DesigningMechatronicSoundSystemsInspiredbySinusoidalMathematics

DanielJ.Reyes1,AjayKapur2

CaliforniaInstituteoftheArts,Valencia,U.S.A.1danielreyes@alum.calarts.edu

2ajay@karmetik.com

Abstract.Thispaperdescribesthreesonicmechatronicsystemsdesignedwiththeinfluenceofsinusoidalpropertiesandtheintentiontoproduceasonicresult.Themainpurposeofthispaperistoillustratethevarioustechniquestakeninconstructingsuchsystemsbyanalyzingtheprocessoftranslatingnaturalphenomenarecurringinnatureintophysicalemulationswithasonicaspect.Thethreemechatronicsystemsdescribedinthispaperarecategorizedintothreeparts:sculpture,musicalinterface,andahybridofthetwo.Post-WaveisaninteractiveroboticinterpretationofRedwoodWave,akineticsculpturebyReubenMargolin.SinuservesasacircularsequencerutilizingphotocellsandLED’stotriggerMIDI.Lastly,CymbalicisahybridsculpturalinterfaceinspiredbythefindingsofmathematicianLeonardEuler.

Keywords:Post-Wave,Sinu,Cymbalic,mechatronic,sculpture,musicalinterface,sinusoid,kinetic

IntroductionWithintheunderstandingofmoderndayacoustics,thesinewaveremainstobethefundamentalrootofitsstudy.(Benson2007)Themathematicalfunctiondescribesapurerepetitiveoscillation.Thisoscillationpatternoccursfrequentlyinnature:lightwaves,waterwaves,shockwaves,radiowaves,X-raysandmostimportantlysoundwaves.(Rossing1990)Althoughsoundwavesareextremelydifferentfromradiowavesoroceanwaves,allwavescontaincertaincommonproperties.Thesimplefactthatthesewavesoccupythespacewithinyouaresittingin,inavarietyofformats,explainsthereasoningbehindadoptingthesinewaveastheoverallaestheticapproachtodesigningthesesystems.

Thepurposeofthispaperistoexploreaestheticsdrivenbymathematicalrelationships.Thefollowingsectionwillfamiliarizethereaderwithsimilarworksinwhichsinusoidsareusedasasourceofinspiration.Section3presentsamechanicalwoodensculpture,Post-Wave,thefirstinaseriesofsinusoidalpieces.Sinu,acircularmusicalinterface,isexplainedindetailinsection4,discussingaestheticmotivesalongwithtechnicalprocedures.Section5completestheserieswiththepresentationofCymbalic.

RelatedWorkThereareseveralkineticartistsandengineerswhohaveusedsimilarmathematicalqualitiesintheirworktorecreatethesenaturallyoccurringphenomenas.ReubenMargolin’sworkisheavilybasedonwaveforms.Initially,inspiredbythecrawlingqualitiesofacaterpillar’smovement,Margolinutilizedmechanizedactuatorstosimulatecomplexmotionsvisibleinnature.(PopTech2014)

TheRoundWave consistsofnine concentric circles, each tied toapulley systemwithapreciseoffset (figure1). Theresultisaphysicalemulationofanexpandingsystemofripplesformedbythedroppingasmallobjectintoapoolofstillwater. Waves by Daniel Palacios is another example of a piece of related artwork. Waves is a sound installationconsistingofalongpieceofropeandtwoturbines(figure2).Thelongpieceofropetakestheshapeofacomplexwaveformastheturbinesrotate.Theresultisathreedimensionalseriesofwavesfloatinginspace.Thephysicalactionoftherope cutting through the air creates thewhistling sound heard by the observers. Not only does the sound sculpturevisualizeaharmonicseries,but italso reactsandchangesshapebasedon thepresenceofpeoplearound it (Palacios

2007). Another examplewith amore tangible aspect is a rotary sequencer inspired by themonomer (figure 3). Thecustom-made rotary sequencer stores a melody on a 10” acrylic disk embedded with small magnets arranged on acirculargrid.Concentriccirclesarecenteredaroundaspindleandrotateoveralineofmagneticsensors.Thedeviceisanattempttoreplicatewhileimprovingthefunctionofatraditionalmechanicalmusicbox(Monomatic2010).

Figure1.TheRoundWavebyReubenMargolin

Figure2.WavesbyDanielPalacios

Figure3.ModularMusicBoxbyLewisSykesandNickRothwell

Mesmerizedbyaconstantrollingballentrenchedwithinametalriminahardwarestorehewalkedpasteverydayonhiswaytowork,Trimpin,afifteenyearoldatthetime,questionedhowaballcancontinuouslyrollaroundtherimifperpetualmotionisnon-existent.Afterfurtherinvestigation,hewasabletodeterminethedisplaywaspneumaticallypowered.Smallholesblewoutapuffofairwhentheballpassedoverit,justenoughtocontinuetomomentumtodrive

itincircles.ThisfascinationstuckwithTrimpin,ashewouldlaterdescribethathe“wasmesmerizedbythesoundandsightofthisperfectlyroundballconstantlyrollinginZen-likerepetition,andfromthatmomentonIwantedtodosomethingwithspheres.Ididn’twanttouseanythingfake-therehadtobeanotherwaytokeepasphereinmotion,probablyusinggravity.”(Focke2011)DerRing’sappearanceisbasedonthedisplayfromthehardwarestore,althoughit’sfunctionalityiscompletelydifferent.Thesculpturefunctionslikeapuppet,its’angleistiltedaroundinacircle,causingtheballstoorbitduetogravity.Eachringisattachedtocablethatrunsthroughasetofpulleysandthenintoasophisticatedgearbox(figure4).

Figure.4DerRingbyTrimpin

Thesemechatronicsystemscollectivelysharesinusoidalandsonicproperties.Eachutilizerepetitivefeaturesfoundinourenvironment,andtranscribetheirdynamicsintoakineticrepresentationoftheirphysicalbehavior.

Post-Wave:MechatronicSculpture

Figure5.Post-Wave

Post-Waveisthefirstinaseriesofsinusoidal-inspiredsoundprojects.Post-WaveishighlybasedonReubenMargolin’sRedwoodWave.Itservedasanintroductorystudyofmechanicsandelectronicdesign,inadditiontobeinganassessmentofcarpentryskills(figure5).

Design

Themechanicaldesign is composedof twometal rods intersectinga setof306”circles,eachoffsetbyoneanother.Thenasetof1”x30”stripstreatedwithdadocutsbeneathsitonapairofcircles.Topreventthewoodenstripsfromslippingoffthecircles,arectangularstripofwoodismountedinthemiddleofthe1”x30”dadostrips,whichisthensecuredbetweentwowoodstripsperpendiculartoitsposition.

Figure6.ProfileSketch

Figure7.SketchofCams

Implementation

ThetworodsarethencoupledtotwoseparateDCgearheadmotors.ThetwomotorsarepoweredandcontrolledwithanArduinoUnousingPolulu’sDualVNH5019MotorDriverShield.AMicrosoftKinect isusedviaProcessingtodetectblobdepthandpositionusingtheSimpleOpenNI library.DepthandX-positiondata issentviaserial toArduino.Thus,depthcontrolsthespeedofthemotors,andX-positiondetermineswhichmotorisbeingcontrolled.

Figure8.InteractionwithSculpture

Theresultisasurprisinglywellexecutedmechanicalacousticemulationofthesoundproducedbywavesonabodyofwater. Thephysical actionof thewood strips slidingbackand forthon topof thewood circlesmakes for a soothinghypnoticaudibleresult.Thewavemovesslowlywhenauserisnotpresent,yetassoonasauserplacehishandontopof thewave, a reaction is immediately identified, engaging the observer to discover his or her ability to control thesculpture(figure8).

Sinu:MusicalInterfaceSinuisanattemptatproducingarenditionofthetraditionalmusicbox.ThisparticularrotarysequencerusesasetofeightphotocellssuspendedaboveasetofeightLEDs.A11”acrylicdiskisdividedinto16slices,eachcontaining8holesfortheLEDtoshinethroughasitpassesthe“playhead”(figure9).Thecirculardiskisplaceontoadirect-driventurntablemotormountedonthemainenclosure.AsetofnineLEDpushbuttons,3rotarypotentiometers,1linearpotentiometer,1LEDpowerindicator,andtwoswitchesarealsomountedontheenclosure.

Figure9.Sinu

Functionality

Figure10.ArduinoPinAssignments

ThefunctionalityofthebuttonsA-Haretoactasstepsofaconventionalsequencer.ButtonIfunctionsasashiftbuttonthatenablestheusertoswitchbetweensequences.Whenfirstpowered,asequencecanbeenteredbypushingbuttonsA-H.Oncethishasbeendone,LEDQwillturnonandthepositionmarkerofthesequencewillflashatposition1(LEDinbuttonA.)AsecondsequencecanbeenteredbyholdingdownbuttonIandselectinganybuttonB-H.WhenasequenceisenteredonanypatternI+A-H,thecorrespondingLEDQ-Xwillturnon. EachpatternI+A-Hcorrespondstoanotein

thescalespecifiedintheprogram.Basically,asLEDQ-XpassesthroughtheplayheadcontainingphotocellQ-X,thepositionmarkerofthesequenceriteratesthroughthesequence.SwitchOturnsonthemotor,andswitchMreversesitsdirection.LinearpotPcontrolsspeedofthemotor.RotarypotsJ-Lcanbemidimappedtocontrolanyparameterdesired.

Implementation

SinuiscontrolledusinganArduinoMegawithanArduinoMotorShieldRev3.ThedesignconsistsoffourcustommadePCBs:aphotocellboard,LEDboard,buttonboard,andacustomArduinoshieldtodirectallconnectionstotheArduinopins.VoltagereadingsfromthephotocellsandbuttonsaredirectedtothemainArduinoshieldwith10conductorribboncable.TheLEDsthatare locatedbeneaththephotocellsarecontrolledthroughthedigitaloutputpinsoftheArduino.The button PCB houses all the buttons, its’ LEDs, switches and pots found on the face of the enclosure. All of theconnections are then routed, via ribbon cables, to themain Arduino shield. An Arduino sketch determines how thesequencerfunctions.Whenatrigger isreceivedfromaphotocell,ArduinosendsamidinotenumbertoChucK.ChucKthensendsoutamidinotemessagethroughanIACMidiBustobereceivedinAbletontocontrolanyMIDIinstrument.Thefinalresultisacircularsequencerwiththeabilitytoperformalternativeformsofmusicalexpression.

Cymbalic:HybridInterface-Sculpture

Figure14.Cymbalic

Cymbalicisthethirdmechatronicsoundsculptureinthisseriesofsinusoidinspiredworks.Therealizationforthisparticularmechanicalsoundstudycamefromthenoveltysciencetoy,Euler’sDisk.Thediskisusedtodemonstratethedynamicsystemofaspinningdiskonaflatsurface.Thesmootheddiskismadeupofthickandheavychrome-platedsteel.Themirroredbasebywhichisspinsonhasaslightconcave,keepingthespinningdiskcenteredontheplate.MathematicianLeonardEulerwasthefirsttostudythisphenomenoninthe18thcentury.(Moffatt2000)ThesculptureisanaiveapproachinrecreatingthemotiontakenbytheEuler’sdisk.

Design

Atthecoreofitsdesign,fourcamsareusedtomanipulatealargerflat16”disk(figure15).Two1/2”metalrods,coupledtoDCgearheadmotors,eachintersectapairofcams.Eachpairconsistsoftwodifferentsizestomakeupfortheoffsetinheightoftheintersectingrods.Onthelargedisksuspendedabove,fourequallyspacedtuningpegseachholdasitarstringwhichthentravelsthroughapulley,downtoacymbal.Thesitarstringsarefedthroughequallyspacedholesinacymbalwhichismountedtothebottommetalframe,andthentiedfirmly.TheintentionistohavethecymbaltakethemotionoftheEulerdisk.

Figure15.CamMechanismSketch

Thedistancefromthecenterofthecircleatwhichtherodintersectsdeterminestheamplitudeofthelinearmotionconvertedfrombyeachcam.Eachrodintersectsthecam7/8”fromthecenterinordertopreventthelargerdiskfromhittingtheoutermetalframe.

Figure16.CamDimensionscircle2=circle1+2offset

7=5+2(1)

Implementation Inordertocreateanelementofhumaninteraction,theincreasingfrequencythattheEulerdiskproduceswhenspunisprogrammed to control the speed of themotors. An ArduinoUno pairedwith a Polulu Dual VNH5019Motor DriverShield isused tooperate the sculpture.Todetect the frequencyof thedisk,Adafruit’sElectretMicrophoneAmplifierMAX4466isplacedundertheconcavemirroredplatform.SincetheArduinoanalogreadratesamplesataboutaquarterofaudiorate,analternativeapproachwastakeninorderachievemorereliableresults.Athresholdontheamplitudeissettoinitiatethestartofthemotors.Thespeediteratesfasteraslongastheamplituderemainsabovethethreshold.Assoonastheamplitudefallsunderthethreshold,themotorscometoasuddenstop.Theelectricmicrophoneprovestobe very sensitive, enough to create a suddenmovement when the disk is touched. This way, the user immediatelyexperiencestheinteraction.

ConclusionThis collection of projects were completed at California Institute for the Arts. Each being a cumulation of the skillslearnedduringthesuccessiveyears.Repetitivemotioncanclearlybeseenasthedrivingforceofinspirationthroughouteachsystem.Circlesareheavilyusedthroughouteachsystemasameanstoconvertortranslatemotion.Theseriesofoffset circles in Post-wave modulate the angle at which its corresponding wooden strip is positioned. When thesuccessionofcircles’phaseisalteredfromoneanother,awavemotionistranslated.InSinu,thecircleistheobjectthatdefinestherecurring featureasequencer functionsby.Asequencerhasthetendencyto loopatacertainpoint in itsarray of values. The circle is the ideal visceral representation of that quality. Cymbalic is an impulsive reaction to afascinatingphenomenon.Anothersetofcircleswithanoffsetinattempttoemulateanaturallyoccurringmotion.Circleshereareameansof translating rotarymotion toa seriesofoffset linearmotions to recreateawobblemotion.Eachtechniquevaries,yetwithasimilar intentiontoutilizea fundamentalsequencesuchas thesinewave,anddevelopauniqueinterpretationsofobjectswhichinhibitarelatablekinetic.

ReferencesBenson,D.J.Music:AMathematicalOffering.Cambridge,UK ;NewYork:CambridgeUniversityPress,2007,13-14.

Moffatt,H.K.“Euler’sDiskandItsFinite-TimeSingularity.”NATURE-LONDON-,no.6780(2000):833.

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Palacios,Daniel.“Waves.”DanielPalacios,20072006.http://www.danielpalacios.info/portfolio/waves/.

Rossing,ThomasD.TheScienceofSound.2nded.Reading,Mass:Addison-WesleyPub.Co,1990,31-32.

Tech,Pop.ReubenMargolin’sKineticArt.AccessedOctober28,2014.http://poptech.org/popcasts/reuben_margolins_kinetic_art.

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