MODELING PROBE-BASED DATA STORAGE DEVICES

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arrayinanareaof3mm 3mm.Indentationsizesandspacingreachassmallas30-40 nmwhichleadstohighstoragedensity. Schlosser etal. [16]andGrif n etal. [5,6]atSchoolofComputerScience,Carnegie MellonUniversitymodelthedataseekandaccessdynamicsonaperrequestbasisfora probe-based(MEMS)storagesystemusing rst-ordermechanics.Theydiscussthedesign ofthephysicalstoragedevice,usesimulationstoexplorehowdifferentphysicalcharac- teristicsimpactthedevice 9sdesigntrade-offsandperformance,anddescribeanumberof interestingarchitecturalusesforMEMS-basedstorageinsystems.Thisthesiswilldiscuss indetailtheirapproachinmodelingthedynamicsofdataaccessandcompareitwithour springmodelapproach. 7 Chapter3 ModelsforProbe-BasedDataStorage Tomakeaccuratedecisionsaboutdataplacementandscheduling,weneedamodelthat preciselydescribestheseektimeforprobe-basedstoragedevices.Thischapterdescribes thedesignandmechanicsofprobe-baseddatastorageandexaminetwomodelsthatrely onmacroscopicphysicalprinciples:thebang-bangmodelandthespringmodel. 3.1DeviceFeaturesandMechanics Probe-basedstoragedevicescanbebuiltwithavarietyofparametersthatyielddiffer- entperformancecharacteristics.Forexample,wecanvarythetipcon guration,number ofsleds,andtypeofmedia.Theprobe-baseddevicediscussedinthisthesisisbasedupon thedesignbyCarleyatCHI PS [2].Inthisdesign,asshowninFigure3.1,theprobe- basedstorageiscomposedoftwoparts.Theupperpart(thegrayparallelograminthe gure)isamovablesledofmagneticdata,andthelowerpartisa xedarrayofread/write 8 Dx Dy ty = # tips (y) assumed to be even tx = #tips (x) assumed to be even bty = #bits per tip (y) btx = # bits per tip (x) Dx Dy Figure3.1 :Probe-BasedStorage tipswhichis100( )by100( ).Themediadensityis50nmperbit.Eachtipcanassess dataatarateof200Kbit/second.Thetiparrayissparse,soeachtipcanmanipulatearect- angleofbitswhilethesledismovingaboveit.ThisrectangleishighlightedinFigure3.2 where and arethedistancesthatonetipcanaccessrespectively. Figure3.2showsanexampledatalayoutscheme.Thisdataaccesspatternisbased upontwoassumptions: rst,onlyonerowoftipsisactiveatatime,andsecond,tipsare capableofreading/writinginbothdirectionsof and .Eachsector 9sbitsarestriped onthemediaalongthe -axis,sothe rstbitisaccessedbythe rsttip,thesecondbythe secondtip,etc.Thusthesectorisread/writtenbyallthetipsofonerowinparallel,while thesledmovesaboveitinthe direction.Afteronerowoftipsaccessesdataalongthe 9 BPS = bits per sector Figure3.2 :MappingDiskSectorstoProbe-BasedStorage -axis,thenexttiprowisactivatedandthesledreversesdirection.Afteralltherowsof tipsonthechipsubstrate nishreading/writingacolumnofbits,thesledrepositionsto thenextcolumnwhichisthentobeaccessed.Tominimizesledmovement,theevenrows accessdatain direction,whiletheoddrowstakethe direction. Themediasledissuspendedacoupleofmicronsabovethetipsubstratebysilicon beamsthatactassprings,andmovedaroundbyforcesgeneratedbylateralresonantmi- croactuators,asshowninFigure3.3.Inthis gure,theshadedpartsmoveandthewhite partsdonot.Electricforcesappliedtothe ngersofthemicroactuatorcombsexertelec- trostaticforcesonthesledthatcauseittomoveinthe and direction,overcomingthe forcesexertedbytheanchorsandbeamsthatkeepitinplace. Toreadorwritedata,thesledhastobemovingoverthetiparrayataspeci edvelocity. Requestsarequeued,andafterservicingonerequest,thesledwillrepositionitselfsothat thetiparraycanaccessthedatarequiredbythenextrequest.Thismovementiscalled 10 \x8\x4\xc\xb\x3\x4\xd\xe \x3\xd\xb\xc \x8\x3\x2\x9 \xa \x4\x7 \xb\xc Figure3.3 :SledandMicroactuators seek .WedescribetwomodelsforthiscomponentofaccesstimeinChapter3.Oncethe sledisinposition,itmovesoverthedatawithaconstantvelocity.The read/writetime is afunctionofthesledvelocityandthedatalayout,andismodeledseparately,asdescribed inChapter5. Intermsofmechanics,therearethreeforcesworkingonthesledwhenitmoves:the externalelectrostaticforceproducedbytheactuator,therestoringforcefromthespring, andthedamping(friction)forcemainlyfromtheair.Eachofthethreeforcesaffectsthe sledmovement.Theelectrostaticforcegeneratedbyasinglecomb ngerisexpressed withtheequation .Here isthepermittivityoffreespace, istheapplied voltage, isthebeamthickness,and isthecomb ngergap. 11 3.2Bang-BangModel Thebang-bangmodel,ourtermforthemodelpresentedinageneralizedformby Schlosser etal. [16],describesthedynamicsofthesledmovementinprobe-baseddata storagesystemaccordingtoNewtonianlawsofmechanics.Thesledmovementinthe bang-bangmodelisdescribedbyitsposition,velocityandacceleration.Inthismodel, theseektime isdecomposedintotwoparts:thepositioningtime andtheset- tlingdowntime .Whenthesledsmovesfromonepositiontoanother,themodel assumesthatitisacceleratingwithmaximumpossibleaccelerationtothemidpointand thendeceleratingtheremaininghalfdistance. First,thecalculationofpositioningtime usesthefollowingstandardformula: (3.1) Herewegettheposition giveninitialposition ,initialvelocity ,time ,andac- celeration .Becausethebang-bangmodelusesconstantaccelerationandinitialvelocity ofzero,wecanderivethefollowingequationforthetimetoreachhalfthedistance: where ishalfofthetotalpositioningtime.With ,theaboveequation yields Inthebang-bangmodel,thetotalpositioningtimedoublesthetimespentonmovinghalf 12 thedistance.Therefore, (3.2) Thesledneedsanadditionaltimetosettledown,whichisthetimerequiredforthe oscillationofthespring-mountedsledtodampenoughfortheprobetiptofunction.The bang-bangmodelsassumesaconstantsettledowntime.Ifincludingthistime,wegetthe totalseektimeasproposedbythebang-bangmodel: (3.3) BaseduponEquation(3.3),thetotalseektimedependsupontheratioofthedistance movedandtheacceleration.Whenthesledmovesalongerdistance,ittakesmoretime with xed.Ontheotherhand,tomovethesamedistance,whenacceleration islarger, ittakeslesstime.AccordingtoEquation(3.3)othermechanicalcharacteristicsofthe probe-basedstoragedevicesuchasspringforceanddampingforcedonotdirectlyaffect theseektime.Theyareabsorbedbyacceleration.Equation(3.3)canbeusedtocalculate thesled 9sseektimeineither or dimension.Whenthesledmovesinbothdirections, theseektimeisthemaximumofthosein and directions. Inpractice,andasourresultswillshow,Equation(3.3)isanaggressiveassumption leadingtounderestimatesofseektime.Accelerationisnotaconstantinsledmovement; itchangesovertime.Andvelocitydoesnotvarylinearlyovertime.Besides,thebang- bangmodeltakesconstantsettledowntimewhileintuitivelytheseektimeissubstantially smallwhenthedistancemovedapproacheszero. 13 IntherecentpaperbyGrif n etal. [6],theabovemodelismodi edsothattheaccel- eration couldchangealongwithposition.Itusesthepiecewise-constantapproximation toallowtheaccelerationtochangeaccordingtothespringforceatdifferentpositionsof thesled.Itbreaksthewholesled 9smovementprocessintoasetofsmaller cchunks d,with thenetaccelerationineachchunkbeingthesumoftheaccelerationduetotheactuators andtheaccelerationduetothesprings.Thiscausestheaccelerationtochangediscon- tinuouslyinthesledmovement.Whenthenumberof cchunks dbecomesin nitelylarge, thisdiscretemodelingpracticeisclosetoacontinuousversionofmodelingapproach, whichisessentiallythespringmodelwedescribeinSection3.3.Thispaperwillfocuson Equation(3.3)whentalkingaboutthebang-bangmodel. Webelievethatthebang-bangmodel,whileusefulforunderstandingsomeofthe characteristicsofprobe-basedstorage,leavesoutimportantfeaturesoftheunderlying architecture.Thespringmodel,whichistobedescribedinSection3.3,isbasedmore closelyuponthemechanicsinvolvedinthesledmovement. 3.3SpringModel Thissectionpresentsourapproachtomodelthesled 9smovementintheprobe-based storagesystem,whichisthemajorcontributionofthethesis.Thespringmodeldescribes thedynamicsoftheprobe-baseddeviceusingclassicmechanicstheory.Inthemodel,the movementofthesledisdeterminedbythefollowingthreeforces:theexternalforce,the springforce,andthedampingforce.Equation(3.4)isthekeyequationofdescribingthe 14 mechanics: (3.4) Here, isthepositionofthesledattime , isthemassofthemovingsled, is thedampingcoef cient, isthecoef cientoftheelasticityoftherestoringforce(spring constant),and isanexternalforce. Theleft-handsideofEquation(3.4)consistsofthreeterms.Theterm describes themovementofsledwithmass accordingtoNewton 9sSecondLawofMotion,where istheaccelerationrate.Thehigher is,thelargertheacceleration,andvicevisa. Thesecondterm describestheimpactofdampinguponthemovement,where is themovementvelocity.Withforcegiven,largedampingorfrictionwillleadtolower velocity.ThethirdtermofEquation(3.4), ,isabouttheeffectoftherestoringsprings whichindicatesthatthestrongerthespringis(larger ),thesmallerdistancethesled couldmove(less )atagivenlevelof .Forexample,itishardertocompressastronger spring. Theactualmovementofthesledovertimeiscomplicatedbecauseatanytimepoint thesledposition,velocityandaccelerationaredeterminedbytheabove-mentionedthree forcessimultaneously.SolvingthisEquation(3.4),whichisaclassicalsecond-ordernon- homogeneousdifferentialequation,twostepsareusedaccordingtoZeldovich etal. [17]. Formoredetailedderivationofthesolution,pleaseseeAppendixA. Section3.3.1andSection3.3.2presentthetwostepstoderivethekeyEquation(3.4) andprovidetheinsightsastohowthephysicalandmechanicalnatureofthesleddevice 15 affectsthewholemovementprocess,itsdynamicbehavior,andseektime. 3.3.1SolutionofSpringModelwithNoExternalForce The rststepconsidersfreeoscillation, i.e. ,thecasewithoutexternalforce .Equa- tion(3.4)thenbecomesthehomogeneousequation: (3.5) ItturnsoutthatthesolutiontoEquation(3.5)dependsuponthesignofthediscrimi- nant , (3.6) When ,thesolutionforEquation(3.5)isasfollows: (3.7) where and areconstantstobedeterminedbytheinitialconditionsofthesled, and and arepositiveandtheyarefunctionsof , ,and (seeAppendixA). Solution(3.7)isapplicablewhenthereexistsastrongdampingforceordensefriction ( isbiggerthan ).Inthissituation,themovingmassdoesnotoscillateatallbut returnstoitsequilibriumasquicklyaspossible.Atypicalexampleinvolvingstrongfric- tionisthecasethatwhenanobjectisdippedintooil,theamplitudeofvibrationdiesout exponentiallyastheenergyofoscillationistransformedintofrictionalheating.However, intheprobe-baseddatastoragemodel,becausethemajordampingfactorisfromair,the caseoflargedampingdoesnotapplyandthesledoscillatesbeforeit nallysettlesdown. 16 When ,thegeneralsolutiontotheEquation(3.5)becomes: (3.8) where (3.9) (3.10) isde nedasresonantfrequency,and and aretwoconstantstobedetermined bytheinitialconditionofthesled.Equation(3.8)isdifferentfromEquation(3.7)because itincludesoscillationparts and . Thissolutioniscalledsmalldampingorslightfriction,inwhichthesledmovement oscillateswhileitgraduallymovestowardsthe nalequilibrium.Thesmaller is,the oscillationplaysabiggerroleduringthemovement.When ,thesolutionis calledcriticaldamping. 3.3.2SolutionofSpringModelwithExternalForce ThesolutiontotheEquation(3.4)isnotyetcomplete,becausewehavenotconsidered theexternalforce.Hereweassume isnon-zeroandthemechanicsofthesledbecomes aforcedoscillation.Tosimplifythemodel,throughoutthispaper,theexternalforce isassumedtobeaconstantduringonesledmovement.Inotherwords,tomoveany particulardistance , hasaconstantvalueanddoesnotchangeovertime.However,to movethesledtoanewposition, musttakeonadifferentvalue. 17 Assuming isindependentoftime ,Equation(3.4)becomes (3.11) Equation(3.11)hasaspeci csolutionwhichoccurswhenthesledstops,namely,the sledreachesitsequilibriumstate.Atthattimebothsledaccelerationandvelocityare zero, i.e. , .Ifwede ne asthat nalpositionthatthesledsettlesdown, Equation(3.11)becomes whichyieldsthefollowingequation: (3.12) AccordingtoZeldovich etal. [17],the nalsolutionforEquation(3.11)isobtainedby addingthesolutionofEquation(3.12)tothegeneralsolutionofEquation(3.8),asgiven byEquation(3.13). (3.13) Here and arecalculatedfromEquation(3.9)andEquation(3.10).Theconstants and canbeobtainedbyusingthefollowingtwoequationsaboutthestartingposition andinitialspeed ofthesled: Solutionoftheabovesystemofequationsisasfollows: 18 Inthisthesisweassume and arezero.Nogeneralityislostbythisassumption becausewhentheinitialpositionisde nedaszero, issimplyde nedasthedistance movedinsteadofcorrespondingtotherealcoordinates.Thusthetwoconstantsbecome thefollowing: ,and wherethevaluesof and canbefoundusingEquation(3.9)and(3.10). Thebehaviorof over couldtakedifferentshapesdependinguponparameters,as illustratedbyFigure3.4athrough3.4d.Hereweusehypotheticalvaluesonlyforthe purposeofillustration.Figure3.4athecaseforverysmalldampingwhere kg, kg/s, N/m,and N.Because issmallrelative to and ,thesledoscillatesfrequentlybeforeit nallystops.Figure4balsouses asmalldampingforcewhere kg, kg/s, N/m,and N.ThedampingforceinFigure3.4bisslightlystrongercomparedwith thatofFigure3.4a.BothFigure3.4aandFigure3.4barebaseduponEquation(3.13). Figure3.4cisacriticaldampingcase, , kg, kg/s, N/s,and N.Finally,Figure3.4dthatisbaseduponEquation(3.7) illustrateslargedampinginwhich , kg, kg/s, N/m,and N.Thesleddoesnotoscillateatallbutslowlyconvergestothe newposition. 19 Equation(3.13)isthekeyequationforthespringmodelinthispaper,whichisin- strumentalforexpressinghowvariousmechanicalandphysicalfactorsaffectthedevice functioning.Italsoenablesusto ndthevelocityandaccelerationrateatany bytaking rst-orsecond-orderdifferentiation.Thevelocityequationisderivedasfollows: andtheaccelerationequationisasbelow: Theessenceofthesledmovementisrepresentedbytheright-handsideofEqua- tion(3.13),wherethe rsttwotermsareoscillationpartswithaconstantresonantfre- quency ,andthethirdtermisbaseduponthe nalpositionofthesled.When islarge enough,the rsttwotermsofEquation(3.13)becomesosmallthatthe nalpositionis solelydeterminedbythethirdterm. 3.3.3ImpactofMechanicalandPhysicalCharacteristics Thissectionexplorestherelationshipamongvariousparametersandhowtheyaffect thesledmovementandseektimeundersmalldamping.Theirrelationshipisbasedon Equation(3.10)whichcomputestheresonantfrequency andEquation(3.12)which describesthesled 9s nalequilibrium.Thepurposeofstudyingtheseinteractionsisto optimizethephysicaldesignofthestoragedeviceinordertoachievethedesiredlevelof performance. 20 IncreaseinValue Description externalforce 0 springcoef cient + sledmass 0 dampingcoef cient 0 Table3.1 :ImpactofParametersuponDistanceMoved,ResonantFrequency,andSeekTimein theSpringModel Table3.1summarizestheseparametersandtheirimpactuponoscillationfrequency , seektime ,anddistance .InTable3.1,a c dindicatesanincrease,a c dindicatesa decrease,and c0 dindicatesnochange. First,astrongerforce willleadtoagreatersledmovement butdoesnotchange , asindicatedbyEquation(3.10).Andbecausethedistanceislonger,seektime increases. Thespringcoef cient iscloselyrelatedtoboth and .Ifotherfactorsareheld constant,ifthespringismoreresilient(higher ),thesledmoveslessdistance accord- ingtoEquation(3.12).Asmaller resultsinlesstimewhenotherparametervaluesare unchanged.Ontheotherhand,thevalueoftheresonantfrequency increases. Anincreaseinthemassofthesled hasnoimpactuponthedistance butitin- creasestheresonantfrequency .Becauseitismoredif culttochangethevelocityofa movingsledifitisheavier,ittakeslongertimetomovetoanewposition,whichresults ingreaterseektime. Finally,ahigherdampingforcecoef cient changesthedynamicsofsledmovement byreducingitsresonantfrequency .Whenthefrictionisintense,thesledislesslikelyto oscillate.Takinganextremecaseasanexample,when islargeenough,thedynamicsof 21 thesledwillswitchfromEquation(3.8)toEquation(3.7),causingtheoscillationbehavior todisappear.Ontheotherhand,changeof doesnotalterthe naldistancemoved . Butseektimeissmallerintheoscillatoryregionbecauseitiseasierforthesledtosettle downunderstrongerfriction. Theforegoinganalysisindicatesthattheimpactsaremulti-dimensionalandvarywith differentparameters.Insmalldamping,fora xedposition,ifafasterrepositiontimeis desired,wecouldcanalowermass ,alarger ,agreater ,orcombinationofthem. Thespringmodeladdressestheseparameters.Basedupontheabovediscussions,the changeoftheseparameterswillaffecttheperformanceofprobe-basedstoragedevices, whichisusefulwhendesigningsuchdevices.However,thebang-bangmodelleavesout theseinteractiveparameterswhendescribingthephysicalmovementofthesledinthe device. Chapter4willpresentourexperimentstofurtherunderstandthetwomodels. 22 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 24 Time (s) Position (m) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0 1 2 x 10 25 Time (s) Position (m) (a)(b) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0 1 2 x 10 25 Time (s) Position (m) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 21.5 21 20.5 0 0.5 1 x 10 25 Time (s) Position (m) (c)(d) Figure3.4 :DynamicofSledMovementinSpringModelwithIncreasingDampingForce:(a) VerySmallDamping( );(b)SmallDamping( )(c)CriticalDamping ( )(d)LargeDamping( ) 23 Chapter4 NumericalSimulation Chapter3showsthatthesledmovementisaprocessdeterminedbyseveralforcesand affectedbyvariousmechanicalandphysicalcharacteristics.Thischapterpresentssimu- lationexperimentstofurtherrecognizethedifferenceinmodelingrepositioningdynamics betweenthespringmodelandthebang-bangmodel.Experimentsaretakenforthepur- posesofcomputingtheseektime,exploringthesensitivitytothevariationofparameters, anddevelopingfurtherinsightsaboutthedynamics. 4.1ExperimentDesign ThekeyequationsusedtocalculatetheseektimeareEquation(3.3)forthebang- bangmodelandEquation(3.13)forthespringmodel.Thesetwoequationsinvolvetwo differentsetsofparameters,whichmakesnumericalcomparisondif cult.Forexample, foragivenacceleration,the naldistancedmoved ischanginginthebang-bangmodel 24 butnotinthespringmodel.Wemakethecomparisonconsistentbyapplyingtheinherent relationshipamongvariousparameters. First,wechooseaseriesof thatthesledisgoingtoreach.Inthebang-bangmodel, thecalculationoftheseektimeisstraightforwardasEquation(3.3)canbeanalytically appliedwithgiven .Howeverinthespringmodel,thedynamicsequationdoesnot directlyinvolve butwecouldvary togeteachvalueof baseduponEquation(3.12). Becausewhenthe naldistanceischosen,theforceneededisdeterminedbyEquation (3.12)inthespringmodel.Eachobtained isthenpluggedintoEquation(3.13)to calculatethedynamicsof andthencalculatetheseektime. Given ,anumericalsolutionisused,whichregeneratesthedynamicsof against time .Byplotting against andimposingatolerancerange,wecancalculatetheseek time. Itisnecessaryforthespringmodeltosetatolerancerangebecausewiththepresence ofrestoringanddampingforces,theslediscontinuouslyoscillating,thoughtheamplitude isdecreasing.Wesaythesledstopswhenitsoscillationamplitudenolongerexceedsa pre-speci edrange,calledtolerance.Becausethedistancebetweentwoneighboringbits is nm,wechooseatolerancevalueas 25nm.Whenrecordingthechangeof along with inthesimulation,wekeepthemostrecenttimewhen isstilloutsidethetolerance range,whichhelpsustodeterminehowlongittakesthesledtosettledown. 25 Parameter Description Values mass kg externalforce N springcoef cient N/m dampingcoef cient kg/s resonantfrequency Hz acceleration m/ settledowntime ms Table4.1 :ParameterValuesintheSimulation 4.2ChoiceofParameterValues Table4.1summarizestheparametersandtheirvaluerangesadoptedinthethesis. First,thevaluefor isapproximatedaccordingtothesizeofthesled.Becausethe polysilicondensityforthesledis2.3g/cm , forthemovingmediasledis kg ( )ifthesledhasthedimensions:length(1cm),width(1cm)and height(1mm).Thesizeofsledcanvaryandsodoesthemass.Thereforewechoosethe followingthreereasonablevaluesformass intheexperiments: kg, kgand kg. Onceweknow and ,force isdeterminedby (4.1) Schlosser etal. [16]setstheacceleration 115m/s .If is kg, is N( ).Thisisanapproximateandweallow tovaryinour experimentstoexploretheeffectofdifferent uponthedynamics. rangesfroma minimumofzerotoamaximumof Nwhichroughlydoublestheestimateof N. 26 Thespringconstant ischosenaccordingtosled 9s nalequilibriumequation(3.12), whichdeterminesthevalueof given and .Themaximumlengththatasledcan moveisabout maccordingtoSchlosser etal. [16]andCarley[2].Thuswecanhave anideaaboutpossiblevaluesfor .Assuming is Nand is m, is N/m.Thisisonereasonableestimate.However,wedeterminetherangeofvaluesused inthisexperimentbylookingathowrealistictheseektimeis.Wewishtolimittheseek timetoseveralmillisecondsatmost.If becomestoosmallcomparedwiththeexternal force ,thesledwillspendalotoftimeoscillatingandtheseektimewillbetoolong. Therefore,wevary intheexperimentsfrom , ,to N/m. Inthespringmodel, issetthroughthefollowingtwomethods.First,accordingto thefrequencyequation(3.10), iscalculatedfromthevaluesof , ,and .Schlosser etal. [16]pointsoutthat is Hzintheir rstgenerationmodel,whichisconsistent withthe -dimensionsettlingdowntimeof1.447ms.Thesamefrequencyof Hzis usedforthetwomodels,whichresultsinthevalueof0.626kg/sfor ,given N/mand kgrespectively. Thesecondmethodofdetermining isbylookingatEquation .With smalldamping,i.e., ,thengiven kgand N/m, shouldbelessthan kg/s( ).Bothmethodssuggestthat shouldbearound0.6kg/sorsmaller,withtheupperlimitdependingupon and . Therefore,throughouttheexperiments, iscalculatedfromEquation(3.10).Whenusing thedefaultvalueofmass kg,itistotakethefollowingthreevalues: 27 0 1 2 3 4 5 6 7 8 0 20 40 60 80 100 120 Time (ms) Position (um) 4.5 5 5.5 6 6.5 7 7.5 8 99.6 99.65 99.7 99.75 99.8 99.85 99.9 99.95 100 100.05 100.1 Time (ms) Position (um) (a)(b) Figure4.1 :DynamicsofSledMovementinSpringModel kg/sfor N/m, kg/sfor N/m,and kg/sfor N/m. Intheexperimentwhenmass changes,ithasdifferentvalueswhichwillbediscussed laterinSection4.3. 4.3DynamicsofModels Figure4.1ashowsthemovementofthesled overtime frominitialpositionto the nalpositionof100 m,wheretheparametersaresetattheirdefaultvalues: N, kg/s, kgand N/mrespectively.All the guresintherestofthispaperusedefaultvaluesoftheparametersunlessexplicitly discussed.Thetwohorizontallinesindicateupperandlowertolerancethatdetermines theseektimeforthespringmodel.Thesetwolinesarewithin nmofeachother. Figure4.1bisamoredetailedpictureofthesled 9sdynamicswhenitisclosetosettling down,i.e., isbetween4.5msand8ms.Theseektime for100 mdistanceisabout 28 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 10 20 30 40 50 60 70 80 90 100 Time (ms) Position (um) Figure4.2 :DynamicsofSledMovementinBang-BangModel msbecauseafterthattimepointthesledoscillationwillnolongerexceedboththe upper(100 m+25nm)andlower(100 m-25nm)tolerancelines. Figure4.2displaysthemovementofthesledovertimeforthebang-bangmodelwhere accelerationequalsitsdefaultvalueinTable3.1and is100 m.Incontrasttothe dynamicsofthespringmodelinFigure4.1,thespeedofthesledislinearlyincreasing duringthe rsthalfoftimeandthenislinearlydecreasingduringthesecondhalf.The overallshapeofthedynamicscurveisconvex rstandthenconcave. Figure4.3aplotstheseektimeestimatedbythetwomodesagainstthedistancemoved. This gureindicatesthatforbothmodelstheseektimeincreasesalongwith butata decreasingrate.Figure4.3aalsoshowthedifferenceintheseektimeestimatedbythetwo models.Morespeci cally,theseektimeestimatedbythebang-bangmodelissmallerfor almosteverypossiblevalueofdistance .Whenthespringmodelestimatesthatittakes 29 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 Distance (um) Time (ms) Spring Model Bang 2Bang Model Modified Bang 2Bang Model 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 Distance (um) Time (ms) Spring Model Bang 2Bang Model Modified Bang 2Bang Model (a)(b) Figure4.3 :PlotofSeekTimeagainstDistanceintwoModels thesled6.6mstomoveadistanceof100 m,thebang-bangestimatestheseektimeto be ms. Figure4.3bgivesusazoom-inviewforFigure4.3a.Wecanseethatwhentheseek distanceisextremelysmall,theseektimeestimatesbybang-bangmodelaregreater.This isbecausethebang-bangmodelassumesaconstantsettlingdowntimeof ms,no matterhowsmallthedistance is,whichisinaccuratebecausepositioningtimeap- proacheszerowhenthedistancemovedisin nitelysmall. ThereisastraightlineinFigure4aand4b,whichisplottedaccordingtoEquation(3.3) butherethevalueofacceleration changes.Itdisplaysthesituationinwhich isre- setalongwiththevariationof accordingto .Thereasonfordoingthisisto indicatethatthetwomodelstakedifferentassumptions.Inbang-bangmodel,acceleration isconstant.Forexample,movingtwicethedistance, inthespringmodeldoubles. However,If inthebang-bangdoublesaswell,theresultwouldbeaconstantseektime 30 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 Distance (um) Time (ms) Tolerance at 10 nm Tolerance at 25 nm Tolerance at 40 nm Bang 2Bang Model Figure4.4 :PlotofSeekTimeagainstDistanceunderDifferentToleranceRange becausetheincreaseof and offseteachother(seeEquation(3.3)andEquation(3.12)). Theaboveexperimentsusetoleranceof 25nm,halfofabitwidth.Whenthetoler- ancevaries,theseektimechangesaswell,asshowninFigure4.4,inwhichthetolerance issetto 10nm, 25nm,and 40nm.Thechangeofthetolerancedoesnotaffectthe estimatedseektimeforthebang-bangmodelwhichusesa xedsettledowntimebased uponresonantfrequency .Forthespringmodel,themorelenientthetolerance,the smalleristheseektime.Buttheestimatedseektimeisstillmuchlongerinthespring modelthanforthebang-bangmodelevenifalenienttoleranceisapplied. Figure4.5studiestheimpactofmassparameter upontheseektime,inwhich is kg, kg,and kg.Forthespringmodel,thechangeofmass leadstodifferentvaluesof accordingtoEquation(3.10).With N/mgiven, kg/sfor kg, kg/sfor kgwhichis 31 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 9 Distance (um) Time (ms) Spring Model (m=0.1g) Bang 2Bang Model (m=0.1g) Spring Model (m=0.2g) Bang 2Bang Model (m=0.2g) Spring Model (m=0.3g) Bang 2Bang Model (m=0.3g) Figure4.5 :PlotofSeekTimeagainstDistanceunderDifferentMass thedefaultvalue,and kg/sfor kgrespectively.Inthebang- bangmodel,accelerationvariescorrespondingly.With N,accordingto Equation(4.1), m/s for kg, m/s (defaultvalue)for kg,and m/s for kg.Theresultsindicatethatfor bothmodels,withotherfactorsunchanged,theheavierthesled,thegreatertheseektime, whichisalsoconsistentwithTable3.1.Theseektimeinthespringmodelishigherthan thatinthebang-bangmodelforeachofthemassvalues. Theimpactofdifferentvaluesof uponseektimeisshowninFigure4.6,where is N/m, N/m,and N/m.Fortheplottingofthespringmodel,theparameters isadjustedwhen changesaccordingtotheresonantfrequency( )Equation(3.10).This isbecausetheresonantfrequencyis xedat220Hzforbothmodels.Different doesnot changetheseektimeforthebang-bangmodelbecauseEquation(3.3)doesnotinvolve 32 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 9 Distance (um) Time (ms) Spring Model (k=700 N/m) Spring Model (k=500 N/m) Spring Model (k=300 N/m) Bang 2Bang Model Figure4.6 :PlotofSeekTimeagainstDistanceunderDifferentSpringCoef cients thisparameter.Figure4.6showsthattheseektimeestimatedbythespringmodelislarger thanthatofthebang-bangestimates,undereachvalueof .When changes,thespring modelsuggeststhatseektime shouldvary,buttheseektimebythebang-bangmodel staysthesame. Intheabove,wehavecomparedthedynamicsandseektimesofthebang-bangmodel andthespringmodel.Thespringmodelmorepreciselymodelstheunderlyingmechanics ofprobe-basedstorageandhasmoreparametersspeci callydescribingthosemechanics. Withconsistentphysicalparameters,seektimesestimatedbythespringmodelarelonger thanthoseofthebang-bangmodelexceptforveryshortseeks.Thisisbecausethebang- bangmodeloptimisticallyusesaconstantmaximumacceleration,andthespringmodel describesthevariationofaccelerationintime.Thespringmodelincorporatessettletime calculationasafunctionofseekdistance,tolerance,andresonantfrequency,whereasthe 33 bang-bangmodelusesaconstantsettletimebasedonestimatesoftheseparameters.Thus, wethinkthespringmodelismoreaccuratethanthebang-bangmodel. 34 Chapter5 Read/WriteTimeforBothModels Thischapteranalyzestheread/writetimeforboththespringandbang-bangmodels.In probe-baseddatastoragesystem,theaccesstimeforadatarequestisthesumoftheseek timeandtheread/writetime.Afterthesledreachestherequiredposition,thetipsstart reading/writingdata. Hereweconsideraconstantread/writespeed.Letusde ne tobethevelocityat whichtipsreadandwrite.Theread/writetime couldbedecomposedintofourparts: actualread/writetime ,turnaroundtime ,tipswitchtime ,and -movementtime asshowninEquation(5.1). (5.1) InEquation(5.1), isderivedbydividingthetotaldistancetraveledbythesledin reading/writingwiththevelocity ; iscalculatedbymultiplyingthetotalnumber ofturnaroundswiththetimeusedononeturnaround; isobtainedbymultiplyingthe 35 0 1 2 3 4 5 6 7 8 x 10 6 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Number of Sectors Time (s) Time w/o Overhead No Settle Time Bang 2Bang Model Spring Model 0 10 20 30 40 50 60 70 80 90 100 0 0.005 0.01 0.015 0.02 0.025 0.03 Number of Sectors Time (s) Time w/o Overhead Time with Overhead (a)(b) Figure5.1 :Read/WriteTime numberoftipswitcheswiththetimespentononetipswitch;and isalsocalculated accordingly. InEquation(5.1),turnaroundtime isnecessarybecausethesled 9sdirectionmay bereversedduringtheprocessofreading/writing.And isthetimeneededforthe activationandswitchoftips.Inourprobe-baseddevice,onlyonerowoftipsisactiveata time.Afteronerow nishesreading/writingdata,thenextrowisactivated. -movement isrequiredforthesledtomovefromonecolumnofbitstothenextcolumnin -dimension whichisaseekactivityandwillinvolvethemechanicsdiscussedinChapter3. 5.1ExperimentsofRead/WriteTime Equation(5.1)isusedtocalculatetheread/writetime.Thecalculationsofthe rst threetermsarethesameforthespringandthebang-bangmodels.Butforthelastterm, i.e. ,the -movementtime,thetwomodelstakedifferentapproachesbecausewemust 36 repositionthesledfromonecolumnofbitstothenextone.Accordingtothedynamics modelsdevelopedinChapter3,thetimeononemovementin -dimensioncanbecalcu- latedbyusingthefollowingthreemethods.Herethedistancemovedis50nm,onebit size. Usingbang-bangmodelandomittingthesettlingdowntime becausewethink itmaynottakeaconstanttimeof1.447mstosettledownforsuchasmalldistance of50nm.Theresultis0.0417ms. Usingthebang-bangmodelwiththesettlingdowntime.Thetimeis1.49ms. Usingthespringmodel,whichdependsuponparametersvalues.Given kg, kg/s, N/m,and N,theresultis0.893 ms. The -movementtimeobtainedthroughtheabovethreemethodsisdifferent.Inour experiments,wetake m/s, m/s .Accordingto rst-ordermechanics, timeforoneturnaroundis ms.Thetipswitchtimeisassumedtobezero becauseittakesalmostzerotimetoactivateandswitchtips. Figure5.1aplotstheread/writetimeneededagainstthenumberofsectorsprocessed whenconsideringfourcases:timewithoutincludingthosespentonturnaroundand - movement;totalread/writetimewiththe -movementtimeusingbang-bangmodelbut omittingsettledowntime;totalread/writetimewiththe -movementtimeusingbang- bangmodel;totalread/writetimewiththe -movementtimeusingspringmodel.The 37 numberofsectorsrangesfrom0to whichisthemaximumnumberthatthesled canaccommodateperrequestaccordingtoourdesign.Weassumethereare512bytesper sectorintheexperiments.The gureshowsthattheread/writetimefor sectors isaround2000seconds.Accordingly,wegetthethroughput1.9Mbyte/secondwhichis consistentwiththeaccessrateof Kbit/secondpertip(thereare100 100tips inthedesign). Figure5.1ashowsthatread/writetimeincreaseslinearlywiththenumberofsectors. Themajoritypartofthetimeisspentonactualreading/writing.Becausetheoverheadtime isquitesmall,thetotaltimeplottedbyusingthreedifferent -movementtimecalculations issimilar(thethreelinesoverlap). Figure5.1bdisplaystheresultswithmuchlessnumberofsectors:1,2,...,upto 100.Hereweplottwolinesrepresentingtheread/writetimewithorwithoutoverhead (turnaroundand -movement).BeingidenticaltoFigure5.1a,this guresuggeststhat overalllinearrelationshipstillholdseventhoughsmallamountofdataareprocessed throughtheprobe-basedsystem. 38 Chapter6 Conclusion Thisthesisdevelopsthespringmodel,anewapproachtoanalyzethedynamicsofsled movementinprobe-basedstoragesystembyusingclassicalphysicsandmechanicstheory, andidenti esmajorfactorsthatcontributetothisprocess. Thespringmodelisdifferentfromthepreviousbang-bangmodelinprobe-storagesys- temmodelingbecauseitdescribeshowaccelerationvariescontinuouslyintime.Itmodels thephysicalsystematalowerlevel,andcanaccommodateawiderrangeofparameters.It providesamoreaccuratecalculationofthesettletime,basedontheseunderlyingphysical andmechanicalparameters.Higher-levelsystemdesigndecisionsdependuponaccurate modelsoflow-levelbehavior,andwehavedescribedimportantdifferencesinthetwo modelingapproachesthatwarrantfurtherinvestigation. Whilewebelievethespringmodelisapromisingapproachforthenewgeneration ofprobe-basedstoragesystem,thereareseveralresearchareasthatcouldbeexamined 39 further.Firstly,theassumptionofexternal beingindependentof canbereconsidered becausetheelectricvoltageoftheprobe-basedstoragesystemmayvarywithtime.So itisinterestingtounderstandhowthealternationofthisassumptionmaychangethedy- namicsprocessandseektimeestimation.Secondly,weneedtostudyhowthetwomodels estimatetheread/writetimewhenconsideringtherepositioningmechanicsinvolvedin -dimensionmovementfromonecolumnofbitstothenextone.Andthirdly,itneedsfur- therexaminationonhowtheestimatedseektimebythetwomodelsdifferinsimulations usingapplicationbenchmarks. 40 Bibliography [1]C.Brown.Microprobespromisesanewmemoryoption. E.E.Times ,1998. [2]L.R.Carley.www.chips.ece.cmu.edu,1999. [3]S.H.Charap,P.L.Lu,andY.He.Thermalstabilityofrecordedinformationathigh densities.In IEEETransactionsonMagnetics,vol.33 ,1994. [4]M.Despont,J.Brugger,U.Drechsler,U.Durig,W.Haberle,M.Lutwyche, H.Rothuzen,R.Stutz,R.Widmer,H.Rohrer,G.Binnig,andP.Vettiger.VLSI- NEMSchipsforAFMdatastorage.In TechnicalDigest.IEEEInternationalMEMS 99Conference.TwelfthIEEEInternationalConferenceonMicroElectroMechani- calSystems ,1999. [5]J.L.Grif nandS.W.Schlosser.Characteristicsandapplicationsofubiquitous smartstorage.Technicalreport,LaboratoryforComputerSystems,CarnegieMellon University(unpublisheddraft),1999. [6]J.L.Grif n,S.W.Schlosser,G.R.Ganger,andD.F.Nagle.Modelingandper- 41 formanceofMEMS-basedstoragedevices.In ProceedingsofACMSIGMETRICS 2000 ,2000. [7]G.T.Sincerboxanded.Selectedpapersonholographicstorage,vol.ms95ofSPIE milestoneseries.In InternalSocietyforOpticalEngineering ,1994. [8]W.Hauser. IntroductoryMechanics .NortheasternUniversityTextbookProgram, 1987. [9]A.HudsonandR.Nelson. UniversityPhysics .HarcourtBraceJovanovich,Inc., 1982. [10]W.A.JohnsonandL.K.Warne.Electrophysicsofmicromechanicalcombactuators. In JournalofMicroelectromechanicalSystems ,1995. [11]H.J.Mamin,B.D.Terris,L.S.Fan,S.Hoen,R.C.Barrett,andD.Rugar.High- densitydatastorageusingproximalprobetechniques.In IBMJournalofResearch andDevelopment ,1995. [12]M.Mehregany,W.H.Ko,A.S.Dewa,andC.C.Liu.IntroductiontoMEMSsystems andthemultiusermemsprocesses.Technicalreport,ElectronicsDesignCenter, DepartmentofElectricalEngineering&AppliedPhysics,CaseWesternReserve University,1993. [13]D.PsaltisandG.W.Burr.Holographicdatastorage.In IEEEComputer ,1998. 42 [14]S.Red eldandJ.Willenbring.Holostoretechnologyforhigherlevelsofmemory hierarchy.In EleventhIEEESymposiumonMassStorageSystems ,1991. [15]C.RuemmlerandJ.Wilkes.Modelingdisks.TechnicalReportHPL-93-68,Hewlett PackardLaboratories,1993. [16]S.W.Schlosser,J.L.Grif n,D.F.Nagle,andG.R.Ganger.Fillingthemem- oryaccessgap:Acaseforon-chipmagneticstorage.Technicalreport,Schoolof ComputerScience,CarnegieMellonUniversity,1999. [17]Ya.B.ZeldovichandA.D.Myskis. ElementsofAppliedMathematics .MIRPub- lishersMoscow,1976. [18]X.ZhangandW.C.Tang.Viscousairdampinginlaterallydrivenmicroresonators. In SensorsandMaterials ,1995. 43 AppendixA SolutiontotheSecond-Order DifferentialEquation Thisappendixprovidesthesolutionforthesecond-orderdifferentialEquation(3.5)where , ,and areparameters.ThedeductionofsolutioncloselyfollowsZeldovich et al. [17].Inmanyaspects,thepropertiesofthisequationaresimilartothoseof rst- orderhomogeneouslinearequations.Forinstance,itiseasytoverifythatif isa particularsolutionof(3.5),thenalsois ,where isanyconstant;thatif and aretwoparticularsolutionsof(3.5),thentheirsum ,isalsoa solutionofthatequation. Fromtheabove,thenifwehavefoundtwoparticularsolutions and , thentheirlinearcombination, (A.1) 44 where and aretwoarbitraryconstants,isalsoasolutionofthatequation.The generalsolutionofasecond-orderdifferentialequationisobtainedviatwointegrations whichresultsintwoarbitraryconstants.Thereforetheexpression(A.1)canbeusedas thegeneralsolutionofEquation(3.5).Itisobviousthat and shouldnotbelinearly dependentwitheachother. Inorderto ndthetwoindependentsolutionsofEquation(3.5),weuseEuler 9sap- proachwhichtakestheform (A.2) where isaconstanttobefound.Substituting(A.2)into(3.5)wegetthefollowing characteristicequation (A.3) Insolvingtheaboveequation,therearedifferentcases,dependinguponthesignof thediscriminant: Ifthefrictionisgreat,namely,if ,then(A.3)hasrealroots: Let and .Then,onthebasisof(A.1)and(A.2),wegetthegeneral solutionof(3.5)intheformofEquation(3.7).Accordingtothisequation,inthecaseof 45 strongfriction,thedeviationofapointfromtheequilibriumtendstozeroexponentially with withoutoscillations. Ifthefrictionissmall,thatistosay,if ,then(A.3)hasimaginaryconjugate roots: where and arerepresentedinEquation(11)and(12).Thenwegetthegeneral solution(A.1) (A.4) Theterms and areperiodicfunctionswithfrequency .Applying Euler 9sformula,wecouldrewriteEquation(A.4)intoEquation(3.13). Inthecasewhen ,wehavetwoequalrealroots,thesolutionwillbecome: Here and aretwoarbitraryconstants.

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