DesigningMechatronicSoundSystemsInspiredby
SinusoidalMathematics
1
DanielJ.Reyes ,AjayKapur
2
CaliforniaInstituteoftheArts,Valencia,U.S.A.
1
[email protected]
2
[email protected]
Abstract.Thispaperdescribesthreesonicmechatronicsystemsdesignedwiththeinfluenceofsinusoidalpropertiesand
theintentiontoproduceasonicresult.Themainpurposeofthispaperistoillustratethevarioustechniquestakenin
constructingsuchsystemsbyanalyzingtheprocessoftranslatingnaturalphenomenarecurringinnatureintophysical
emulationswithasonicaspect.Thethreemechatronicsystemsdescribedinthispaperarecategorizedintothreeparts:
sculpture,musicalinterface,andahybridofthetwo.Post-WaveisaninteractiveroboticinterpretationofRedwood
Wave,akineticsculpturebyReubenMargolin.SinuservesasacircularsequencerutilizingphotocellsandLED’sto
triggerMIDI.Lastly,CymbalicisahybridsculpturalinterfaceinspiredbythefindingsofmathematicianLeonardEuler.
Keywords:Post-Wave,Sinu,Cymbalic,mechatronic,sculpture,musicalinterface,sinusoid,kinetic
Introduction
Withintheunderstandingofmoderndayacoustics,thesinewaveremainstobethefundamentalrootofitsstudy.
(Benson2007)Themathematicalfunctiondescribesapurerepetitiveoscillation.Thisoscillationpatternoccurs
frequentlyinnature:lightwaves,waterwaves,shockwaves,radiowaves,X-raysandmostimportantlysoundwaves.
(Rossing1990)Althoughsoundwavesareextremelydifferentfromradiowavesoroceanwaves,allwavescontain
certaincommonproperties.Thesimplefactthatthesewavesoccupythespacewithinyouaresittingin,inavarietyof
formats,explainsthereasoningbehindadoptingthesinewaveastheoverallaestheticapproachtodesigningthese
systems.
Thepurposeofthispaperistoexploreaestheticsdrivenbymathematicalrelationships.Thefollowingsectionwill
familiarizethereaderwithsimilarworksinwhichsinusoidsareusedasasourceofinspiration.Section3presentsa
mechanicalwoodensculpture,Post-Wave,thefirstinaseriesofsinusoidalpieces.Sinu,acircularmusicalinterface,is
explainedindetailinsection4,discussingaestheticmotivesalongwithtechnicalprocedures.Section5completesthe
serieswiththepresentationofCymbalic.
RelatedWork
Thereareseveralkineticartistsandengineerswhohaveusedsimilarmathematicalqualitiesintheirworktorecreate
thesenaturallyoccurringphenomenas.ReubenMargolin’sworkisheavilybasedonwaveforms.Initially,inspiredbythe
crawlingqualitiesofacaterpillar’smovement,Margolinutilizedmechanizedactuatorstosimulatecomplexmotions
visibleinnature.(PopTech2014)
The Round Wave consists of nine concentric circles, each tied to a pulley system with a precise offset (figure 1). The
resultisaphysicalemulationofanexpandingsystemofripplesformedbythedroppingasmallobjectintoapoolofstill
water. Waves by Daniel Palacios is another example of a piece of related artwork. Waves is a sound installation
consistingofalongpieceofropeandtwoturbines(figure2).Thelongpieceofropetakestheshapeofacomplexwave
formastheturbinesrotate.Theresultisathreedimensionalseriesofwavesfloatinginspace.Thephysicalactionofthe
rope cutting through the air creates the whistling sound heard by the observers. Not only does the sound sculpture
visualize a harmonic series, but it also reacts and changes shape based on the presence of people around it (Palacios
2007). Another example with a more tangible aspect is a rotary sequencer inspired by the monomer (figure 3). The
custom-made rotary sequencer stores a melody on a 10” acrylic disk embedded with small magnets arranged on a
circulargrid.Concentriccirclesarecenteredaroundaspindleandrotateoveralineofmagneticsensors.Thedeviceisan
attempttoreplicatewhileimprovingthefunctionofatraditionalmechanicalmusicbox(Monomatic2010).
Figure1.TheRoundWavebyReubenMargolin
Figure2.WavesbyDanielPalacios
Figure3.ModularMusicBoxbyLewisSykesandNickRothwell
Mesmerizedbyaconstantrollingballentrenchedwithinametalriminahardwarestorehewalkedpasteverydayonhis
waytowork,Trimpin,afifteenyearoldatthetime,questionedhowaballcancontinuouslyrollaroundtherimif
perpetualmotionisnon-existent.Afterfurtherinvestigation,hewasabletodeterminethedisplaywaspneumatically
powered.Smallholesblewoutapuffofairwhentheballpassedoverit,justenoughtocontinuetomomentumtodrive
itincircles.ThisfascinationstuckwithTrimpin,ashewouldlaterdescribethathe“wasmesmerizedbythesoundand
sightofthisperfectlyroundballconstantlyrollinginZen-likerepetition,andfromthatmomentonIwantedtodo
somethingwithspheres.Ididn’twanttouseanythingfake-therehadtobeanotherwaytokeepasphereinmotion,
probablyusinggravity.”(Focke2011)DerRing’sappearanceisbasedonthedisplayfromthehardwarestore,although
it’sfunctionalityiscompletelydifferent.Thesculpturefunctionslikeapuppet,its’angleistiltedaroundinacircle,
causingtheballstoorbitduetogravity.Eachringisattachedtocablethatrunsthroughasetofpulleysandthenintoa
sophisticatedgearbox(figure4).
Figure.4DerRingbyTrimpin
Thesemechatronicsystemscollectivelysharesinusoidalandsonicproperties.Eachutilizerepetitivefeaturesfoundin
ourenvironment,andtranscribetheirdynamicsintoakineticrepresentationoftheirphysicalbehavior.
Post-Wave:MechatronicSculpture
Figure5.Post-Wave
Post-Waveisthefirstinaseriesofsinusoidal-inspiredsoundprojects.Post-WaveishighlybasedonReubenMargolin’s
RedwoodWave.Itservedasanintroductorystudyofmechanicsandelectronicdesign,inadditiontobeingan
assessmentofcarpentryskills(figure5).
Design
The mechanical design is composed of two metal rods intersecting a set of 30 6” circles, each offset by one another.
Thenasetof1”x30”stripstreatedwithdadocutsbeneathsitonapairofcircles.Topreventthewoodenstripsfrom
slippingoffthecircles,arectangularstripofwoodismountedinthemiddleofthe1”x30”dadostrips,whichisthen
securedbetweentwowoodstripsperpendiculartoitsposition.
Figure6.ProfileSketch
Figure7.SketchofCams
Implementation
ThetworodsarethencoupledtotwoseparateDCgearheadmotors.Thetwomotorsarepoweredandcontrolledwith
anArduinoUnousingPolulu’sDualVNH5019MotorDriverShield.AMicrosoftKinectisusedviaProcessingtodetect
blobdepthandpositionusingtheSimpleOpenNIlibrary.DepthandX-positiondataissentviaserialtoArduino.Thus,
depthcontrolsthespeedofthemotors,andX-positiondetermineswhichmotorisbeingcontrolled.
Figure8.InteractionwithSculpture
Theresultisasurprisinglywellexecutedmechanicalacousticemulationofthesoundproducedbywavesonabodyof
water. The physical action of the wood strips sliding back and forth on top of the wood circles makes for a soothing
hypnoticaudibleresult.Thewavemovesslowlywhenauserisnotpresent,yetassoonasauserplacehishandontop
of the wave, a reaction is immediately identified, engaging the observer to discover his or her ability to control the
sculpture(figure8).
Sinu:MusicalInterface
Sinuisanattemptatproducingarenditionofthetraditionalmusicbox.Thisparticularrotarysequencerusesasetof
eightphotocellssuspendedaboveasetofeightLEDs.A11”acrylicdiskisdividedinto16slices,eachcontaining8holes
fortheLEDtoshinethroughasitpassesthe“playhead”(figure9).Thecirculardiskisplaceontoadirect-driven
turntablemotormountedonthemainenclosure.AsetofnineLEDpushbuttons,3rotarypotentiometers,1linear
potentiometer,1LEDpowerindicator,andtwoswitchesarealsomountedontheenclosure.
Figure9.Sinu
Functionality
Figure10.ArduinoPinAssignments
ThefunctionalityofthebuttonsA-Haretoactasstepsofaconventionalsequencer.ButtonIfunctionsasashiftbutton
thatenablestheusertoswitchbetweensequences.Whenfirstpowered,asequencecanbeenteredbypushingbuttons
A-H.Oncethishasbeendone,LEDQwillturnonandthepositionmarkerofthesequencewillflashatposition1(LEDin
buttonA.)AsecondsequencecanbeenteredbyholdingdownbuttonIandselectinganybuttonB-H.Whenasequence
isenteredonanypatternI+A-H,thecorrespondingLEDQ-Xwillturnon. EachpatternI+A-Hcorrespondstoanotein
thescalespecifiedintheprogram.Basically,asLEDQ-XpassesthroughtheplayheadcontainingphotocellQ-X,the
positionmarkerofthesequenceriteratesthroughthesequence.SwitchOturnsonthemotor,andswitchMreversesits
direction.LinearpotPcontrolsspeedofthemotor.RotarypotsJ-Lcanbemidimappedtocontrolanyparameter
desired.
Implementation
SinuiscontrolledusinganArduinoMegawithanArduinoMotorShieldRev3.Thedesignconsistsoffourcustommade
PCBs:aphotocellboard,LEDboard,buttonboard,andacustomArduinoshieldtodirectallconnectionstotheArduino
pins.VoltagereadingsfromthephotocellsandbuttonsaredirectedtothemainArduinoshieldwith10conductorribbon
cable.TheLEDsthatarelocatedbeneaththephotocellsarecontrolledthroughthedigitaloutputpinsoftheArduino.
The button PCB houses all the buttons, its’ LEDs, switches and pots found on the face of the enclosure. All of the
connections are then routed, via ribbon cables, to the main Arduino shield. An Arduino sketch determines how the
sequencerfunctions.Whenatriggerisreceivedfromaphotocell,ArduinosendsamidinotenumbertoChucK.ChucK
thensendsoutamidinotemessagethroughanIACMidiBustobereceivedinAbletontocontrolanyMIDIinstrument.
Thefinalresultisacircularsequencerwiththeabilitytoperformalternativeformsofmusicalexpression.
Cymbalic:HybridInterface-Sculpture
Figure14.Cymbalic
Cymbalicisthethirdmechatronicsoundsculptureinthisseriesofsinusoidinspiredworks.Therealizationforthis
particularmechanicalsoundstudycamefromthenoveltysciencetoy,Euler’sDisk.Thediskisusedtodemonstratethe
dynamicsystemofaspinningdiskonaflatsurface.Thesmootheddiskismadeupofthickandheavychrome-plated
steel.Themirroredbasebywhichisspinsonhasaslightconcave,keepingthespinningdiskcenteredontheplate.
MathematicianLeonardEulerwasthefirsttostudythisphenomenoninthe18thcentury.(Moffatt2000)Thesculpture
isanaiveapproachinrecreatingthemotiontakenbytheEuler’sdisk.
Design
Atthecoreofitsdesign,fourcamsareusedtomanipulatealargerflat16”disk(figure15).Two1/2”metalrods,
coupledtoDCgearheadmotors,eachintersectapairofcams.Eachpairconsistsoftwodifferentsizestomakeupfor
theoffsetinheightoftheintersectingrods.Onthelargedisksuspendedabove,fourequallyspacedtuningpegseach
holdasitarstringwhichthentravelsthroughapulley,downtoacymbal.Thesitarstringsarefedthroughequallyspaced
holesinacymbalwhichismountedtothebottommetalframe,andthentiedfirmly.Theintentionistohavethecymbal
takethemotionoftheEulerdisk.
Figure15.CamMechanismSketch
Thedistancefromthecenterofthecircleatwhichtherodintersectsdeterminestheamplitudeofthelinearmotion
convertedfrombyeachcam.Eachrodintersectsthecam7/8”fromthecenterinordertopreventthelargerdiskfrom
hittingtheoutermetalframe. Figure16.CamDimensions
circle2=circle1+2offset
7=5+2(1)
Implementation
Inordertocreateanelementofhumaninteraction,theincreasingfrequencythattheEulerdiskproduceswhenspunis
programmed to control the speed of the motors. An Arduino Uno paired with a Polulu Dual VNH5019 Motor Driver
Shield is used to operate the sculpture. To detect the frequency of the disk, Adafruit’s Electret Microphone Amplifier
MAX4466isplacedundertheconcavemirroredplatform.SincetheArduinoanalogreadratesamplesataboutaquarter
ofaudiorate,analternativeapproachwastakeninorderachievemorereliableresults.Athresholdontheamplitudeis
settoinitiatethestartofthemotors.Thespeediteratesfasteraslongastheamplituderemainsabovethethreshold.As
soonastheamplitudefallsunderthethreshold,themotorscometoasuddenstop.Theelectricmicrophoneprovesto
be very sensitive, enough to create a sudden movement when the disk is touched. This way, the user immediately
experiencestheinteraction.
Conclusion
This collection of projects were completed at California Institute for the Arts. Each being a cumulation of the skills
learnedduringthesuccessiveyears.Repetitivemotioncanclearlybeseenasthedrivingforceofinspirationthroughout
eachsystem.Circlesareheavilyusedthroughouteachsystemasameanstoconvertortranslatemotion.Theseriesof
offset circles in Post-wave modulate the angle at which its corresponding wooden strip is positioned. When the
successionofcircles’phaseisalteredfromoneanother,awavemotionistranslated.InSinu,thecircleistheobjectthat
definestherecurringfeatureasequencerfunctionsby.Asequencerhasthetendencytoloopatacertainpointinits
array of values. The circle is the ideal visceral representation of that quality. Cymbalic is an impulsive reaction to a
fascinatingphenomenon.Anothersetofcircleswithanoffsetinattempttoemulateanaturallyoccurringmotion.Circles
here are a means of translating rotary motion to a series of offset linear motions to recreate a wobble motion. Each
technique varies, yet with a similar intention to utilize a fundamental sequence such as the sine wave, and develop a
uniqueinterpretationsofobjectswhichinhibitarelatablekinetic.
References
Benson,D.J.Music:AMathematicalOffering.Cambridge,UK ;NewYork:CambridgeUniversityPress,2007,13-14.
Moffatt,H.K.“Euler’sDiskandItsFinite-TimeSingularity.”NATURE-LONDON-,no.6780(2000):833. Monomatic.“ModularMusicBox|Monomatic.”ModularMusicBox,October1,2010.
http://www.monomatic.net/modular-music-box/.
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.
Trimpin,.,&Focke,A.(2011).Trimpin:Contraptionsforartandsound.Seattle,UniversityofWashingtonPress.