http://en.wikipedia.org/wiki/Oscillation 1/5

Anundampedspringmasssystemisanoscillatorysystem

OscillationFromWikipedia,thefreeencyclopedia

Oscillationistherepetitivevariation,typicallyintime,ofsomemeasureaboutacentralvalue(oftenapointofequilibrium)orbetweentwoormoredifferentstates.Familiarexamplesincludeaswingingpendulumandalternatingcurrentpower.Thetermvibrationissometimesusedmorenarrowlytomeanamechanicaloscillationbutissometimesusedasasynonymof"oscillation".

Oscillationsoccurnotonlyinmechanicalsystemsbutalsoindynamicsystemsinvirtuallyeveryareaofscience:forexamplethebeatinghumanheart,businesscyclesineconomics,predatorpreypopulationcyclesinecology,geothermalgeysersingeology,vibratingstringsinmusicalinstruments,periodicfiringofnervecellsinthebrain,andtheperiodicswellingofCepheidvariablestarsinastronomy.

Contents

1Simpleharmonicoscillator2Dampedanddrivenoscillations3Coupledoscillations4Continuoussystemswaves5Mathematics6Examples

6.1Mechanical6.2Electrical6.3Electromechanical6.4Optical6.5Biological6.6Human6.7Economicandsocial6.8Climateandgeophysics6.9Astrophysics6.10Quantummechanical6.11Chemical

7Seealso8References9Externallinks

Simpleharmonicoscillator

http://en.wikipedia.org/wiki/Oscillation 2/5

Thesimplestmechanicaloscillatingsystemisamassattachedtoalinearspringsubjecttoonlyweightandtension.Suchasystemmaybeapproximatedonanairtableoricesurface.Thesystemisinanequilibriumstatewhenthespringisstatic.Ifthesystemisdisplacedfromtheequilibrium,thereisanetrestoringforceonthemass,tendingtobringitbacktoequilibrium.However,inmovingthemassbacktotheequilibriumposition,ithasacquiredmomentumwhichkeepsitmovingbeyondthatposition,establishinganewrestoringforceintheoppositesense.Ifaconstantforcesuchasgravityisaddedtothesystem,thepointofequilibriumisshifted.Thetimetakenforanoscillationtooccurisoftenreferredtoastheoscillatoryperiod.

Systemswheretherestoringforceonabodyisdirectlyproportionaltoitsdisplacement,suchasthedynamicsofthespringmasssystem,aredescribedmathematicallybythesimpleharmonicoscillatorandtheregularperiodicmotionisknownassimpleharmonicmotion.Inthespringmasssystem,oscillationsoccurbecause,atthestaticequilibriumdisplacement,themasshaskineticenergywhichisconvertedintopotentialenergystoredinthespringattheextremesofitspath.Thespringmasssystemillustratessomecommonfeaturesofoscillation,namelytheexistenceofanequilibriumandthepresenceofarestoringforcewhichgrowsstrongerthefurtherthesystemdeviatesfromequilibrium.

Dampedanddrivenoscillations

Allrealworldoscillatorsystemsarethermodynamicallyirreversible.Thismeanstherearedissipativeprocessessuchasfrictionorelectricalresistancewhichcontinuallyconvertsomeoftheenergystoredintheoscillatorintoheatintheenvironment.Thisiscalleddamping.Thus,oscillationstendtodecaywithtimeunlessthereissomenetsourceofenergyintothesystem.Thesimplestdescriptionofthisdecayprocesscanbeillustratedbyoscillationdecayoftheharmonicoscillator.

Inaddition,anoscillatingsystemmaybesubjecttosomeexternalforce,aswhenanACcircuitisconnectedtoanoutsidepowersource.Inthiscasetheoscillationissaidtobedriven.

Somesystemscanbeexcitedbyenergytransferfromtheenvironment.Thistransfertypicallyoccurswheresystemsareembeddedinsomefluidflow.Forexample,thephenomenonofflutterinaerodynamicsoccurswhenanarbitrarilysmalldisplacementofanaircraftwing(fromitsequilibrium)resultsinanincreaseintheangleofattackofthewingontheairflowandaconsequentialincreaseinliftcoefficient,leadingtoastillgreaterdisplacement.Atsufficientlylargedisplacements,thestiffnessofthewingdominatestoprovidetherestoringforcethatenablesanoscillation.

Coupledoscillations

Theharmonicoscillatorandthesystemsitmodelshaveasingledegreeoffreedom.Morecomplicatedsystemshavemoredegreesoffreedom,forexampletwomassesandthreesprings(eachmassbeingattachedtofixedpointsandtoeachother).Insuchcases,thebehaviorofeachvariableinfluencesthatoftheothers.Thisleadstoacouplingoftheoscillationsoftheindividualdegreesoffreedom.Forexample,twopendulumclocks(ofidenticalfrequency)mountedonacommonwallwilltendtosynchronise.ThisphenomenonwasfirstobservedbyChristiaanHuygensin1665.[1]Theapparentmotionsofthecompoundoscillationstypicallyappearsverycomplicatedbutamoreeconomic,computationallysimplerandconceptuallydeeperdescriptionisgivenbyresolvingthemotionintonormalmodes.

Morespecialcasesarethecoupledoscillatorswhereenergyalternatesbetweentwoformsofoscillation.WellknownistheWilberforcependulum,wheretheoscillationalternatesbetweenanelongationofaverticalspringandtherotationofanobjectattheendofthatspring.

Continuoussystemswaves

http://en.wikipedia.org/wiki/Oscillation 3/5

Twopendulumswiththesameperiodfixedonastringactaspairofcoupledoscillators.Theoscillationalternatesbetweenthetwo.

Oscillationofasequence(showninblue)isthedifferencebetweenthelimitsuperiorandlimitinferiorofthesequence.

Asthenumberofdegreesoffreedombecomesarbitrarilylarge,asystemapproachescontinuityexamplesincludeastringorthesurfaceofabodyofwater.Suchsystemshave(intheclassicallimit)aninfinitenumberofnormalmodesandtheiroscillationsoccurintheformofwavesthatcancharacteristicallypropagate.

Mathematics

Themathematicsofoscillationdealswiththequantificationoftheamountthatasequenceorfunctiontendstomovebetweenextremes.Thereareseveralrelatednotions:oscillationofasequenceofrealnumbers,oscillationofarealvaluedfunctionatapoint,andoscillationofafunctiononaninterval(oropenset).

Examples

Mechanical

DoublependulumFoucaultpendulumHelmholtzresonatorOscillationsintheSun(helioseismology),stars(asteroseismology)andNeutronstaroscillations.QuantumharmonicoscillatorPlaygroundswing

Electromechanical

CrystaloscillatorLoudspeakerMicrophone

Optical

Laser(oscillationofelectromagneticfieldwith

frequencyoforder1015Hz)OscillatorTodaorselfpulsation(pulsationof

Economicandsocial

BusinesscycleGenerationgapMalthusianeconomicsNewscycle

Climateandgeophysics

AtlanticmultidecadaloscillationChandlerwobbleClimateoscillation

http://en.wikipedia.org/wiki/Oscillation 4/5

StringinstrumentsTorsionalvibrationTuningforkVibratingstringWilberforcependulumLeverescapement

Electrical

AlternatingcurrentArmstrong(orTicklerorMeissner)oscillatorAstablemultivibratorBlockingoscillatorButleroscillatorClapposcillatorColpittsoscillatorDelaylineoscillatorDow(orultraaudion)oscillatorElectronicoscillatorHartleyoscillatorOscillistorPierceoscillatorRelaxationoscillatorRLCcircuitRoyeroscillatorVakoscillatorWienbridgeoscillator

outputpoweroflaserat

frequencies104Hz

106Hzinthetransientregime)Quantumoscillatormayrefertoanopticallocaloscillator,aswellastoausualmodelinquantumoptics.

Biological

CircadianrhythmCircadianoscillatorLotkaVolterraequationNeuraloscillationOscillatinggeneSegmentationoscillator

Human

NeuraloscillationInsulinreleaseoscillationsgonadotropinreleasinghormonepulsationsPilotinducedoscillationVoiceproduction

ElNioSouthernOscillationPacificdecadaloscillationQuasibiennialoscillation

Astrophysics

NeutronstarsCyclicModel

Quantummechanical

NeutrinooscillationsQuantumharmonicoscillator

Chemical

BelousovZhabotinskyreactionMercurybeatingheartBriggsRauscherreactionBrayLiebhafskyreaction

Seealso

AntiresonanceBeat(acoustics)BIBOstabilityCriticalspeedCycle(music)DynamicalsystemEarthquakeengineering

OscillatorphasenoisePeriodicfunctionPhasenoiseReciprocatingmotionResonatorRhythmSeasonality

SignalgeneratorSqueggingStrangeattractorStructuralstabilityTunedmassdamperVibrationVibrator(mechanical)

http://en.wikipedia.org/wiki/Oscillation 5/5

FeedbackFrequency

SelfoscillationHiddenoscillation

References

1. ^Strogatz,Steven.Sync:TheEmergingScienceofSpontaneousOrder.Hyperion,2003,pp106109

Externallinks

Vibrations(http://www.lightandmatter.com/html_books/3vw/ch01/ch01.html)achapterfromanonlinetextbook

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