k)$jasafunctionofkwithintheithringdownsegment.Importantly,themeasurementnoisecontributiontothetimingjitterdoesnotvanishask(0,whichisusedforestimatingthemeasurementnoiseintensityM.Incontrast,thecorrespondingcontributionsofthermalandfrequencynoisesourcestothetimingjitterarebothk!dependent,buthavedi!erentmagnitudesindi!erentringdownregimesduetotheirdi!eringamplitudedependence.termin##T2(i,j,k)$j,extractedfromtheringdownsegments,asafunctionoftheresonatoramplitude,andÞttingtheresultingcurvewiththepowerlawformgiveninEq.(3.56);seeFig.3.13a.Next,byÞttingthek!dependentpartof##T2(i,j,k)$jwithapolynomialof3rdorder,forsu"cientlysmallk,andisolatingtheterm.k,wecanextracttheresonatorphasedif-fusionconstantdeterminedbytheintensitiesofthermalandfrequencynoisesources.Asmentionedabove,thethermalnoisecontributiontotheresonatorphaseßuctuationsand,consequently,thetimingjitter,arebothamplitude-dependent,whiletheassociatedcontri-butionoffrequencynoiseisindependentoftheresonatoramplitude.Thisdi!erencebetweenthermalandfrequencynoisesourcesallowsustoindependentlyestimatetheintensitiesofadditive,Df,andmultiplicative,D&,noisesourcesbycomparingthephasedi!usioncon-stant,extractedfrommultipleringdownsegments,againsttheanalyticalpredictionfromEq.(3.56);seeFig.3.13b.Followingthisnoisemeasurementprotocol,wehaveestimated81(a)Extractednon-vanishingcontributionofmeasurementnoisetothetimingjitterasafunctionoftheresonatoramplitude(fromdif-ferentringdownsegments).Discretedotsrep-resentextractedcontributionsofmeasurementnoisetothetimingjitter,whilethesolidlineistheleast-squaresÞtwiththemodelinEq.(3.56).(b)Behavioroftheleading-orderk!dependentpartof#,T2(i,j,k)$jasafunctionoftheres-onatoramplitudealongthemeasuredring-downresponse.Discretedotsrepresentex-tractedcontributionofmeasurementnoisetothetimingjitter,whilethsolidlineistheleast-squareÞtwiththemodelinEq.(3.56).Figure3.13:Characterizationofmeasurement,thermalandfrequencynoisesources.theintensitiesofmeasurement,additive,andmultiplicativenoisesourcesfortheDA-DETFresonatorunderstudy,andtheresultsarepresentedinTable3.3.Fig.3.13aillustratesbothqualitativeandquantitativeagreementofthemodelwiththeexperimentallyobtaineddata,andthereisagoodqualitativeagreementbetweenthetheoreticalandexperimentalresultsinFig.3.13b.TheredoesexistaquitenoticeablespreadofexperimentalresultsaroundtheÞttedcurveinFig.3.13b,whichisbelievedtobeduetotwofactors:First,measurementnoise,ifitisnotweak,cancontributetotheterm.kinEq.(3.56),andthise!ecthasnotbeenaccountedinthepresentwork.Second,thee!ectsofthermalandfrequencynoiseisofhigher-order,ascomparedwiththemeasurementnoisecontribution,whichmakesthecharacterizationprocedureforthesenoisesourcesmoresensitivetotheringdownpartitioning,therebya!ectingthetheaccuracyofextractingthecoe"cientoftheterm.k.82Table3.3:Extractedintensitiesofthemeasurement,frequency,andthermalnoisesources.ThermalnoiseintensityDfisexpressedintermsoftheroot-mean-squareoftheresonatorvibrationamplitudeinthermalequilibrium:Df=4#1!20##a2$th.NoisecharacterizationwasperformedinÞveindependentringdownmeasurementsatVDC=30VandT=!40"Cwithinitialamplitudea0=265mV.Ringdown1M,mVD&,Hz%##a2$th,mV10.646*10!30.6820.6312*10!30.5830.645.1*10!30.5840.636.7*10!30.6550.635.4*10!30.753.5OutlookInthischapterwehaveintroducedacomprehensivemethodthatallowsonetocharacterizemanyimportantmodelparametersforMEMSresonantorsusingtransientringdownmeasure-ments.Averyimportantadvantageofthischaracterizationmethodisthattheelectronicsthatareresponsiblefortheresonatordrivedonota!ecttheringdownprocessand,asaresult,donotcontributeuncertaintiestothecharacterizationprocess.Intermsofdeterministicmodelparameters,wehaveillustratedhowtoextractdeter-ministicsti!nessanddampingparametersforthesymmetricvibrationalmodeofaMEMSusingasingleringdownmeasurement.Theseincludevaluesoflinearandnonlinear(cubic)frictioncoe"cients,obtainedfromtheshapeofthevibrationamplitudeenvelope,aswellasthemodalnaturalfrequencyandconservativeDu"ngandquinticsti!nessnonlinearities,obtainedusingthezero-crossingtimesintheringdown.Akeytothemethodisthatthevibrationalamplitudeisa!ectedonlybythedissipationparameters,whilethefrequencyandphasearea!ectedbytheresonatorconservativeparameters,therebyuncouplingthispartofthecharacterizationprocess.Furthermore,wehavedevelopedanoveltechniquefordirect83revelationofmeasurement,thermal,andfrequencynoisesourcesbyperformingastatisticalanalysisonthezero-crossingpoints.Weshowedthatthesenoisesourceshavequalitativelydi!erentsignaturesontheresonatortimingjitterandthatbyconsideringspeciÞcfeaturesofthejitter,oneisabletoestimatetheintensitiesofthesenoisesources.ThemethodsfordeterminingbothdeterministicandnoiseparametersweresuccessfullytestedonresonatordataprovidedbycollaboratorsatStanford.Theestimationofdetermin-isticparameterswasverysuccessful.Theestimationofnoiseintensitieswasverypromising,butcorruptedbyrelativelystrongmeasurementnoise.Futureworkinthisareaincludesdevelopmentoftechniquesforsystemswithasymmetricnonlinearities,furthertestingofthetechniqueondatawithlessmeasurementnoise,andextensionofthemethodtoaccountformoreinformationaboutthenoisesources.84Chapter4ImprovingthesensitivityofMEMSring/diskresonatinggyroscopesInthischapterweexaminetheself-inducedparametricampliÞcationinMEMSringanddiskresonatinggyroscopesanddescribehowthisnonlinearphenomenona!ectstheperformanceofthesegyroscopeswhenemployedasangularratesensors.Studiesofthelinearandnonlin-earvibrationsofsystemswithcircularsymmetryhasalonghistoryrelevanttothisproblem,anditincludespapersonthetransversevibrationofplates,shells,membranes,rods,andtubesaswellasthein-planevibrationsofplatesandrings;see[100Ð103]andtheresearchcitedthereinforasamplingoftheseworks.Thisclassofsystemshasapplicationsinanum-berofareasincludingvibrationsofantennae,pipes,and,mostrelevanttothepresentwork,wineglassvibratorygyroscopesthatuseCoriolise!ectstomeasurespinrates[104,105].Inrecentdecadestherehasbeenadesiretodevelopsmallerversionsoftheseratesensors,spurredbytechnologicaladvancementsinfabricationtechniquesandbyincreasingdemandsincommercialandmilitaryapplications,andthishasledtoanumberofimportantadvancesinthistechnology[100,106,107].ProminentamongthesearedevelopmentsinMEMSvibra-torygyroscopes,whichhaveshowngreatpotentialduetotheirsmalldimensions,favorablepowerconsumption,andhighqualityfactors[108,109].Generally,suchdevicesarebasedonamicro-mechanicalresonatorwithatleasttwomatchedresonantmodesthatinteractvia85Coriolise!ects[105],asfollows:theresonatorisforcedtooscillateinoneofitsvibrationalmodes,calledthedrivemode,andanexternalrotationat$givesrisetoCorioliscouplingbetweenthismodeanditssymmetricpartner,thesensemode,whichisnotdrivenbyanexternalinput.Theresponseofthesensemodethushasanamplitudeproportionalto$(atleastwhenitissmallcomparedtothevibrationfrequency),sothatbycalibratingandmeasuringtheamplitudeofthereadoutsignalfromthesensemode,onecanestimate$.ImprovingtheprecisionandaccuracyofMEMSvibratorygyroscopesinvolvesseveralchallengingtasksincludingtheprecisematchingofhigh-Qmodalfrequencies[38,110,111],compensationofquadratureerrorsthatarisefromcouplingofthedriveandsensemodes[112Ð114],andoptimizingthegeometryoftheresonatorinordertoachievehighQfac-tors[115,116],tonameafew.Also,allsuchdevicesaregenerallyoperatedinthelinearoperatingregime,soastoavoidfrequencyshiftsassociatedwithnonlinearity.Inthislight,ßexural-modering[19,112]anddisk[117Ð119]vibratorygyroscopeso!ersigniÞcantadvan-tagesduetotheinherentsymmetryintheirgeometriesand,consequently,symmetryoftheirdriveandsensemodes.Recentworkondiskresonatinggyroscopes(DRGs)hasex-perimentallydemonstratedthatthegyroscopesensitivitytotheexternalangularratecanincreasesigniÞcantlywhenthegyroscopeisdrivenintoanonlinearoperatingregime[27].TheauthorshypothesizedthattheobservedphenomenonisduetoparametricampliÞca-tion[120Ð122]arisingfromnonlinearelasticcouplingbetweenthedriveandsensemodesofthedevice,whichhavenearlyequalfrequencies.Inclassicalnonlinearvibrations,thisisanexampleofautoparametricresonance[123Ð125].ThesensitivitySofarategyroscope,thatis,theratiooftheamplitudeofthesensesignaltotheangularrate$,isonethemostimportantcharacteristicsofsensorperformance,sinceit,andthenoiselevelsofthedevice,quantifytheresolutionofthesensorintermsof86thelowerendoftheangularvelocitiesthatcanbedetected[105].Thus,thereisstrongmotivationtounderstand,fromafundamentalpointofview,theadvantageouse!ectsoftheself-inducedampliÞcationofthesensesignalobservedin[27],especiallysinceitappearstobethefortuitousresultofpassivenonlinearbehavior,requiringnoadditionalsensingorenergysource.Understandingandtakingfulladvantageofthisandothernonlineare!ectsarethemaingoalsoftheinvestigationdescribedinthischapter.Nonlinearmodalcouplingisawell-knownphenomenoninthetheoryofnonlinearvi-brationsandithasbeenthoroughlystudiedinawidevarietyofsystems[65],includingmicromechanicalsystems[76,126Ð128].Itgenerallyoccursinresonatorsexperiencingvi-brationamplitudesatwhichnonlinearstrain-displacementrelationships,orothernonlineare!ects,coupletwoormorevibrationalmodes.Thiscouplinghasitsmostdramatice!ectswhenthecoupledmodeshavecommensuratefrequenciesandarelightlydamped,whichpromotesresonantinteractionsbetweenthemodes.SpeciÞcresearchonthenonlinearvibrationsofspinningring-likegeometrieshasil-lustratedtherichdynamicsassociatedwiththein-planeßexuralmodesofthesestruc-tures[129Ð132].Hereweanalyzethedynamicbehavioroftheellipticalmodesinring/diskresonatinggyroscopestoexplainandexploreself-inducedparametricampliÞcationthathasbeenexperimentallyobservedinthesesystems[27].Inparticular,weuseamodeloftheres-onatorconsistingofathinringspinningaboutitsaxisofradialsymmetry,consideringbothmechanicale!ectsaswellaselectrostaticforcesarisingfromcapacitiveactuation/sensingschemes.UsingÞnitedeformationkinematics,weshowthattheellipticaldriveandsensemodesarenonlinearlycoupledthroughbothsti!ness(includingelectrostaticcontributions)andinertialterms.Next,weshowthatthegeneralcaseofmode-coupleddynamicscanbesimpliÞedbyneglectingtheback-actionofthesensemodemotiononthedrivemode(due87totheirdi!eringamplitudes),andprovideconditionsforwhichthisapproximationholds.InthissimpliÞedpicture,wediscussthee!ectsofinertialnonlinearitiesonthedrivemodedynamicsandshowhownonlinearmodalinteractionsleadtoparametricampliÞcationofthesensemode,andthustoanincreaseinthegyroscopesensitivity.Additionally,weexamineseveralmethodsonecanutilizeinordertomanipulategyroscopenonlinearities,includingtheimportantintermodalcouplingstrengthandthemodalDu"ngconstant,forimprovingtheratesensorperformance.ModalcouplingstrengthdeterminesthelevelofparametricampliÞcationonecanexpectforagivenoperatingamplitudeofthedrivemode,whiletheDu"ngnonlinearityisthekeyparameterdeterminingtheupperlimitofthegyroscopedy-namicrange,speciÞcallyforthedrivemodeinthisapplication.Inthislight,usingourmodelofathinspinningring,weillustratehowcleverdesignofthegyroscopeandelectrostatictun-ingcanbeusedformaximizingthestrengthofthenonlinearmodalcoupling,orminimizingindividualmodalDu"ngnonlinearities,andoptimizingtheircombinede!ects.Theremainderofthechapterisorganizedasfollows.InSection4.1,weformulateamodelforresonatorgeometriesthatsupportapairofdegenerate(equalfrequency)n=2(nisthemodalwavenumber)radialmodesandprovidemaintheoreticalconceptsusedinthesubsequentanalysis.InSection4.2weconsiderthenonlinearin-planeßexuralvibrationsofathinspinningringinthepresenceofelectrostaticactuation.SpeciÞcally,Section4.2.1containsastep-by-stepderivationofgoverningequationsofmotionforthegyroscopicdriveandsensemodes.AdetailedanalysisofthedynamicbehaviorofthedriveandsensemodesisgiveninSections4.2.2and4.2.3,respectively.InSection4.2.4weillustratetheappli-cabilityofourresultstoamodeloftherepresentativeringresonatinggyroscopereportedin[19].Finally,inSection4.3weconsidershapeoptimizationtechniquesandelectrostatictuningmethodsthatcanbeusedtomanipulatethemodalcouplingstrengthandtheDu"ng88nonlinearityand,asaresult,opitmizeitsdynamicbehavior.ConcludingremarksforthechapteraregiveninSection4.4.4.1GeneraldynamicalmodelInthissectionwepresentananalyticalframeworkwhichcanbeusedtoderiveequationsofmotionforthein-planevibrationmodesofinterestforring/diskresonatinggyroscopes.Suchaformulationisadvantageoussinceitcanbeappliedtosystemswithrelativelysimplegeometries,suchasathinringorasolidcircularplate,aswellastoMEMSgyroscopeswithlesstrivialgeometries;see,forexample,[133].Westartouranalysisbyintroducingacylindricalcoordinatesystem(r,+,z)andconsideragyroscopewithgenericgeometrythatsupportsapairofdegenerateellipticalmodeseachofwhichhastwonodaldiameters,whichwedenotewithmodalcoordinatesAandBandmodeshapesdescribedby&A(r,+)=.(r)cos2+and&B(r,+)=.(r)sin2+,sothattheirnodaldiametersareseparatedby)/4[27].Withoutlossofgenerality,wedesignatetheseasthedrive(A)andsense(B)modes,respectively.Duringgyroscopeoperation,theresonatorconÞgurationisdescribedbydisplacementsintheradial,u=u(r,+,t),andcircumferential,v=v(r,+,t),directions,whichweassumetobeindependentoftheout-of-planecoordinatez.ThesedisplacementscanbeexpressedintermsofthemodalcoordinatesA(t)andB(t)inamannerthatdependsonthegyroscopegeometry,asu=u(r,+,A(t),B(t))andv=v(r,+,A(t),B(t)).Generally,bothuandvcanbenonlinearinA(t)andB(t).Explicitexpressionsforu(r,+,A(t),B(t))andv(r,+,A(t),B(t))canbeobtainedwithcertainassumptionsforrelativelysimplestructureslikeathinring,seeSection4.2,orasolidcircularplate[101,134,135],whileinthecaseofnon-trivialgeometries,89computingsimilarexpressionsgenerallyrequirestheuseofÞniteelementmethods.InthisworkweutilizeLagrangeÕsmethodtoderivetheequationsthatgovernthedriveandsensemodesofthegyroscope,usinggeneralizedcoordinatesq1,2=A,Btoexpressthenonlinearequationsofmotionforthegyroscopicellipticalmodes.Thekineticenergyofthesystemiscomputedusingstandardmethods[95,134]andisgivenbyT=12,,,V%[(úu!v$)2+(úv+(r+u)$)2]rdrd+dz,(4.1)where$istheexternalangularrateaboutthez!axis,%=%(r,+)istheresonatormaterialmassdensity,whichisassumedtobeuniforminz,andVisthevolumeoftheresonatorbody.ThepotentialenergyofthegyroscopeconsistsofelasticUdandelectrostaticUecomponents.Theelectrostaticpartresultsfromelectrostaticinteractionoftheresonatorbodywiththedrive/senseelectrodes.Generally,theelectrodegapsize%ismuchsmallerthattheouterradiusoftheresonatorRoandthezthicknessoftheresonator,sothatonecanneglectthecurvatureoftheelectrodesandapplyalocalparallel-plateapproximation.Wealsoassumethat%isuniformalongthegyroscopecircumference.Inthiscase,theelectrostaticpotentialenergybecomesUe=!-02,2(0d+b(Ro,+)Ro(VDC+VAC(+))2%!u(Ro,+),(4.2)whereb(Ro,+)isthezthicknessproÞleofthegyroscopebodyalongitscircumference,VDCandVACrepresentmagnitudesofthebiasandperiodicvoltagesusedfortheelectrostaticactuationandsensingand-0=8.85*10!12F/misthevacuumpermittivity.MechanicaldeformationsoftheresonatorbodyandtheassociatedstressescontributetotheelasticpotentialenergyintheformofadeformationpotentialUd.Forin-plane90vibrations,Udcanbeexpressed,[134],Ud=12,,,V(3rr-rr+3-----+3r--r-)rdrd+dz,(4.3)where-ij=-ij(u,v)and3ij=3ij(u,v)arethestrainandstressinthebody.HookeÕslawestablishesthewell-knownrelationshipsbetweenthesequantities,3rr=E+(-rr+&---),3--=E+(---+&-rr),3r-=G-r-,(4.4)whereE+=E(1!&2)!1andGarethee!ectivenormalandshearmoduli,respectively.Inordertoanalyzethenonlineardynamicbehaviorofthegyroscopicradialmodes,onenecessarilyhastoaccountforhigher-ordertermsinthestraintensor-ij(u,v).UsingÞnitedeformationtheory(seeAppendixC)onecanshowthatthenonlinearstrain-displacementrelationships,uptosecondorderinuandv,aregivenby-rr=,u,r+12r2#,,rvr$2,(4.5a)---=ur+,vr,++12#,ur,+$2+ur,vr,+,(4.5b)-r-=,v,r+,ur,+!vr!uvr2!ur,ur,+!,ur,+,vr,++vr,u,r+ur,v,r!,u,r,v,r.(4.5c)SincetheresonatordisplacementsuandvarefunctionsofthemodalcoordinatesAandB,theLagrangianforthesystembecomes,afterintegrationovertheresonatorvolume,L=L(A,B,úA,úB).BysubstitutingthisformofLintoLagrangeÕsequations,onecan91immediatelyobtainequationsofmotionforthegyroscopicellipticmodesofinterest.InSection4.2weapplythisproceduretoasimplemodelofathininextensiblering,which,asweshow,issu"cienttodemonstratetheself-inducedampliÞcationphenomenon,and,fortunately,isamenabletodetailedanalysis.Asnotedabove,thisanalyticalapproachissu"cientlygeneraltobeusedforanalyzinggyroscopeswithdi!erentgeometries,suchasringswithsupportingspringelements[19],circularplatesofnon-uniformthicknessinthez!direction,andothercomplexgeometries,solongastheyhavesimilarcircularsymmetry[27,133].4.2Non-linearforcedvibrationsofathinspinningring4.2.1Gyroscopedynamicswithfully-coupledmodesWeapplythegeneralformulationofSection4.1foranalysisofthenonlinearin-planevi-brationsoftheellipticalmodesofauniform(%,b,h,and%areconstants)circularringrotatingataconstantspeed$aboutthez!axisinthepresenceofelectrostaticforcesfromelectrodes,asdepictedinFig.4.1.Hereafter,weemployathinringapproximation,i.e.,h%R,wherehandRaretheringradialthicknessanditsmid-lineradius,respectively.Inthiscasewecanapplyresultsforthevibrationsofshallowshells[134],andneglectthestressintheradialdirection,3rr=0,aswellastheshearstress,3r-=0.TheapplicationoftheseassumptionsinEq.(4.4)yields3--=E---,whichisthesameexpressionasthatforthelongitudinalstress/strainrelationshipforthetransversebendingofanEuler-Bernoullibeam.Followingthebendingtheoryforthinshells,weexpresstheradialandcircumferential92Figure4.1:Schematicrepresentationofthesystemunderstudy:auniformcircularringrotatingataconstantangularrate$aboutthez!axiswithsegmentedelectrodesrepre-sentingthemeansforelectrostaticactuationandreadout.Segmentationofelectrodesisanessentialfeatureofthedevice,necessaryforproducingspatially-dependentdrivingforcesthroughVAC(+,t),andfortuningthegyroscopicdriveandsensemodesviaanon-uniformdistributionofthebiasvoltagesVDC(+)[27].displacementsofanypointoftheringasu(r,+,t)=u(+,t),v(r,+,t)=v0(+,t)+#v1(+,t),(4.6)where#=r!Ristheradialcoordinaterelativetotheringmid-line,v0isthecircumferentialdisplacementofapointontheringmid-line,andv1istheslopeofthetangentialdisplacementproÞleacrossh.Similartov,weapproximatethestrainÞeldinthe+directionasalinearfunctionof#,i.e.,---=-(0)--+#-(1)--,where-(0)--representsthemid-linestretchingofthe93ring,while-(1)--representsthestrainduetoringbending.Itisknownthattheringmid-linestretchinghasanegligiblee!ectontheringdynamics,solongasthewavelengthofthevibrationmodeislargeascomparedwithitsthicknessh[129,134,136].Sinceweareinterestedinthedynamicbehaviorofelliptical(n=2)gyroscopicmodes,thisconditionissatisÞedforathinring.Therefore,tosimplifytheanalysis,wemakethereasonableassumptionthattheringisinextensiblealongitsmid-line,sothat-(0)--=0.Applyingtheseassumptions,wecanwritetheelasticpotentialenergyoftheresonatorbodyas,Ud=EI2R3,2(0d+2u+,2u,+2!12R#,u,+$232,(4.7)whereI=bh3/12isthesecondmomentofareaoftheringcross-section[95].ConsideringUeforthisgeometry,weassumethatradialringdeßectionsaresmallcom-paredtothegapsize,u%%,whichisfrequentlythecaseforcapacitively-drivenMEMSresonators.UsingthisassumptionweobtaintheapproximateexpressionfortheelectrostaticcontributiontothesystempotentialenergyUe'!-0bR2%4!n=0,2(0d+(VDC+VAC(+,t))2un(+,t)%n,(4.8)wherewehaveexpandedthedenominatoruptothefourthorderinu/%inordertoconsis-tentlyincludenonlineartermsinUeandUd.Notethatinpracticetheelectrodesareonlyontheoutsideofthering,whichresultsinaslightexpansionoftheringinitsradialdirection.However,inlightofourassumptionthattheringisinextensible,weneglectthiscorrectioninouranalysis.Inthecaseofthethinring,themodeshapesfortheellipticalmodesbecomeindependent94ofrandaregivenby'A(+)=cos2+and'B(+)=sin2+,Fig.4.2.Usingthesemodeshapes,Figure4.2:Degenerateellipticalmodesoftheuniformcircularringunderstudy.wecanexpresstheradialdeßectionoftheringbodyasu(+,t)=A(t)cos2++B(t)sin2++C(t),(4.9)wherethetime-dependentfunctionC(t)isincludedinordertoensureperiodicityofv(+,t)in+undertheinextensibilitycondition[129,131].NotethattheexpressionforC(t)isobtainedbysolving-(0)--=0,seeEq.(4.5b),andhastheformC(t)'!(A2+B2)/R.Weusethisformfortheradialdisplacement,andassumethatthespatialdistributionoftheoscillatingactuationvoltageacrosstheelectrodesisthesameasthatofthedrivemode,thatis,VAC(+,t)=VAC(t)cos2+,with|VAC(t)|%VDC.1Wealsoassumethattheelectrodegapissmallcomparetothesizeofthedevice,%%R.Undertheseassumptions,LagrangeÕsmethodisappliedusingthekineticenergyinEq.(4.1)andthepotentialenergyU=Ue+Ud,resultinginnonlinearequationsofmotiongoverningthen=2ellipticalmodesofthering.Afterdividingthroughbythemodalmassforthethinring,mA,B=(5/4))%bhR,these1Thisassumptionisconvenientbutnotnecessary,sinceageneraldistributioncanbeprojectedontothemodesofinterest.95equationstaketheform¬A21+35R2(11A2+B2)3+úA22#A+65BúBR23+A2!20+$2#115!3710B2R2$+2B2R2+335úA2R2+315úB2R2+345B¬BR23+A3R22"!3310$23!165úBR2$A2=85$úB#1!2B2R2$+FA(A,B,t),(4.10a)¬B21+35R2(11B2+A2)3+úB22#B+65AúAR23+B2!20+$2#115!3710A2R2$+2A2R2+335úB2R2+315úA2R2+345A¬AR23+B3R22"!3310$23!165úAR2$B2=!85$úA#1!2A2R2$+FB(A,B,t),(4.10b)where!20=15%#3Eh2R4!4-0V2DCh%3$,"=2=6R25%#Eh2R6!-0V2DCh%5$,FA(A,B,t)=-0VAC(t)VDC5%h%2#4+3B2%2+9A2%2$,FB(A,B,t)=-0VAC(t)VDC5%h%4AB.(4.11)Here!0isthenaturalfrequencyofthemodes,"isthee!ectivemodalDu"ngcoe"cient,and2isthestrengthoftheintermodaldispersivecoupling,allofwhichaccountforbothelasticandelectrostaticsti!nessesandarenormalizedbythemodalmass.Notethatcoe"cients2and"areequalforthethinring,butwekeeptheirdesignationdistinctsincetheyhavedi!erente!ectsonthesystemresponseandmaydi!erforothergeometries.TermsFAandFBrepresentthetime-periodicexcitationactingonthedriveandsensemodes,respectively,andnotethatthedrivemodehasbothdirectand(nonlinear)parametriccomponents,whilethesensemodeisdriveninapurelynonlinearparametricmanner,sincetheexcitationistakentobeperfectlyalignedwiththe(linear)drivemode.ThenonlineartermsinthemodalforcesFA,Bresultfromthenonlineartreatmentoftheelectrostaticpotentialenergy.Inorder96tocompletethemodel,wehaveintroducedphenomenologicallineardissipationcoe"cients#Aand#Bforthemodes.TheanalysisofEqs.(4.10a)and(4.10b)intheirfullformisquitechallenging,duetothefactthattheequationsarenonlinearandcoupledthroughmultipleterms,includingelastic,inertial,andeventheexternaldrivingterms;seeEq.(4.11).Inordertoobtainfurtherinsightintothegyroscopedynamicsandobtainabetterun-derstandingoftheself-inducedampliÞcationphenomenon,weusethefactthatthedrivemodeisdirectlydriventoanamplitudethatismuchlargerthantheamplitudethatwillbeexperiencedbythesensemode.Infact,thesensemodeisdrivenbythevibrationsofthedrivemodethroughtheCoriolistermproportionalto$úA(theÞrsttermontherighthandsideofEq.(4.10b)),andalsoparametricallythroughcouplingtermslike2BA2/R2.Inthiscasetherelativephasebetweenthesetwotermsis)/4,whichindicates,followingtheanalysisin[120],thatthesensemoderesponsewillbeampliÞedbythedrivemodevibrationsregardlessoftheiramplitude.Whentheparametricdriveisweakenough,meaningthatAissu"cientlysmall,theresponseofthesensemodeduetotheCoriolise!ectremainsstableandtheseparametrictermscanamplifyorattenuatethesensemoderesponse,dependingontherelativephasebetweenthedirectandparametricdrives[121,122,137].Consequently,ifthegyroscopeisexposedtoangularratesthatsatisfy$#!0(allrategyrosaresode-signed),wecanassumethatthesensemodeoperatesinitslinearrangewellbelowtheonsetofnonlinearity.Notethat$#!0alsoallowsustoneglecttermsproportionalto$2inEqs.(4.10a)and(4.10b).Incontrast,thegyroscopicdrivemodecanoperateatamplitudeswherenonlineare!ectscomeintoplay.Infact,thismustbethecaseinordertoachievethedesiredampliÞcationofthesensemode.Undertheseconditions,thebackactionofthesensemodeonthedrivemodecanbeneglected,andwecananalyzethedynamicsofthedrivemodeindependently.Afterobtainingthe(nonlinear)solutionforthedrivemode,wecan97analyzetheresponseofthegyroscopicsensemodeandstudytheself-inducedampliÞcationandassociatedincreaseofthegyroscopesensitivity.Inessence,wecanemployamodelwithone-waycouplingforsu"cientlylow$.4.2.2DynamicsoftheDriveModeHerewestudythedynamicbehaviorofthedrivemodeoftheringandanalyzethee!ectsofinertialnonlinearitiesandnonlinearforcingtermsonitsbehavior.Usingtheassumptionsderivedabove,weneglectcouplingtothesensemodeinEq.(4.10a)andassumerelativelyslowexternalrotation,$#!0,toobtainthefollowingnonlinearmodelforthedrivemodebehavior¬A#1+µA2R2$+2#AúA+A#!2A+µúA2R2+"A2R2$=Fcos(!t+/F)#1+CFAA2%2$,(4.12)whereµisthestrengthoftheinertialnonlinearityandCFArepresentsthenonlinearcor-rectiontothemodalforcing.Additionally,thedrivefrequencyisnearthemodalnaturalfrequency,thatis,!=!A+#!with#!%!A.NotethatinEq.(4.12)wekeepallcoe"-cientsinagenericformtokeeptheformulationgeneral,butapplytheresultsfortheringgeometrybelow.InordertoanalyzeEq.(4.12),wenotethatthesystemislightlydamped(typicaldampingratiosareintherange10!5!10!4),resonantlydriven,andhascubicsti!nessandinertialnonlinearities,sotheproblemistreatedinthestandardway.WestartbyrepresentingthemodaldisplacementintheformA(t)=a(t)Rcos(!t+/A(t)),where(a(t),/A(t))arethenon-dimensionalvibrationamplitudeandthephaseofthedrivemoderesponse.Byemployingthemethodofaveraging[89],weassumethat(a(t),/A(t))changeslowlyover98times&!!1AandletúA(t)'!!a(t)Rsin(!t+/A(t)).ByusingtheseexpressionsforA(t)andúA(t)inEq.(4.12)anddisregardingfast-oscillatingterms,weobtainequationsgoverning(a,/A)ontheslowtimescale&#!1A.Thesteady-stateresponsesfoundfromtheseequationscanbesolvedtoobtainthefollowingexpressionthatrelatesdrivemoderesponseamplitudetothesystemandinputparameters,#!(a)'3a28!A#"!23µ!2A$1!14µa2±44444#2+3CFAa2R24#2+1CFAa2R244446F2(4#2+CFAa2R2)28a2R2#4!2A!#2A1!14µa2.(4.13)Asexpected,thenonlinearforcingtermdoesnota!ecttheshapeofthemodalbackbonecurve,representedbytheÞrstterminEq.(4.13),butalterstheshapeofthefrequencyresponsebranchesandrenormalizesthee!ectivemodalforcingamplitude.SimpliÞcationofEq.(4.13)canbemadewhenthemodalamplitudeissmallcomparedwiththeelectrodegapsize,thatis,whena%%/R,asiscommoninapplications(toavoidpull-in[97,138]).Inthiscase,thenonlinearcorrectiontothedrivemodeforcingcanbesafelyneglected,CFA=0,anassumptionweemployinthefollowingdevelopment.Itisimportanttorecognizethattheamplitude-dependentshiftofthefreevibrationfre-quencyofthemodeshasthefollowingsources:thesti!nessDu"ngnonlinearity"="d+"e,where"dand"earethecontributionsfromelasticandelectrostaticsti!nesse!ects,respec-tively(seeEq.(4.11)),andinertialnonlinearitieswhichhaveane!ectiveDu"ngnonlinearity"i=!23µ!2A(seeEq.(4.13)).Theinertialnonlineare!ectshavethesameorigin(Þnitedefor-mationkinematics)asthenonlinearitiesintheelasticdeformationpotential,andwecombinethesee!ectsintoasinglemechanicalcontributiontothemodalDu"ngconstant,denoted"m="d+"i.Formoderatevibrationamplitudes,thesteady-stateamplituderesponseis99thatofanequivalentDu"ngsystemandcanbeexpressedas!(a)'!A+3a28!A("m+"e)±6#F2aR!A$2!#2A,(4.14)examplesofwhichareshowninFig.4.3.Figure4.3:Representativesteady-statefrequencyresponsecurvesoftheringdrivemodedescribedbyEq.(4.14)fordi!erentvaluesoftheforcingamplitudeF.Theblue,red,andblackcurvescorrespondtoforcingmagnitudesF0,2F0,4F0.ResponsesareobtainedundertheassumptionthatelectrostaticforcesdominatetheDu"ngnonlinearity,i.e.,|"e|)|"m|.Solidanddashedcurvesrepresentstableandunstableresponseamplitudes.Analysisshowsthatinertialnonlinearitiesmustbetakenintoaccountwhen|"e|"|"m|,whichcanbethecaseinresonatorswithlargemechanicalsti!ness,likecircularplates.Interestingly,whentheelectrostaticpotentialprovidesonlysmallcorrectionstobothlinearandnonlinearsti!nessconstants,inertialnonlinearitieshaveadominante!ectonthemodalfrequencyresponseandcausesubstantialsofteningoftheresonatorfrequency.SpeciÞcally,forthethinring"d=2!2Aand"m=!125!2A.Notethatasimilarsituationoccursin100cantilevertyperesonatorswhereinertialnonlinearitiesessentiallydominatethedynamicbehaviorofthefundamentalmodeandcausethevibrationfrequencytosoftenasafunctionofvibrationamplitude[139,140]2.Ontheotherhand,when|"e|)|"m|,electrostatice!ectsdominatethenonlineardynamicsofthesystem,sothatthenonlineartermsarisingfrommechanics,bothinertialandelasticsti!ness,canbeneglected,inwhichcaseweusetheapproximation"m'0,whichwewilluseinSection4.2.4.4.2.3DynamicsoftheSenseMode:ParametricAmpliÞcationInthissectionweanalyzetheresponseofthegyroscopicsensemodeB,usingthedriveresponseasane!ectiveexcitation.ThisexcitationhascomponentsfromCorioliscouplingfromtheexternalangularrate$,andfromnonlineardispersivecouplingfromelastic,iner-tial,andelectrostatice!ects.Inwhatfollows,weassumethatthedrivemodemotioncanberepresentedas,A(t)=aRcos(!t+/A),whereaand/Aarethedrivemodesteady-stateamplitudeandphase,respectively.When$issmallascomparedwiththegyroscopeoper-ationfrequency,i.e.,$#!,andtheparametricpumpingdoesnotdestabilizethesenseresponse,thedynamicbehaviorofthesensemodeisgovernedbythefollowingequationofmotion,obtainedfromEq.(4.10b),¬B+úB#2#B+C!AúAR2$+B#!2B+CdA2R2+C1úA2R2+C2A¬AR2$=C$$úA,(4.15)wherewehaveemployedlinearizeddynamicsforB(justiÞedinSection4.2.1).Thismodelhasdirect(Coriolis)excitationfromúAandparametricexcitationfromnonlinearcombinationsof(A,úA,¬A).TheCjareconstantsthatdependonthegeometryofthegyroscopebody.Since2Thecantileverisanothersystemmodeledusinganinextensibilityassumption.101Eq.(4.15)containsbothdirectandparametricresonantdrivingterms,itisconvenienttorepresentthesensemoderesponseintheformB=R(bexp[i!t]+c.c.)andapplymethodofaveraginginthemannerin[120].Afteraveragingandsomemanipulationsweobtainthefollowingexpressionforthesteady-stateamplitudeofthesensemode|b|=|C$|$a!254!2#2B+(!2n!!2+$)2|4!2#2B+(!2n!!2)2!$2|(4.16)where!2n=!2B+12a2(Cd+!2(C1!C2))(4.17)isthee!ectivevibrationfrequencyofthesensemode,whichismodiÞedbynonlinearcouplingtothedrivemodeatamplitudea,whichstemsfromtheDCcomponentsofthedrivemodenonlinearcouplingtermsterms.Similarly,theACcomponentsofthesetermsproducethecoe"cient$=14a2(Cd+!2(C!!C1!C2))(4.18)whichrepresentsthestrengthoftheparametricpumpingarisingfromthedrivemode.Notethatboth!2nand$aredeterminedbythesystemnonlinearities,thedrivevibrationampli-tudea,andthegyroscopeoperationfrequency!.TheexpressioninEq.(4.16)describestheamplitudeofthesensemodeandcapturestheinteractionofthee!ectsofthedirect(Coriolis)driveandtheparametricpumpingfromnonlinearcoupling,anditrevealssomeimportantfeatures.First,whenthedrivemodevibrationamplitudeissu"cientlysmall,suchthatonecanneglectthee!ectoftheparametric102pumping,thatis,$canbeneglected,theexpressionforthesensemodeamplitudebecomes|b|l=a!|C$|$25(!2B!!2)2+4!2#2B,(4.19)whichrepresentsthecasewhenbothmodesbehavelikelinearresonators.Inthislight,itisconvenienttoexpressthegyroscopicsensemodeamplitudeinthemoregeneralcaseasfollows|b|par=G|b|l,G=5[4!2#2B+(!2n!!2+$)2][(!2B!!2)2+4!2#2B]|4!2#2B+(!2n!!2)2!$2|,(4.20)whereGistheampliÞcationofthesensemode,i.e.,thegain,thatarisesfromtheparametriccouplingtothedrivemode.ThisgainfromthecouplingisillustratedinFig.4.4,whichshowsthesensemoderesponseamplitudeforthecasewherethecouplingisignored(blackdashedlines)andfortwolevelsofcoupling(redandbluelines).HeretheparametricampliÞcationisevident,asisthefrequencyshift(softeningduetothepresenceofelectrostaticcouplingforces)thatarisesfromthecoupling;seeEq.(4.15).Amorecompleterepresentationofthegainisconsideredbelow.AnotherfeatureassociatedwithEq.(4.16)isthatthesystemgainGor,equivalently,thesensemodeamplitude|b|pardivergeswhenthedenominatorinEq.(4.20)vanishes,i.e.,74!2#2B+(!2n!!2)2!$28(0.Bysolvingthisequation,oneobtainstheparametricinstabilityconditionexpressedintermsofthedriveparametersas(a+,!+),correspondingtoG(-.This(a+,!+)conditioncorrespondstothecasewheretheparametriccouplingtermsinEq.(4.15)resultininstabilityofthesensemode[65].Asthisinstabilityisapproached,thelinearizedversionofthesensemodemodel,givenbylinearizingEq.(4.15),isinsu"cient103Figure4.4:E!ectofself-inducedparametricampliÞcationonfrequencyresponsesoftheringsensemodedescribedbyEq.(4.16)fordi!erentvaluesofthedispersivemodalcouplingcoe"cientCd,whereweconsiderthecaseCd/!2B)C1,C2,C!,sothattheparametricpumpingcoe"cient$isessentiallyproportionaltoCd.FrequencyresponsesareobtainedforaR/%=0.2and$/!B=2*10!4.Thedashedcurveisthenon-ampliÞedresponse(Cd=0),whiletheredandbluecurvescorrespondtothesensemodefrequencyresponseswithCd/!2B=!0.5*104,!1.1*104respectively;thesenumbersarechosensuchthattheanalyticalresultsofSection4.2.3remainvalid,thatis,sothatthestatedapproximationshold.SignalampliÞcationfromtheintermodalcouplingisevident.todescribethesensemodedynamicsandthefullcoupledformoftheequationsofmotionEqs.(4.10a)and(4.10b)mustbeused.Inthiswork,however,werestrictouranalysistothecasewheretheresponseofthesensemoderemainsinitslinearrangeanditsvibrationamplituderemainsproportionalto$.Infact,thisistherangeofpracticalinterest.4.2.4ExampleGenerallyspeaking,onecanadapttheapproachdevelopedheretoavarietyofgyroscopeconÞgurationsthatexploitcircularsymmetry.Hereweillustratetheapplicabilityoftheresultsbyusingparametersderivedforthepolysiliconringgyroscopereportedin[19].First,weconsidertheidealizedcaseofafree(nosuspension)gyroscopering.Thegyroscopeparametersareasfollows:themid-lineradiusoftheringisR=550µm,theradialthickness104ish=4µm,theelectrodegapsizeis%=1.4µm,theestimatedqualityfactorisQ=1200,andthebiasvoltageistakentobeVDC=3V(weintentionallytakethisvalueofVDC,ascomparedto7Vin[19],toavoidtheelectrostaticpull-ine!ect).Asaresult,thegyroscopedynamicparametersbecome!B/2)=12.5kHz,#B/!B=1/2400andtheelectrostaticpotentialstronglydominatesthestrengthofthedispersivemodalcoupling,Cd/!2B&!105,whiletheotherconstantsdeÞnedinEq.(4.10b)satisfyC!,C1,C2%Cd/!2B.Figures4.5aand4.5bshowtheself-inducedparametricgainGasafunctionofthenormalizeddrivefrequency!/!BandthenormalizedvibrationamplitudeofthedrivemodeaR/%(leftpanel)andthenormalizedstrengthofthesti!nesscouplingCd/!2B(rightpanel).Thesolidredcurvedepictstheinstabilitycondition,(a+,!+)inbothpanels,wherethevalueofthegainapproachesinÞnity,thatis,itistheArnoldtongueforthesensemode[141,142].ThemeshedregiononbothpanelscorrespondstothesetofoperatingconditionswherethesolutionfoundinEq.(4.16)isunstable.Inordertodescribethegyroscopedynamicsintheseregions,onemustanalyzethefullformofEqs.(4.10a)and(4.10b),sinceinthisregionnonlineare!ectsthathavebeenignoredwillcomeintoplay.AsfollowsfromconsideringtheresultsofFig.4.5a,inordertoachievesigniÞcantgainG,thedrivemodeshouldbeoperatedatfrequenciesslightlylessthan!B.ThiscanbeeasilysatisÞedsincetheelectrostaticforcesdominatethenonlinearitiesofthegyroscopicdrivemodeanditsfrequencyresponseexhibitssofteningbehavior.Figure4.5b,ontheotherhand,illustratesthebehaviorofthegainGasafunctionoftheoperatingfrequencyandthestrengthoftheintermodaldispersivecoupling.Importantly,themagnitudeoftheintermodalcouplingcanbecontrolledbyadjustingthebiasvoltageVDCappliedtotheresonatorbodyorattendantelectrodes,thusallowingonetotunetheamountofself-inducedparametricampliÞcation,whichincreasesgyroscopesensitivitySinthevicinityoftheinstabilityregion.105(a)Dependenceofthestrengthoftheself-inducedparametricampliÞcationGonthescaledoperationfrequency!/!Bandthevi-brationamplitudeofthedrivemodeaR/#.Thesti"nesscouplingstrengthisCd/!2B=!95*103.(b)Dependenceofthestrengthoftheself-inducedparametricampliÞcationonthescaledoperationfrequency!/!Bandthenonlin-eardispersivecouplingstrengthCd/!2B.ThedrivemodevibrationamplitudeischosentobeaR/#=0.1.Figure4.5:Increaseofthesensitivityofthegyroscopering(withoutsuspension)duetononlinearmodalcouplingasafunctionofsystemanddriveparameters.Thesolidredlinerepresentsthea+!!+curvewherethegainGdivergesaccordingtothelinearmodeldescribedbyEq.(4.15);thisistheprimaryArnoldtongueforthesensemode[141].ThemeshedregionisthesetofoperatingconditionswherethesolutionfoundinEq.(4.16)isunstable.Furthermore,thecriticalvalueofthedrivemodeamplitude,whereG(-,decreasesasC!1/2d.Physically,asexpected,thisimpliesthatforstrongerdispersiveintermodalcoupling,smallerdrivevibrationamplitudesarerequiredtoachievethesamelevelofgain.Theresultsobtainedforthecaseoftheunsuspendedgyroscoperingcanbeeasilyex-tendedtoaccountforsupportingspringsintheformofsemicircleswiththemid-lineradiusRs=235µmandradialthicknessh=4µm;see[19]fordetails.NumericalanalysisshowsthatforthepolarizationvoltageVDC=7V(thevoltageusedbyAyazietal.intheirexper-106iments),electrostaticforcesstilldominatethesystemnonlinearities,includingthedispersivemodalcouplingstrength,Cd/!2B&!104.Asaresult,thedependenceofthegyroscopesensitivityonthesystemparametersisqualitativelythesameasinthecaseofthefreering;seeFigs.4.5aand4.5b.Ouranalysisofagyroscopewithasuspensionshowsthattheinclusionofthesemi-circularsuspendingspringsinthemodelchangetheresonatorkineticandpotentialenergies,wherethelatterisa!ectedthroughitselasticcomponentonly,sincethereisnointeractionofthespringswiththeelectrodes.Duetothesymmetryofthegyroscopeellipticalmodes,thesuspensionspringsareequivalenttofouradditionalringsofradiusRs,whereonepairbelongstothedrivemodeandtheothertothesensemode.Inthiscase,however,whencalculatingthekineticenergywealsohavetoaccountforthemotionofthespringmass,whichcontributestothee!ectivemodalmass.Furthermore,analysisshowsthatthegyroscopesuspensionhasconsiderablee!ectontheindividualmodalsti!nessparameters,a!ectingboththelinearnaturalfrequencyandtheDu"ngnonlinearity.ThesecontributionscanbecalculatedinastraightforwardwayfollowingthemethoddescribedinSection4.2.1.Inparticular,themechanicalcontributionstothemodalnaturalfrequencyandtheDu"ngmodalnonlinearityincreasebyfactorsof5and12,respectively.Additionally,thesuspensionspringsalsoa!ectthedispersivemodalcouplingstrength;infact,themechanicalcomponentof2increasesbynearlyafactoroftwo.Thiscontributiontothemodalcouplingstrengthistheresultofthenonlinearnatureofthestrain-displacementrelationshipsinEqs.(4.5a)to(4.5c).Similarresultsformorecomplicatedgeometries,suchastheDRGin[27],canbeobtainedusingÞniteelementmethodsadaptedforcomputingnonlinearcoe"cientsformechanics[43,143],althoughdoingsuchcalculationsfornonlinearelectrostatice!ectsisstillbeingdeveloped.1074.3Manipulatinggyroscopenonlinearities:geometricandelectrostaticoptimizationmethodsInthissectionweanalyzethreedi!erentmethodsformanipulatingthemagnitudeofnon-lineardispersivecoupling|2|andthemodalDu"ngnonlinearity"inordertomaximizethesensitivityofthemicromechanicalgyroscopewithrespecttotheexternalangularrate$andincreasethegyroscopedynamicrange.SpeciÞcally,inthefollowingsubsectionsweconsidertailoringoftheaforementionedgyroscopenonlinearitiesviaanon-uniformdistributionofthebiasvoltageVDCamongthegyroscopeelectrodes,anangle-dependentelectrodegapsize%=%(+),andanangle-dependentthicknessofthegyroscoperingh=h(+),respectively.NotethatinallthesemethodsweconsideronlysuchmodiÞcationsofthegyroscopebiasvoltage,theelectrodegapsize,andtheradialthicknessoftheresonatorringthathaveatleast8-foldrotationsymmetryrelativetothez!axisofthegyroscope.Importantly,thisconstraintonthegyroscopeparametersmaintainstheinherentsymmetryofthegyroscopicvibrationalmodes,whichallowsonetoavoidadditionalproblemsassociatedwithmatchingthemodalnaturalfrequencies.Duetothedispersivenatureoftheintermodalcouplingbetweenthegyroscopicdriveandsensemodes,thegyroscopesensemoderesponsedependsonboth2and",andthisdependenceisgenerallycomplicated.Givenourgoalsofmaximizingthegyroscopeangularratesensitivityandthegyroscopedynamicrangeatthesametime,itisconvenienttochoosetheobjectivefunctionforouroptimizationproblemtobethestrengthoftheparametricpumpingwhenthegyroscopedrivemodeisoperatingattheinßectionpointonitsfrequencyresponse(onsetofnonlinearity).Theobjectivefunctionisformulatedasfollows:Wenotethatthedrivemodecriticalamplitudeacr."!1/2andthepumpingstrengthis2a2cr,108suggestinganobjectivefunctionofK=|2/"|.WewillassumethatthemodalnaturalfrequenciesarestillmatchedandthatthedampingconstantsdependweaklyonthechangesemployedforthegyroscopegeometryandDCbias.BeforeweproceedtospeciÞcoptimizationmethods,itisusefultounderstandthee!ectofthechosenobjectivefunctionKonthegyroscopeperformance.Inordertodothis,weÞrstrewriteEq.(4.16)asfollows|b|a=acr=|C$|$,acr!,26!,2Q2B+#1!2KQB13!!,2$24444!,2Q2B+#1!4K3QB13!!,2$2!427Q2BK24444,(4.21)where!,=!cr/!B=1!132QBisthegyroscopeoperatingfrequency(which,ofcourse,coin-cideswiththecriticalvibrationfrequencyofthedrivemode),$,=$/!Bisthescaledexternalangularrate,andwehaveassumed,givenourexampleinSection4.2.4,that|Cd|/!2B)|C1|,|C2|,|C!|.FromEq.(4.21)itfollowsthatthegyroscopeperformancedependsonbothacrand,consequently,"andK,wherethelattere!ectofourobjectivefunctionKonthesensemodeamplitude|b|canbeconvenientlyexpressedusingthefollowingÞgureofmerit,F.O.M.=6!,2Q2B+#1!2KQB13!!,2$24444!,2Q2B+#1!4K3QB13!!,2$2!427Q2BK24444.(4.22)Figure4.6illustratesthedependenceoftheÞgureofmeritonthevalueoftheobjectivefunction.AsfollowsfromthisÞgure,inordertoincreasethegyroscopesensitivity,itisdesirabletoimplementsuchmodiÞcationstothegyroscopegeometryandelectrostaticsetup109Figure4.6:DependenceoftheÞgureofmeritinEq.(4.22)onthevalueoftheobjectivefunctionK.QualityfactorsofthegyroscopedriveandsensemodesareassumedtobeQA=QB=1200[19].thatwillmaximizeKwhilekeeping|"|lowduetothedependenceofthegyroscopesensemoderesponseonacr.Thisformsthebaselineforoursubsequentdevelopmentoftheoptimizationmethodsinfollowingsectionsofthisdissertation.4.3.1Nonlinearelectrostatictuningbyanon-uniformbiasvoltageWestartwiththerelativelysimplemethodofelectrostatictuning,whichisfrequentlyusedformodifyingthevibrationfrequencyofMEMSresonatorsbyalteringthebiasvoltage.Inthepresentcase,however,weareinterestedinmodifyingthegyroscopenonlinearparameters2and".Withoutanyalterationsofthegyroscopeelectrostaticactuation/sensingsetup,2and",aredeÞnedinEq.(4.11),2="=6R25%#Eh2R6!-0V2DCh%5$,fromwhereitfollowsthatauniformincrease/decreaseofthebiasvoltagechangesbothcoe"cientssimultaneously.Ourgoal,however,istomaximize|2|whileminimizing|"|.110Inordertoachievethis,belowweconsiderthecaseofanon-uniformbiasvoltage,i.e.VDC=VDC(+).Frequently,MEMSring/diskresonatinggyroscopesaredesignedtohave4Nelectrodes(N32)alongthecircumferenceinordertobeabletosupportsin2+andcos2+in-planeellipticalvibrationalmodes.Inthiscase,thenonlinearelectrostatictuningmethodresultsindi!erentvaluesofthebiasvoltageappliedtodi!erentelectrodes,i.e.VDC=VDC(k),wherekistheelectrodenumber.Keepinginmindthesymmetryofthegyroscopicmodes,itbecomesclearthatthe8-electrodeconÞgurationcansupportonlyuniformdistributionofthebiasvoltageVDC,which,aswediscussedabove,isoflimitedinterest.Asaresult,inordertoshowthee!ectofVDC(k)ontheresonatornonlinearities,wefurtherassumethatthegyroscopedesignhas8Nelectrodes(N32).InthiscasetheelectrostaticpotentialenergybecomesUe'!-0bR28N!k=14!n=0,-k++-k!+d+(VDC(k)+VAC(k,t))2un(+)%n+1,(4.23)where+kistheangularpositionofthekthelectrodecenter,21isthecircumferentiallengthofeachelectrode(inradians),andbothVDCandVACarenowfunctionsoftheelectrodenumber.SincethegyroscopekineticenergyinEq.(4.1)isindependentofVDC,anon-uniformbiasvoltagedistributiononlya!ects2and"throughtheirelectrostaticcontributions,2eand"e,determinedbythecoe"cientsoftheterms.A2B2and.A4inEq.(4.23).Inordertoillustratethenonlinearelectrostatictuningmethod,weconsiderthespeciÞccasewhenN=2,asin[38],andassumethefollowingdistributionofthebiasvoltageVDC(k)=VDC(1+(!1)krDC),k=1,...,16.Restrictingourselvestothecaseofthecommonlyused111uniformelectrodegapsize,weobtain2(DC)e2(0)e=1!rDC12R2)(3R2!12R%+8%2)+r2DC'1!4)rDC+r2DC,(4.24a)"(DC)e"(0)e=1+rDC4R2)(3R2!12R%+8%2)+r2DC'1+43)rDC+r2DC,(4.24b)where2(0)e="(0)earethevaluesofthecouplingstrengthandDu"ngnonlinearityintheabsenceofthenonlinearelectrostatictuning,i.e.,whenrDC=0.Thedependenceof2e,"e,andKeonthevariationinthegyroscopebiasvoltagerDCpresentedinEq.(4.24a)andEq.(4.24b)isshowninFig.4.7.Figure4.7:Behavioroftheelectrostaticcomponentsofthedispersivecouplingstrength(blackdashedcurve),theDu"ngnonlinearity(blacksolidcurve),andtheobjectivefunction(bluesolidcurve)asafunctionofthevariationinthebiasvoltagerDC.Inring/diskresonatinggyroscopes,theelectrostaticforcesfromtheactuation/sensingschemefrequentlydominatethenonlineardynamicbehaviorofthegyroscopicvibrationalmodes,i.e.,2'2eand"'"e.Inthiscase,Fig.4.7clearlyshowsthatwhenrDC<0,onecanachieveupto3-foldincreasein|2|.Atthesametime,|"|decreasesforrDC0(!4/3(,0),butincreases,althoughslowerthan|2|,forrDC0(!1,!4/3().Assuming,forinstance,112thatwewouldliketonotdecreasethegyroscopedynamicrange,wethenhavetochooserDC=!4/3(,whichcorrespondsto'72%increaseofthemodalcouplingstrengthand,asaresult,theobjectivefunctionK.Suchanincreaseof|2|shouldresultinapproximately2-foldincreaseoftheÞgureofmerit,seeFig.4.6.Notethatthenonlinearelectrostatictuningmethoddescribedherealsoshiftsthenaturalfrequenciesofthegyroscopicvibrationalmodes.However,sincethegyroscopelinearsti!nessisdominatedbythemechanicalelastice!ects,thiscorrectiontothemodalfrequenciesisrathersmall,seenbyconsidering,forexample,thatthecoe"cientoftheterm.A2inEq.(4.23),andfollowingthesamelineofthoughtthattheelectrostaticcorrectiontothemodallinearsti!nessis.r2DC.4.3.2E"ectofanon-uniformelectrodegapsizeNowweassumethatVDC=constandconsiderthee!ectofanon-uniformelectrodegapsizeonthemagnitudeofthedispersivemodalcouplingstrengthandmodalDu"ngnonlinearity.Unlikeinthepreviousmethod,theelectrodegapsizecanbevariedinacontinuousfashionalongthegyroscopecircumference,i.e.,%=%(+)andthegyroscopeelectrostaticpotentialenergydeterminingthesystemsti!nessparametersbecomesUel'!-0bR2V2DC4!n=0,2(0d+un(+)%n+1(+).(4.25)Aswementionedbefore,thevariationintheelectrodegapsizeshouldpossessatleast8-foldrotationsymmetryin+inordertoensurethatthegyroscopicmodesremaindegenerate%(+)=-!k=0%kcos8k+,(4.26)113where%karesomeconstants.Inthiswork,weconsiderthesimplestcaseyieldinganon-trivialresult:%(+)=%0(1+r#cos8+).Fromthephysicalstandpoint,thevariationr#mustsatisfy|r#|<1;inpractice,however,thisconditionfor|r#|becomesevenstrongerduetoadditionalconstraintsofthechosenfabricationprocess.Assumingthatthegapsbetweenindividualelectrodesaresmall,weexpandthedenominatorinEq.(4.25)inTaylorseriesuptothesecondorderinr#andobtaintheelectrostaticcorrectionstothecoe"cientsofthedispersivecouplingandtheDu"ngnonlinearityintheform2(#)e2(0)e'1+52r#+152r2#,(4.27a)"(#)e"(0)e'1!56r#+152r2#.(4.27b)Thedependenceoftheelectrostaticcomponentstothegyroscopeintermodalcouplingstrength,theindividualmodalDu"ngnonlinearityandtheobjectivefunctionKonthevariationintheelectrodegapsizer#isshowninFig.4.8a.FromthisÞgureitimmediatelyfollowsthatthecasewithr#>0becomesofsigniÞcantimportanceforthegyroscopeswhosedynamicbehaviorinthenonlinearregimeisdominatedbyelectrostatice!ects.Indeed,whenr#>0,thedispersivecouplingstrength|2e|growsfasterthantheDu"ngnonlinearity|"e|.Furthermore,"(#)e/"(0)e<1forr#0(0,1/9).Assuming,asbefore,thatwewouldlikenottosupressthegyroscopedynamicrange,wechooser#=1/9andachieve'37%increaseof|2e|byasimplealterationoftheshapeofthegyroscopeelectrodesatthedesignstage;seeFig.4.8b.AsitfollowsfromFig.4.6,thisincreaseofthecouplingstrengthandtheobjectivefunctionshouldleadto'32%increaseofthegyroscopesensemoderesponsewhenthedrivemodeisdrivenontheonsetofmodalnonlinearity.114(a)Dependenceoftheelectrostaticcomponentsofthedispersivecouplingstrength(blackdashedcurve),theDu%ngnonlinearity(blacksolidcurve),andtheob-jectivefunction(bluesolidcurve)asfunctionsofthevariationintheelectrodegapsizer!.(b)Schematicrepresentationoftheringresonatinggyroscopewiththenon-uniformelectrodegapsizewithr!=0.4.Thesolidcirclerepresentstheouterboundaryofthegyroscopebody;thedashedwavylooprepresentstheinnerboundaryoftheattendantelectrodes.Figure4.8:Manipulatinggyroscopenonlinearitiesviaanon-uniformelectrodegapsize4.3.3ShapeoptimizationofthegyroscopebodyLastly,weshowhowonecanalterthemechanicalcontributionstothedispersivemodalcou-plingstrengthandtheindividualmodalDu"ngnonlinearityinmicromechanicalring/diskresonatinggyroscopesbymodifyingtheshapeofthegyroscopebodywhilekeepingtheelec-trodegapthesamealongthegyroscopecircumference.Hereweillustratethistechniquebymanipulatingthenonlinearsti!nessparametersofathinspinningring,discussedinSec-tion4.2,bymodifyingtheringradialthicknessalongthegyroscopecircumference.SimilartotheelectrodegapsizeinSection4.3.2,weenforcetheringradialthicknesstobeperiodicin+withtheperiodbeingatmost)/4inordertopreservethesymmetryofthedriveand115sensemodes.Mathematicallyweexpressh(+)ash(+)=-!k=0hkcos8k+,(4.28)wherehkaresomeconstants.Inclusionofh(+)inthedynamicmodelfortheringresonatinggyroscopemodiÞesobtainedearlierexpressionsforthegyroscopekineticandmechanicalpotentialenergiesasfollowsT(u,v,úu,úv)=%bR2,2(0d+[(úu!v$)2+(úv+(r+u)$)2]h(+),(4.29a)Um=Eb24R3,2(0d+2u+,2u,+2!12R#,u,+$232h(+)3.(4.29b)AswementionedinSection4.2,modiÞcationsintheresonatorgeometrye!ectthegy-roscopeintermodalcouplingandDu"ngnonlinearityintwoways.First,asitfollowsfromEq.(4.29a),thenon-uniformityoftheringradialthicknessa!ectsthee!ectivemodalmass,thecoe"cientoftheterm.úA2inthegyroscopekineticenergy.Additionally,h(+)changessti!nesscoe"cientsoftheterms.A2B2and.A4inthegyroscopepotentialenergyduetoelasticdeformationsofthering.Byaccountingforthesetwoe!ects,weexpressmechanicalcontributionsto2and"as2m=2E%R692(0d+(1+sin24+)h3(+)92(0d+(1+3cos22+)h(+),(4.30a)"m=2E3%R692(0d+(1+4sin22++4sin42+)h3(+)92(0d+(1+3cos22+)h(+),(4.30b)fromwhereitimmediatelyfollowsthatiftheringradialthicknessisconstant,i.e.,h(+)=h0,then2m="m,asexpected.116Todemonstratethee!ectofanon-uniformringradialthickness,weconsiderarathersimplecasewhenhk=0fork32inEq.(4.28).Inthiscasewehave2(h)m2(0)m=1!12rh+32r2h!18r3h,(4.31a)"(h)m"(0)m=1+16rh+32r2h+124r3h,(4.31b)whererh=h1/h0.Figure4.9aillustratesthedependenceof2mand"monrh.Herewe!1.0!0.50.00.51.01.01.52.02.53.0rh"m!h"#"m!0",#m!h"##m!0"(a)Dependenceofmechanicalcomponentsofthedis-persivecouplingstrength(blackdashedcurve)andtheDu%ngnonlinearity(blacksolidcurve)ontheringthicknessmodiÞerrh.Sincethegyroscopedynamicsisdominatedbyelectrostaticforces,theobjectivefunc-tionisessentiallydeterminedby.eand/eand,thus,notshownhere.(b)Representativegeometryoftheringresonatinggyroscopewithnon-uniformradialthicknesswithrh>0.Electrodesareomittedforclarity.Figure4.9:Manipulatingnonlinearparametersofathinspinningringbymodifyingtheradialringthicknessh(+)assume,aswedidpreviously,thatthegyroscopenonlineardynamicsisdominatedbytheelectrostaticrestoringforces,i.e.,|2e|>2mand|"e|>"m.Since|2|=|2e|!2mand|"|=|"e|!"m,itisclearthatinordertoincrease|2|anddecrease|"|,weneedtominimize2mandmaximize"m.FromFig.4.9aitfollowsthatbothgoalscanbeachievedifwe117modifytheringradialthicknesswithrh>0,seeFig.4.9b.Physically,thecasewhenrh>0correspondstoaddingthemasstotheareasoftheringthatexperiencelargestdeßectionsintheradialdirectionasitvibratesinthedriveandsensemodeswhileremovingthematerialintheotherareas.Inparticular,2(h)m<2(0)mwhenrh0(0,6!412)withitsminimumreachedatr+h=4!2%11/3.Inthiscase2(h)mreducesby'5%and"(h)mincreasesby'7%.Whiletheseresultsareobviouslylessimpressiveascomparedtotheonesweobtainedusingnonlinearelectrostatictuningmethods,seeSections4.3.1and4.3.2,thismechanicaloptimizationmethodshouldnotbeoverlookedwhenoneattemptstooptimizetheperformanceofmicromechanicalringresonatinggyroscopeswithrelativelysimplegeometriesoftheresonatorbody[19].4.4OutlookInthischapterwehaveanalyzedthephenomenonofself-inducedparametricampliÞcationofin-planeßexuralvibrationsofdegenerateellipticalmodesinring/diskresonatinggyroscopes.ThemostimportantfeatureofthisampliÞcationisagaininsensitivitythatisachievedfromthenaturallyoccurringdynamicsofthesystem.Thisisaprimeexampleofwherenonlinearbehaviorprovidesanopportunityforimprovedperformanceofapracticaldevice.Byutilizingthemodelofathinspinningringinthepresenceofelectrostaticactuation/sensing,wehavedemonstratedthat,inadditiontothelinearCorioliscouplingthatisthebasisforoperationasanangularratesensor,thedriveandsensemodesarecouplednonlinearlythroughelastic,inertial,andelectrostatice!ects.Wehavefurtherillustratedthatthismodalcouplingresultsinparametricpumpingofthesensemodebythedrivemode,whichcanleadtoasigniÞcantimprovementinthegyroscopesensitivitywithrespecttotheexternalangularrate,aswas118experimentallyobservedin[27].Wehavealsoexaminedthee!ectsofthedriveconditionsontheperformanceofthesensor,andillustratedthesee!ectsfortworepresentativemicro-mechanicalringresonatinggyroscopes.Inouranalysiswehavefocusedourattentiononthecasewhen$%!0(avalidassumptionforhighprecisiongyroscopes),whichallowedustoneglecttheback-actione!ectofthesensemodevibrationsonthedrivemodedynamics,signiÞcantlysimplifyingtheanalysis.Ourfuturee!ortsinthisdirectionwillbedevotedtodevelopingabetterunderstandingoftheinternalresonancethatnaturallyoccursinthesesystems,whichhavetwovibrationalmodeswithclosenaturalfrequencies.Thisleadstoveryrichdynamics,includinghighsensitivitytocertaine!ects,suchaslinearmodalcoupling(quadratureerror),whichneedtobebetterunderstoodinordertofullyexploitthenonlinearmodalcoupling.Additionally,wehavedescribedthreedi!erentmethodsonecanusetomanipulatenon-linearparametersofring/diskresonatinggyroscopesinasystematicwaytoimprovethegyroscopeperformanceasaratesensor.Inthisstudyourmaingoalwastomaximizetheintermodaldispersivecouplingstrengthbetweenthedriveandsensemodesinordertoen-hancetheself-inducedparametricampliÞcationandthegyroscoperatesensitivity,andtosimultaneouslysuppresstheindividualmodalDu"ngnonlinearityofthedrivemodeinordertomaximizethegyroscopelinearrange.TwooutofthreemethodsinvolvemodiÞcationsinthegyroscopeelectrostaticsetup,whicharefoundtobehighlye!ectiveformanipulatingtheresponse,andthethirdtechniquedealswithgeometrymodiÞcationsofthegyroscopemass,whichisfoundtobelesse!ective.SpeciÞcally,wehaveshownthatifthegyroscopeactuation/sensingschemehasatleast16electrodes,itispossibletomodifytheintermodalcouplingstrengthandmodalDu"ngconstantbyapplyinganon-uniformdistributionofthebiasvoltage.Additionally,wedemonstratedthatanon-uniformelectrodegapsizealso119a!ectsthemagnitudeofthegyroscopenonlinearcoe"cientsandcanbeusedtotailorthegyroscopenonlineardynamicbehavior.Finally,weillustratedtheapplicabilityoftheshapeoptimizationmethodinmanipulatingnonlinearsti!nessparametersofathinspinningringbyalteringtheringradialthicknessalongtheringcircumference.TheseexamplesclearlyillustratetheimpactofoptimizationmethodsforimprovingtheperformanceofMEMSgyroscopesandprovidesadditionalmotivationfortheirfurtherex-plorationforothertypesofMEMSresonators.Anobviousextensiontofurtherimprovetheperformanceofthering/diskresonatinggyroscopeistocombinetheaboveapproaches.Forexample,assumingthatthenonlinearbehaviorofthegyroscopeisprimarilydeterminedbyelectrostaticforces,i.e.,|2e|)|2m|and|"e|)|"m|,onecanapplymodiÞcationsofboththeelectrodegapsizeandthebiasvoltagealongthegyroscopecircumference.Inthiscase,followingtheresultsofSections4.3.1and4.3.2,oneshouldbeabletopreservethegyroscope[19]dynamicrangewhileachievingK'2.36,wherethelatterresultsinnearly12-foldincreaseofthegyroscopesensitivity,asseenfromFig.4.6.Usingobtainedresultsasastartingpoint,weplantocontinueourworkonoptimizationoftheperformanceofringanddiskresonatinggyroscopes.Particulardirectionswouldincludeidentifyingtheappropriatemetric(objectivefunction)thatwouldcharacterizetheoverallperformanceofthegyroscopeandanalysisofthewaysthatwouldoptimizetheselectedobjectivefunction.Additionally,weplantotestourhypothesesandverifyanalyticalÞndingswithourcollaborators.Theanalyticalresultspresentedherecanbeusedforpredictingthenonlinearbehaviorofexistinggyroscopesandproposedgyroscopemodels,and,moreimportantly,fordesign-ingring/diskresonatinggyroscopeswithoptimizedperformanceandmaximizedsensitivityusingtheirinherentdynamics.However,tomakefulluseoftheseideasonemustemploycomputationaltoolsthatallowforoptimizationusingmanymoredesignvariables.Such120methodscanbeappliedtotherestrictedproblemsproposedhere,combinationsofthem,ortoentirelynewconÞgurations,possiblysuggestedbyshapeandtopologyoptimizationtechniques.121Chapter5ConclusionsInthisdissertationwehaveconsideredthreetopicsrelatedtononlinearandnoise-induceddynamicsinmicro-scaleand,toacertainextent,innano-scaleelectromechanicalresonators.Ingeneral,ourperspectiveonnonlinearitiesandnoiseinMEMSistoacceptthemasintegralpartsofvirtuallyanyMEMSresonatormodel,toÞndwaystounderstandtheire!ects,todevelopmethodsforcharacterizingthem,and,wherepossible,toemploythemforspeciÞcapplications.ThespeciÞcgoalswere:(i)toextendourunderstandingofPoissonnoiseinparametricallydrivenMEMSresonators;(ii)todevelopcomprehensiveandrobustmethodsforquantitativecharacterizationofnonlinearitiesandnoiseinMEMS;and(iii)todevelopmodelsforthenon-linearself-inducedparametricampliÞcationobservedinMEMSringanddiskresonatinggyroscopesandusethesemodelstoproposenewdesignsthatimprovetheirperformanceasangularratesensors.InChapter2westudiedtheprobabilitydistributionofaparametricallydrivenresonatordrivenbymodulatedPoissonpulsesinthepresenceofweakthermalnoise.Thepulsesareactuallycomposedofshortburstsofharmonicsignalatonehalfofthedrivingfrequencywiththedurationofeachpulsemuchlongerthantheresonatorvibrationperiodbutmuchsmallerthantheresonatortypicalrelaxationtime.Intherotatingframeofreferencetheseburstsappearasessentiallyinstantaneouspulsesthatcausejumpsinthelocationofthestateinthatframe.WiththeabilitytotunethedirectionofthesePoissonpulses,andtochoosethemeasurementquadratureintherotatingframe,wehaveshownthat,forsu"-122cientlysmallpulserates,theprobabilitydistributionexhibitsapower-lawsingularityneartheresonatorequilibriuminboththeoverdampedandunderdampedregimesoftherotatingframe.Additionally,wehavedescribedthedependenceofthecorrespondingexponentonthePoissonmeanpulserateandthesystemdecayrateintherotatingframe.Wealsofoundadditionalpeak(s)inthedistributionawayfromtheoriginandspeciÞedtheirpositionsandtheconditionsfortheirappearance.Inparticular,wedemonstratedthattheprobabilitydistributionisstronglyasymmetricintheoverdampedregime,whileintheunderdampedregimeithasaself-similarstructure.Weakthermalnoisea!ectsthesystembysmoothen-ingthesingularpeaksinthedistributionandbymakingthedistributionGaussianinthevicinityofthesepeaks.Ouranalyticalresultsforthepower-lawexponentsnearthepeaksoftheprobabilitydistribution,andforthepositionsofpeaksintherotatingframe,areinexcellentagreementwithexperimentalmeasurementsfromamicromechanicalresonatortakenbycollaboratorsattheHongKongUniversityofScienceandTechnology.Importantly,ourresultscanbeimmediatelyextendedtounderstandthedynamicsofaMEMS/NEMSresonatorsubjecttoamultiplicativenoiseofPoissontype,whichisfrequentlythecaseinmasssensingapplications.InChapter3weintroducedanddevelopedacomprehensivemethodforcharacterizingmodelparametersforMEMSresonatorswhosevibrationaregovernedbyasinglemode.Themethodreliesentirelyonmeasurementstakenfromasingleringdownsignalofalightlydampedsystem.Thisapproachhastheadvantagethattheresonatorresponseisnota!ectedbyanydrivetransduction,whichreducesmodelinguncertainty;onemustonlyaccountforreadouttransductiontoseparatetheunderlyingresonatordynamics.Themodelparametersofinterestincludecoe"cientsoflinearandnonlineardampingandsti!nessterms,aswellasthemostimportanttypesofnoisesources.Inparticular,wehaveillustratedhowone123canestimatevaluesfordeterministicsti!nessanddampingparametersforthesymmetricvibrationalmodeofaMEMSresonator.Akeytothemethodisthatthevibrationalam-plitudeisa!ectedonlybythedissipationparameters,whilethefrequencyandphasearea!ectedbytheresonatorconservativeparameters,therebyuncouplingthecharacterizationprocess.First,thelinearandnonlinearfrictioncoe"cientsareobtainedfromtheshapeofthevibrationenvelope,Þttedtoamodelfortheslowlyvaryingamplitudeobtainedfromthemethodofaveraging.Additionally,wehavedemonstratedhowonecanestimatethemodalnaturalfrequencyandconservativecubicandquintic(sti!ness)nonlinearitiesusingamea-suredsequenceofzero-crossingtimesintheringdownresponse,whicharea!ectedbytheinstantaneousfrequencyofvibration,whichisdirectlylinkedtothesti!nessmodel.Notethatthisringdown-basedcharacterizationmethod,unlikeconventionalfrequencysweeping,isuniquesinceallowsdirectrevealationofthenonlineardissipationinthesystemofinterest.Furthermore,wehavedevelopedanoveltechniquefordirectdeterminationoftheintensitiesofmeasurement,thermal,andfrequencynoisesourcesintheresonatoranditsreadoutsignal,byperformingastatisticalanalysisofthelocationsofzero-crossingpointsinthemeasuredringdownresponse.Wehaveshownthatthesenoisesourceshavequalitativelydi!erentsig-naturesonthejitteroftheresonatortiming.Therefore,usingthesespeciÞcfeaturesofthejitterwearetoestimatetheintensitiesofthesethreeimportantnoisesources.Possiblefuturedirectionsincludeadaptationoftheringdown-basedmethodforparametercharacterizationofsystemswithasymmetricrestoringforces,e.g.beamresonatorswithoneelectrode,andcoupled-modesystems.Additionale!ortscanbedirectedtowardsimprovingaccuracyofthenoisecharacterizationinthesystemswithstrongmeasurementnoiseandgainingmorecomprehensiveunderstandingoftheoriginofdi!erentnoisesourcesbystudyinghigh-orderstatisticsoftheresonatorphaseintheringdownresponse.124InChapter4weanalyzedthephenomenonofself-inducedparametricampliÞcationofin-planeßexuralvibrationsofdegenerateellipticalmodesinring/diskresonatinggyroscopes.Thisstudywasmotivatedbyexperimentalobservationsdescribedin[27].Themostimpor-tantfeatureofthisampliÞcationisagaininsensitivitythatisachievedfromthenaturallyoccurringdynamicsofthesystem,whichimprovesbothdevicesensitivityanditssignal-to-noiseratio(SNR).Thisisaprimeexampleofwherenonlinearbehaviorprovidesanopportunityforimprovedperformanceofapracticaldevice.Byutilizingthemodelofathinspinningringinthepresenceofelectrostaticactuation/sensing,wehavedemonstratedthat,inadditiontothelinearCorioliscouplingthatisthebasisforoperationasanangularratesensor,thedriveandsensemodesarecouplednonlinearlythroughelastic,inertial,andelec-trostatice!ects.Wehavefurtherillustratedthatthismodalcouplingresultsinparametricpumpingofthesensemodebythedrivemode,whichcanleadtoasigniÞcantimprovementinthegyroscopesensitivitywithrespecttotheexternalangularrate,aswasexperimentallyobservedin[27].Wehavealsoexaminedthee!ectsofthedriveconditionsontheperfor-manceofthesensorandillustratedthesee!ectsfortworepresentativemicro-mechanicalringresonatinggyroscopes.Additionally,weusedourmodelingresultstodevelopmethodsonecanusetomanipu-latethenonlinearsti!nessparametersofthesegyroscopesinasystematicwayinordertoimprovetheirperformanceasratesensors.Ourmaingoalwastomaximizetheintermodaldispersivecouplingstrengthbetweenthedriveandsensemodes,inordertoenhancetheself-inducedparametricampliÞcationandthegyroscoperatesensitivity,whileatthesametimetoreducetheindividualmodalDu"ngnonlinearitytermsinordertomaximizethegyroscopelinearrange.ThisledtoanobjectivefunctiondeÞnedastheratioofthedisper-sivemodalcouplingstrengthandtheindividualmodalDu"ngnonlinearity.Physically,this125objectivefunctionrepresentstheamountoftheparametricpumpingthatisinducedbythedrivemodeonthesensemode,whentheformerisoperatedattheonsetofitsnonlinearregime.TheoptimizationproblemforthesedevicesiseasilyformulatedintermsofFourierseries,sincethedevicemustmaintainitscircularsymmetry.ThemethodsdevelopedinvolvemodiÞcationsinthegyroscopearounditscircumference.Theseincludealteringtheelectro-staticsetup,boththegapdistribution,whichistypicallyuniform,andtheuseofdi!erentvoltagesinsegmentedelectrodes,aswellasthegeometryoftheresonatorbody,whichaltersitsmechanicalproperties,bothsti!nessandinertial.SpeciÞcally,wehaveshownthatifthegyroscopeactuation/sensingschemehasatleast16electrodes,itbecomespossibletomodifytheintermodalcouplingstrengthandthemodalDu"ngconstantbyapplyinganon-uniformdistributionofthebiasvoltage.Additionally,wedemonstratedthatanon-uniformelectrodegapsizealsoa!ectsthemagnitudeofthegyroscopenonlinearcoe"cientsandcanbeusedtotailorthegyroscopenonlineardynamicbehavior.Finally,weillustratedtheapplicabilityoftheshapeoptimizationmethodinmanipulatingnonlinearsti!nessparametersofathinspinningringbyalteringtheringradialthicknessalongtheringcircumference.AnalysisshowsthatusingpropermodiÞcationstothegyroscope/electrodegeometryaswellasthedistributionofthebiasvoltage,onecansigniÞcantlyincreasethevalueoftheintermodaldispersivecouplingstrengthwhilepreservingthegyroscopedynamicrange.Inthiscase,thegyroscoperatesensitivityisexpectedtoincreasebymorethananorderofmagnitude.Futureworkinthisareaincludesexperimentaldemonstrationoftheapproach,aswellastheuseofmoresophisticatedoptimizationtoolsthatallowformoregeneraldesigns.Thelatterwillbeessentialforcomplexgeometries,suchasthoseindiskresonatinggyros(DRGs).Theanalyticalresultspresentedherecanbeusedforpredictingthenonlinearbehaviorofexistinggyroscopesandproposedgyroscopemodels,and,moreimportantly,fordesigning126ring/diskresonatinggyroscopeswithoptimizedperformanceandmaximizedsensitivityus-ingtheirinherentdynamics.Possiblefuturedirectionsincludeadaptingthetuningmethodspresentedhereforapplicationinrealring/diskresonatinggyroscopesinattempttoimprovetheirdynamicrangeandangularratesensitivity.Insummary,thetopicsofthisdissertationdemonstratethefollowingpointforMEMSresonators:(i)theutilityofsystematicmodeling,(ii)theimportanceofnonlinearbehavior,(iii)theimportanceofaccountingfornoise,and(iv)theutilityofthepreviousitemsincharacterizationanddesign.127APPENDICES128AppendixAPoissonnoiseintheresonatoreigenfrequencyTheanalysisoftheresonatorquadratureprobabilitydistributionpresentedinSection2.3canbeextendedtoaccountforthecasewhentheresonatoreigenfrequencyßuctuationshavePoissonstatistics.Thisisofparticularinterest,forexample,insystemswhereattach-ment/detachmentofmoleculestoaresonatorcausesßuctuationsoftheresonatore!ectivemassand,asaresult,oftheresonatoreigenfrequency[51].Inthiscasethegoverningequationofmotionfortheresonator,Eq.(2.2),canbewrittenas¬++2#ú++(!20+.!(t)+hcos!t)++"+3=0,(A1)where.(t)istheregularPoissonnoiseoftheform.!(t)=g1j#(t!tj)andwedisregardthermalnoiseforsimplicity.Ofcourse,itse!ectcanbeaccountedforinasimilarwaytothatinSection2.3.3.FromEq.(A1),itfollowsthattheresonatordynamicsintherotatingframecannowbedescribed,similarlytoEq.(2.7),asúq=K(q)+lq.!(0),(A2)129wherelq=(Y,!X)Tisthepulsedirectionintherotatingframeasdeterminedbythecurrentstateofthesystemand.!(0)isthePoissonnoisewiththemeanpulserate&,=&/#andpulseareag,=g/!0.TostudythequadratureprobabilitydistributionwithweakmultiplicativenoiseofPois-sontype,oneneedstolinearizeEq.(A2)aboutitsdeterministicstationarysolutionq0.Inthiscaselq=(Y0,!X0)Tandinordertoobservethee!ectof.!,onehastopumpthesystemaboveitsthresholdtoensurethatq0/=0.TherestoftheanalysisisthensimilartothatpresentedinSection2.3.TheimportantfeatureoftheresonatorwithmultiplicativePoissonnoise,ascomparedwiththesystemdescribedindetailinChapter2,isthatthedirectionofthePoissonpulsesintherotatingframelqisnowpredeterminedbytheloca-tionoftheresonatoroperatingpointq0,whichisgenerallymoredi"culttocontrolintheexperiment.Nevertheless,onestillhasfreedomtochoosedi!erentdirectionsforthemea-surementquadratureqmandtoanalyzetheresonatorquadratureprobabilitydistributioninbothoverdampedandunderdampedregimes.130AppendixBMechanicalandelectrostatice"ectsinclamped-clampedbeamresonatorsInthissectionweformallyderivemechanicalandelectrostaticcontributionstotherestoringforceofMEMSresonatorswithÒclamped-clampedbeamÓtopology.Byincludingbothmechanicalandelectrostaticforces,discussedinSection3.3.3andactingontheresonatorperunitlength,wecanobtainthefollowingmodelfortheunforcedßexuralvibrationsoftheDA-DETFresonatorunderstudy[45],%S¬y+c(y,úy)úy=!EIy(4)+Fs(x,t)+Fe(x,t),(B.1)wherey=y(x,t)istheresonatorphysicaldisplacement(xisthecoordinatealongtheresonatorlength),%,E,andIaretheresonatormassdensity,YoungÕsmodulusandthecross-sectionalmomentofinertiaandc(y,úy)úyisthephenomenologicalresonatordampingforce,whichcanbelinearornonlinear[35].TheÞrsttermontheright-handsideofEq.(B.1)representsthee!ectofbeambending,thetermFs(x,t)=ESLy,,,L0dx751+(y,)2!18(B.2)131accountsfortheresonatormid-linestretchinge!ect,andFe(x,t)istheelectrostaticforceduetotheÞnitebiasvoltagebetweentheresonatorbodyandattendantelectrodes.Sincetheelectrodegapsized%L,whereListheresonatorlength,wecansafelyapplyalocalparallelplateassumptionandwritetheelectrostaticforceactingonasymmetricallybiasedclamped-clampedbeammicromechanicalresonatorasFe(x,t)=2-(d!y)!2!(d+y)!2.=42dy(d2!y2)!2,(B.3)where2=-0wV2b/2isthestrengthoftheelectrostaticforce,whichdependsontheresonatorthicknesswinthedirectionperpendiculartotheresonatorvibrationsandthebiasvoltageVb.Notethat-0=8.85*10!12F/misthevacuumpermittivity.Inordertoexaminethemechanicalandelectrostatice!ectsonthedynamicsofapar-ticularvibrationalmode,onehastoperformtheGalerkinprojectionofEq.(B.1)ontothemodeofinterest.Thisiscommonlydonebyseparatingtemporaltandspatialxvariablesasy(X,t)=q(t)+(X)withX=x/L,whereq(t)isthetime-dependentmodalcoordinateand/(X)isthemodeshape.Findinganexactexpressionfor+(X)isgenerallyachallengingproblem,anditcanbesolvedintheclosedformonlyforcertainsystemsintheirlinearrange.Inpractice,however,modalnonlinearitiesaswellasdeviationsofthedeviceboundarycon-ditionsfromanidealclamped-clampedmodelresultinthedistortionoflinearmodeshapes.Inthiscaseweemployverypracticalmethodofassumedmodes,andsinceherewefocusontheprimaryßexuralmodeoftheDA-DETFresonator,wechoose+(X)=sin2)X,whichcloselyresemblestheactualdeformationproÞleofaclamped-clampedbeam.Projecting132Eq.(B.1)on+(X)yields¬q+2#(q,úq)úq+!20mq+Cm22)E#!)2q2L$!13q+Ce21(1+qd)3/2!1(1!qd)3/23=0,(B.4)where!0m=4)2%EI/3%SL4isthelinearmechanicalmodalfrequencyduetobendinge!ects,Cm=4)2E/3%L2isthestrengthofthemechanicalpotentialduetothemid-linestretching,Ce=2-0wV2b/3%Sd2representsthestrengthofelectrostaticcontributiontothesystemrestoringforceandq0=q/'isthemodalcoordinatescaledwiththeappropriatelengthscale,dorL.Forthesakeofgenerality,wekeepthemodaldecayrate#(q,úq)initsgeneralform,sincedissipationdoesnota!ectthemodalvibrationfrequencyintheresonatorringdownresponse[35].Thee!ectoftheelectrostaticactuation/sensingschemerepresentedinEq.(B.4)bytheterm.Cecontributestobothlinearandnonlinearcomponentsoftheresonatorrestor-ingforce.Inthelowestorder,.q,theelectrostaticforcesa!ecttheresonatorvibrationfrequencyas!0=5!20m!3Ce/d,thewell-knownphenomenonfrequentlyusedfortun-ingmodalfrequenciesinMEMSresonantdevices,e.g.formatchingmodalfrequenciesinMEMSvibratorygyroscopes.Whenthemodalvibrationamplitudeexceedstheresonatorlinearrange,itsvibrationfrequencybecomesamplitude-dependentduetothee!ectoftheresonatormid-linestretchingandanharmonictermsintheelectrostaticpotential.Afterthelinearrange,theresonatorentersitsDu"ngrange,wherethesystemdynamicbehaviorisdeterminedbyquadraticandcubictermsintheresonatorrestoringforce.Insymmetricallybiasedresonators,however,theresonatorrestoringforcedoesnotcontainquadraticterms,aswellashigher-ordernonlineartermswithevenpowersofq,whichcanbeeasilyveriÞedbyexpandingnonlineartermsinEq.(B.4).AstheresonatoramplitudeincreasesintheDu"ng133range,theresonatorfrequencyeitherincreases,seeFig.3.3,ifthemechanicale!ectofthemid-linestretchingdominatesoverelectrostaticnonlinearitiesordecreasesintheoppositescenario.FromFig.3.3itisclear,however,thattheresonatorfrequencyisanon-monotonicfunctionofthemodalamplitude,whichcannotbeexplainedbyrelativelysimpleDu"ngmodel.Inthiscase,higher-orderquinticterms(wedisregardquartictermsduetothesym-metryreasons)inthemodalrestoringforcemustbetakenintoaccountinordertocaptureandexplaintheexperimentalobservation.Mechanicalandelectrostaticcontributionstotheresonatoramplitude-frequencyrelation-shipcanbedemonstratedifwewritetheresonatormodaldisplacementasin??.Inthelightofourpreviousdiscussion,wefocusourattentiononthedynamicsoftheslowlyvaryingresonatorphase/(t);byapplyingthemethodofaveraging[144]toEq.(B.4),weobtain,ú/=%!m(a)+%!e(a),(B.5a)%!m(aL)=Cm2!0aL283)4aL#E2(Q)(4)2a2L!2)!K2(Q)(1!)2a2L+51+)2a2L)+2E(Q)K(Q)(1!2)2a2L+51+)2a2L)!aL3,(B.5b)%!e(ad)=!2Ce)!0d1a2d(1!a2d)3/2*2E#2adad!1$51!a2d(%1!ad+%1+ad)!K#2adad!1$%1+ad(1!a2d)3,(B.5c)where%!m(a)and%!e(a)arethemechanicalandelectrostaticamplitude-dependentshiftsofthemodalvibrationfrequencyduetheresonatormid-linestretchingandelectrostaticinteractionoftheresonatorwiththeattendantelectrodesoftheactuation/sensingscheme,respectively.Inaboveexpressions,aL=a/L,ad=a/d,Q=12!1251+)2a2L,andK(...)andE(...)arecompleteellipticintegralsoftheÞrstandsecondkind,respectively.Asit134followsfromEqs.(B.5b)and(B.5c),mechanicalandelectrostaticcorrectionstotheresonatorfrequency,writtenintheirclosedfrom,havequitecomplicatedformand,asaresult,weusetheirTaylorexpansionsinSection3.3.3inordertoapplytheringdown-basedmethodforindependentcharacterizationofmechanicalandelectrostaticcontributionstotheresonatorsti!nessparameters.Nevertheless,Eqs.(B.5b)and(B.5c)areusefulwhenonewouldliketostudytheresonatordynamicbehaviorwithinalargerangeofvibrationalamplitudes.135AppendixCDerivationofnonlinearstrain-displacementrelationshipsinafreeringHerewederivethenonlinearstrain-displacementrelationshipsfor-ij(u,v)presentedinEqs.(4.5a)to(4.5c).Inordertodoso,weconsideraninÞnitesimalsegmentofthegyroscopebody,designatedbyKLMNinwithcoordinatesrand+andhavingradialthicknessdrandangularlengthrd+;seeFig.C.1.ThissegmentcanbeconvenientlydeÞnedintermsofthecoordinatesofitscornerpointsasK=(r,+),L=(r+dr,+),M=(r+dr,++d+),N=(r,++d+).(C.1)Duringoperationthebodyexperienceselasticdeformationsandthesegmentdeformsinto136FigureC.1:DeformationofthegyroscopesegmentKLMNintoK1L1M1N1.K1L1M1N1,whichwe,inturn,expressasK1=#r+u(r,+),++v(r,+)r$,L1=#r+dr+u(r+dr,+),++v(r+dr,+)r+dr$,M1=#r+dr+u(r+dr,++d+),++d++v(r+dr,++d+)r+dr$,N1=#r+u(r,++d+),++d++v(r,++d+)r$.(C.2)FromFig.C.1itisclearthatthestrain-displacementrelationships-ij(u,v)aregivenby-rr=K1L1!KLKL,---=K1N1!KNKN,-r-='+(.(C.3)GiventhecoordinaterepresentationsofthesegmentcornerpointsinEqs.(C.1)and(C.2),137wehaveKL=dr,K1L1'dr6#1+,u,r$2+#1+ur$2#,v,r$2,(C.4a)KN'rd+,K1N1'rd+6#1+ur$2#1+,vr,+$2+1r2#,u,+$2,(C.4b)''r+u1+1u/1r,,r#vr$,('1u/1-(r+u)(1+1v/r1-).(C.4c)Finally,byusingEqs.(C.4a)to(C.4c)inEq.(C.3)andexpandingtheresultingexpressionsuptothesecondorderinuandv,weobtainthenonlinearstrain-displacementrelationshipsgiveninEqs.(4.5a)to(4.5c).138BIBLIOGRAPHY139BIBLIOGRAPHY[1]DarioAntonio,Dami«anHZanette,andDanielL«opez.Frequencystabilizationinnonlinearmicromechanicaloscillators.Naturecommunications,3:806,2012.[2]EyalKenig,MCCross,LGVillanueva,RBKarabalin,MHMatheny,RonLifshitz,andMLRoukes.Optimaloperatingpointsofoscillatorsusingnonlinearresonators.PhysicalReviewE,86(5):056207,2012.[3]AlexandraNafari,DavidKarlen,CristinaRusu,KristerSvensson,HûakanOlin,andPeterEnoksson.MEMSsensorforinsituTEMatomicforcemicroscopy.Microelec-tromechanicalSystems,Journalof,17(2):328Ð333,2008.[4]GPrakash,SHu,ArvindRaman,andRReifenberger.Theoreticalbasisofparametric-resonance-basedatomicforcemicroscopy.PhysicalReviewB,79(9):094304,2009.[5]TOuisse,MStark,FredericoRodrigues-Martins,BBercu,SHuant,andJChevrier.TheoryofelectricforcemicroscopyintheparametricampliÞcationregime.PhysicalReviewB,71(20):205404,2005.[6]BScottStrachan,StevenWShaw,andOlegKogan.Subharmonicresonancecascadesinaclassofcoupledresonators.JournalofComputationalandNonlinearDynamics,8(4):041015,2013.[7]KRQalandar,BSStrachan,BGibson,MSharma,AMa,SWShaw,andKLTurner.Frequencydivisionusingamicromechanicalresonancecascade.AppliedPhysicsLet-ters,105(24):244103,2014.[8]ANClelandandMLRoukes.Noiseprocessesinnanomechanicalresonators.JournalofAppliedPhysics,92(5):2758Ð2769,2002.[9]AkihisaIchiki,YukihiroTadokoro,andMIDykman.Singularprobabilitydistributionofshot-noisedrivensystems.PhysicalReviewE,87(1):012119,2013.[10]JSAldridgeandANCleland.Noise-enabledprecisionmeasurementsofaDu"ngnanomechanicalresonator.Physicalreviewletters,94(15):156403,2005.[11]RichardPFeynman.ThereÕsplentyofroomatthebottom.Engineeringandscience,23(5):22Ð36,1960.140[12]HarveyCNathanson,WilliamENewell,RobertWickstrom,JohnRansfordDavisJr,etal.Theresonantgatetransistor.ElectronDevices,IEEETransactionson,14(3):117Ð133,1967.[13]KurtEPetersen.Siliconasamechanicalmaterial.ProceedingsoftheIEEE,70(5):420Ð457,1982.[14]TMattila,JKiiham¬aki,TLamminm¬aki,OJaakkola,PRantakari,AOja,HSepp¬a,HKattelus,andITittonen.A12MHzmicromechanicalbulkacousticmodeoscillator.SensorsandActuatorsA:Physical,101(1):1Ð9,2002.[15]VilleKaajakari,JukkaKKoskinen,andTomiMattila.Phasenoiseincapacitivelycou-pledmicromechanicaloscillators.Ultrasonics,Ferroelectrics,andFrequencyControl,IEEETransactionson,52(12):2322Ð2331,2005.[16]ClaudeAudoinandBernardGuinot.Themeasurementoftime:time,frequencyandtheatomicclock.CambridgeUniversityPress,2001.[17]ArkadyPikovsky,Micha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