Oscillation - Wikipedia

Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value or between two or more different states. Oscillation FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Repetitivevariationofsomemeasureaboutacentralvalue "Oscillator"redirectshere.Forotheruses,seeOscillator(disambiguation).Thisarticlerelieslargelyorentirelyonasinglesource.Relevantdiscussionmaybefoundonthetalkpage.Pleasehelpimprovethisarticlebyintroducingcitationstoadditionalsources.Findsources: "Oscillation" – news ·newspapers ·books ·scholar ·JSTOR(November2016)Anundampedspring–masssystemisanoscillatorysystem Oscillationistherepetitiveorperiodicvariation,typicallyintime,ofsomemeasureaboutacentralvalue(oftenapointofequilibrium)orbetweentwoormoredifferentstates.Familiarexamplesofoscillationincludeaswingingpendulumandalternatingcurrent.Oscillationscanbeusedinphysicstoapproximatecomplexinteractions,suchasthosebetweenatoms. Oscillationsoccurnotonlyinmechanicalsystemsbutalsoindynamicsystemsinvirtuallyeveryareaofscience:forexamplethebeatingofthehumanheart(forcirculation),businesscyclesineconomics,predator–preypopulationcyclesinecology,geothermalgeysersingeology,vibrationofstringsinguitarandotherstringinstruments,periodicfiringofnervecellsinthebrain,andtheperiodicswellingofCepheidvariablestarsinastronomy.Thetermvibrationispreciselyusedtodescribeamechanicaloscillation. Contents 1Simpleharmonic 2Two-dimensionaloscillators 2.1Anisotropicoscillators 3Dampedoscillations 4Drivenoscillations 4.1Resonance 5Coupledoscillations 6Smalloscillationapproximation 7Continuoussystems –waves 8Mathematics 9Examples 9.1Mechanical 9.2Electrical 9.3Electro-mechanical 9.4Optical 9.5Biological 9.6Humanoscillation 9.7Economicandsocial 9.8Climateandgeophysics 9.9Astrophysics 9.10Quantummechanical 9.11Chemical 9.12Computing 10Seealso 11References 12Externallinks Simpleharmonic[edit] Mainarticle:Simpleharmonicmotion Thesimplestmechanicaloscillatingsystemisaweightattachedtoalinearspringsubjecttoonlyweightandtension.Suchasystemmaybeapproximatedonanairtableoricesurface.Thesystemisinanequilibriumstatewhenthespringisstatic.Ifthesystemisdisplacedfromtheequilibrium,thereisanetrestoringforceonthemass,tendingtobringitbacktoequilibrium.However,inmovingthemassbacktotheequilibriumposition,ithasacquiredmomentumwhichkeepsitmovingbeyondthatposition,establishinganewrestoringforceintheoppositesense.Ifaconstantforcesuchasgravityisaddedtothesystem,thepointofequilibriumisshifted.Thetimetakenforanoscillationtooccurisoftenreferredtoastheoscillatoryperiod. Thesystemswheretherestoringforceonabodyisdirectlyproportionaltoitsdisplacement,suchasthedynamicsofthespring-masssystem,aredescribedmathematicallybythesimpleharmonicoscillatorandtheregularperiodicmotionisknownassimpleharmonicmotion.Inthespring-masssystem,oscillationsoccurbecause,atthestaticequilibriumdisplacement,themasshaskineticenergywhichisconvertedintopotentialenergystoredinthespringattheextremesofitspath.Thespring-masssystemillustratessomecommonfeaturesofoscillation,namelytheexistenceofanequilibriumandthepresenceofarestoringforcewhichgrowsstrongerthefurtherthesystemdeviatesfromequilibrium. Inthecaseofthespring-masssystem,Hooke'slawstatesthattherestoringforceofaspringis: F = − k x {\displaystyleF=-kx} ByusingNewton'ssecondlaw,thedifferentialequationcanbederived. x ¨ = − k m x = − ω 2 x {\displaystyle{\ddot{x}}=-{\frac{k}{m}}x=-\omega^{2}x} , where ω = k m {\displaystyle\omega={\sqrt{\frac{k}{m}}}} Thesolutiontothisdifferentialequationproducesasinusoidalpositionfunction. x ( t ) = A cos ⁡ ( ω t − δ ) {\displaystylex(t)=A\cos(\omegat-\delta)} whereωisthefrequencyoftheoscillation,Aistheamplitude,andδisthephaseshiftofthefunction.Thesearedeterminedbytheinitialconditionsofthesystem.Becausecosineoscillatesbetween1and-1infinitely,ourspring-masssystemwouldoscillatebetweenthepositiveandnegativeamplitudeforeverwithoutfriction. Two-dimensionaloscillators[edit] Intwoorthreedimensions,harmonicoscillatorsbehavesimilarlytoonedimension.Thesimplestexampleofthisisanisotropicoscillator,wheretherestoringforceisproportionaltothedisplacementfromequilibriumwiththesamerestorativeconstantinalldirections. F = − k r → {\displaystyleF=-k{\vec{r}}} Thisproducesasimilarsolution,butnowthereisadifferentequationforeverydirection. x ( t ) = A x cos ⁡ ( ω t − δ x ) {\displaystylex(t)=A_{x}\cos(\omegat-\delta_{x})} , y ( t ) = A y cos ⁡ ( ω t − δ y ) {\displaystyley(t)=A_{y}\cos(\omegat-\delta_{y})} , [...] Anisotropicoscillators[edit] Withanisotropicoscillators,differentdirectionshavedifferentconstantsofrestoringforces.Thesolutionissimilartoisotropicoscillators,butthereisadifferentfrequencyineachdirection.Varyingthefrequenciesrelativetoeachothercanproduceinterestingresults.Forexample,ifthefrequencyinonedirectionistwicethatofanother,afigureeightpatternisproduced.Iftheratiooffrequenciesisirrational,themotionisquasiperiodic.Thismotionisperiodiconeachaxis,butisnotperiodicwithrespecttor,andwillneverrepeat.[1] Dampedoscillations[edit] Mainarticle:Harmonicoscillator Seealso:Anti-vibrationcompound Allreal-worldoscillatorsystemsarethermodynamicallyirreversible.Thismeanstherearedissipativeprocessessuchasfrictionorelectricalresistancewhichcontinuallyconvertsomeoftheenergystoredintheoscillatorintoheatintheenvironment.Thisiscalleddamping.Thus,oscillationstendtodecaywithtimeunlessthereissomenetsourceofenergyintothesystem.Thesimplestdescriptionofthisdecayprocesscanbeillustratedbyoscillationdecayoftheharmonicoscillator. Dampedoscillatorsarecreatedwhenaresistiveforceisintroduced,whichisdependentonthefirstderivativeoftheposition,orinthiscasevelocity.ThedifferentialequationcreatedbyNewton'ssecondlawaddsinthisresistiveforcewithanarbitraryconstantb.Thisexampleassumesalineardependenceonvelocity. m x ¨ + b x ˙ + k x = 0 {\displaystylem{\ddot{x}}+b{\dot{x}}+kx=0} Thisequationcanberewrittenasbefore. x ¨ + 2 β x ˙ + ω 0 2 x = 0 {\displaystyle{\ddot{x}}+2\beta{\dot{x}}+\omega_{0}^{2}x=0} , where 2 β = b m {\displaystyle2\beta={\frac{b}{m}}} Thisproducesthegeneralsolution: x ( t ) = e − β t ( C 1 e ω 1 t + C 2 e − ω 1 t ) {\displaystylex(t)=e^{-\betat}(C_{1}e^{\omega_{1}t}+C_{2}e^{-\omega_{1}t})} , where ω 1 = β 2 − ω 0 2 {\displaystyle\omega_{1}={\sqrt{\beta^{2}-\omega_{0}^{2}}}} Theexponentialtermoutsideoftheparenthesisisthedecayfunctionandβisthedampingcoefficient.Thereare3categoriesofdampedoscillators:under-damped,whereβω0;andcriticallydamped,whereβ=ω0. Drivenoscillations[edit] Inaddition,anoscillatingsystemmaybesubjecttosomeexternalforce,aswhenanACcircuitisconnectedtoanoutsidepowersource.Inthiscasetheoscillationissaidtobedriven. Thesimplestexampleofthisissspring-masssystemwithasinusoidaldrivingforce. x ¨ + 2 β x ˙ + ω 0 2 x = f ( t ) {\displaystyle{\ddot{x}}+2\beta{\dot{x}}+\omega_{0}^{2}x=f(t)} ,where f ( t ) = f 0 cos ⁡ ( ω t + δ ) {\displaystylef(t)=f_{0}\cos(\omegat+\delta)} Thisgivesthesolution: x ( t ) = A cos ⁡ ( ω t − δ ) + A t r cos ⁡ ( ω 1 t − δ t r ) {\displaystylex(t)=A\cos(\omegat-\delta)+A_{tr}\cos(\omega_{1}t-\delta_{tr})} , where A = f 0 2 ( ω 0 2 − ω 2 ) + 2 β ω {\displaystyleA={\sqrt{\frac{f_{0}^{2}}{(\omega_{0}^{2}-\omega^{2})+2\beta\omega}}}} and δ = tan − 1 ⁡ ( 2 β ω ω 0 2 − ω 2 ) {\displaystyle\delta=\tan^{-1}({\frac{2\beta\omega}{\omega_{0}^{2}-\omega^{2}}})} Thesecondtermofx(t)isthetransientsolutiontothedifferentialequation.Thetransientsolutioncanbefoundbyusingtheinitialconditionsofthesystem. Somesystemscanbeexcitedbyenergytransferfromtheenvironment.Thistransfertypicallyoccurswheresystemsareembeddedinsomefluidflow.Forexample,thephenomenonofflutterinaerodynamicsoccurswhenanarbitrarilysmalldisplacementofanaircraftwing(fromitsequilibrium)resultsinanincreaseintheangleofattackofthewingontheairflowandaconsequentialincreaseinliftcoefficient,leadingtoastillgreaterdisplacement.Atsufficientlylargedisplacements,thestiffnessofthewingdominatestoprovidetherestoringforcethatenablesanoscillation. Resonance[edit] Resonanceoccursinadampeddrivenoscillatorwhenω=ω0,thatis,whenthedrivingfrequencyisequaltothenaturalfrequencyofthesystem.Whenthisoccurs,thedenominatoroftheamplitudeisminimized,whichmaximizestheamplitudeoftheoscillations. Coupledoscillations[edit] Twopendulumswiththesameperiodfixedonastringactaspairofcoupledoscillators.Theoscillationalternatesbetweenthetwo. Mainarticle:injectionlocking ExperimentalSetupofHuygenssynchronizationoftwoclocks Theharmonicoscillatorandthesystemsitmodelshaveasingledegreeoffreedom.Morecomplicatedsystemshavemoredegreesoffreedom,forexample,twomassesandthreesprings(eachmassbeingattachedtofixedpointsandtoeachother).Insuchcases,thebehaviorofeachvariableinfluencesthatoftheothers.Thisleadstoacouplingoftheoscillationsoftheindividualdegreesoffreedom.Forexample,twopendulumclocks(ofidenticalfrequency)mountedonacommonwallwilltendtosynchronise.ThisphenomenonwasfirstobservedbyChristiaanHuygensin1665.[2]Theapparentmotionsofthecompoundoscillationstypicallyappearsverycomplicatedbutamoreeconomic,computationallysimplerandconceptuallydeeperdescriptionisgivenbyresolvingthemotionintonormalmodes. Thesimplestformofcoupledoscillatorsisa3spring,2masssystem,wheremassesandspringconstantsarethesame.ThisproblembeginswithderivingNewton'ssecondlawforbothmasses. m 1 x ¨ 1 = − ( k 1 + k 2 ) x 1 + k 2 x 2 {\displaystylem_{1}{\ddot{x}}_{1}=-(k_{1}+k_{2})x_{1}+k_{2}x_{2}} , m 2 x ¨ 2 = k 2 x 1 − ( k 2 + k 3 ) x 2 {\displaystylem_{2}{\ddot{x}}_{2}=k_{2}x_{1}-(k_{2}+k_{3})x_{2}} , Theequationsarethengeneralizedintomatrixform. F = M x ¨ = k x {\displaystyleF=M{\ddot{x}}=kx} , where M = [ m 1 0 0 m 2 ] {\displaystyleM={\begin{bmatrix}m_{1}&0\\0&m_{2}\end{bmatrix}}} , x = [ x 1 x 2 ] {\displaystylex={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}} ,and k = [ k 1 + k 2 − k 2 − k 2 k 2 + k 3 ] {\displaystylek={\begin{bmatrix}k_{1}+k_{2}&-k_{2}\\-k_{2}&k_{2}+k_{3}\end{bmatrix}}} Thevaluesofkandmcanbesubstitutedintothematrices. m 1 = m 2 = m {\displaystylem_{1}=m_{2}=m} , k 1 = k 2 = k 3 = k {\displaystylek_{1}=k_{2}=k_{3}=k} , M = [ m 0 0 m ] {\displaystyleM={\begin{bmatrix}m&0\\0&m\end{bmatrix}}} , k = [ 2 k − k − k 2 k ] {\displaystylek={\begin{bmatrix}2k&-k\\-k&2k\end{bmatrix}}} Thesematricescannowbepluggedintothegeneralsolution. ( k − M ω 2 ) a = 0 {\displaystyle(k-M\omega^{2})a=0} [ 2 k − m ω 2 − k − k 2 k − m ω 2 ] = 0 {\displaystyle{\begin{bmatrix}2k-m\omega^{2}&-k\\-k&2k-m\omega^{2}\end{bmatrix}}=0} Thedeterminantofthismatrixyieldsaquadraticequation. ( 3 k − m ω 2 ) ( k − m ω 2 ) = 0 {\displaystyle(3k-m\omega^{2})(k-m\omega^{2})=0} ω 1 = k m {\displaystyle\omega_{1}={\sqrt{\frac{k}{m}}}} , ω 2 = 3 k m {\displaystyle\omega_{2}={\sqrt{\frac{3k}{m}}}} Dependingonthestartingpointofthemasses,thissystemhas2possiblefrequencies(oracombinationofthetwo).Ifthemassesarestartedwiththeirdisplacementsinthesamedirection,thefrequencyisthatofasinglemasssystem,becausethemiddlespringisneverextended.Ifthetwomassesarestartedinoppositedirections,thesecond,fasterfrequencyisthefrequencyofthesystem.[1] Morespecialcasesarethecoupledoscillatorswhereenergyalternatesbetweentwoformsofoscillation.Well-knownistheWilberforcependulum,wheretheoscillationalternatesbetweentheelongationofaverticalspringandtherotationofanobjectattheendofthatspring. Coupledoscillatorsareacommondescriptionoftworelated,butdifferentphenomena.Onecaseiswherebothoscillationsaffecteachothermutually,whichusuallyleadstotheoccurrenceofasingle,entrainedoscillationstate,wherebothoscillatewithacompromisefrequency.Anothercaseiswhereoneexternaloscillationaffectsaninternaloscillation,butisnotaffectedbythis.Inthiscasetheregionsofsynchronization,knownasArnoldTongues,canleadtohighlycomplexphenomenaasforinstancechaoticdynamics. Smalloscillationapproximation[edit] Inphysics,asystemwithasetofconservativeforcesandanequilibriumpointcanbeapproximatedasaharmonicoscillatornearequilibrium.AnexampleofthisistheLennard-Jonespotential,wherethepotentialisgivenby: U ( r ) = U 0 [ ( r 0 r ) 12 − ( r 0 r ) 6 ] {\displaystyleU(r)=U_{0}\left[\left({\frac{r_{0}}{r}}\right)^{12}-\left({\frac{r_{0}}{r}}\right)^{6}\right]} Theequilibriumpointsofthefunctionarethenfound. d U d r = 0 = U 0 [ − 12 r 0 12 r − 13 + 6 r 0 6 r − 7 ] {\displaystyle{\frac{dU}{dr}}=0=U_{0}[-12r_{0}^{12}r^{-13}+6r_{0}^{6}r^{-7}]} ⇒ r 0 ≈ r {\displaystyle\Rightarrowr_{0}\approxr} Thesecondderivativeisthenfound,andusedtobetheeffectivepotentialconstant. γ e f f = d 2 U d r 2 | r = r 0 = U 0 [ 12 ( 13 ) r 0 12 r − 14 − 6 ( 7 ) r 0 6 r − 8 ] {\displaystyle\gamma_{eff}={\frac{d^{2}U}{dr^{2}}}\vert_{r=r_{0}}=U_{0}[12(13)r_{0}^{12}r^{-14}-6(7)r_{0}^{6}r^{-8}]} γ e f f = 114 U 0 r 2 {\displaystyle\gamma_{eff}={\frac{114U_{0}}{r^{2}}}} Thesystemwillundergooscillationsneartheequilibriumpoint.Theforcethatcreatestheseoscillationsisderivedfromtheeffectivepotentialconstantabove. F = − γ e f f ( r − r 0 ) = m e f f r ¨ {\displaystyleF=-\gamma_{eff}(r-r_{0})=m_{eff}{\ddot{r}}} Thisdifferentialequationcanbere-writtenintheformofasimpleharmonicoscillator. r ¨ + γ e f f m e f f ( r − r 0 ) = 0 {\displaystyle{\ddot{r}}+{\frac{\gamma_{eff}}{m_{eff}}}(r-r_{0})=0} Thus,thefrequencyofsmalloscillationsis: ω 0 = γ e f f m e f f = 114 U 0 r 2 m e f f {\displaystyle\omega_{0}={\sqrt{\frac{\gamma_{eff}}{m_{eff}}}}={\sqrt{\frac{114U_{0}}{r^{2}m_{eff}}}}} Or,ingeneralform[3] ω 0 = d 2 U d U 2 | r = r 0 {\displaystyle\omega_{0}={\sqrt{{\frac{d^{2}U}{dU^{2}}}\vert_{r=r_{0}}}}} Thisapproximationcanbebetterunderstoodbylookingatthepotentialcurveofthesystem.Bythinkingofthepotentialcurveasahill,inwhich,ifoneplacedaballanywhereonthecurve,theballwouldrolldownwiththeslopeofthepotentialcurve.Thisistrueduetotherelationshipbetweenpotentialenergyandforce. d U d t = − F ( r ) {\displaystyle{\frac{dU}{dt}}=-F(r)} Bythinkingofthepotentialinthisway,onewillseethatatanylocalminimumthereisa"well"inwhichtheballwouldrollbackandforth(oscillate)between r m i n {\displaystyler_{min}} and r m a x {\displaystyler_{max}} .ThisapproximationisalsousefulforthinkingofKeplerorbits. Continuoussystems –waves[edit] Mainarticle:Wave Asthenumberofdegreesoffreedombecomesarbitrarilylarge,asystemapproachescontinuity;examplesincludeastringorthesurfaceofabodyofwater.Suchsystemshave(intheclassicallimit)aninfinitenumberofnormalmodesandtheiroscillationsoccurintheformofwavesthatcancharacteristicallypropagate. Mathematics[edit] Mainarticle:Oscillation(mathematics) Oscillationofasequence(showninblue)isthedifferencebetweenthelimitsuperiorandlimitinferiorofthesequence. Themathematicsofoscillationdealswiththequantificationoftheamountthatasequenceorfunctiontendstomovebetweenextremes.Thereareseveralrelatednotions:oscillationofasequenceofrealnumbers,oscillationofareal-valuedfunctionatapoint,andoscillationofafunctiononaninterval(oropenset). Examples[edit] Mechanical[edit] Doublependulum Foucaultpendulum Helmholtzresonator OscillationsintheSun(helioseismology),stars(asteroseismology)andNeutron-staroscillations. Quantumharmonicoscillator Playgroundswing Stringinstruments Torsionalvibration Tuningfork Vibratingstring Wilberforcependulum Leverescapement Electrical[edit] Mainarticle:Electronicoscillator Alternatingcurrent Armstrong(orTicklerorMeissner)oscillator Astablemultivibrator Blockingoscillator Butleroscillator Clapposcillator Colpittsoscillator Delay-lineoscillator Electronicoscillator Extendedinteractionoscillator Hartleyoscillator Oscillistor Phase-shiftoscillator Pierceoscillator Relaxationoscillator RLCcircuit Royeroscillator Vačkářoscillator Wienbridgeoscillator Electro-mechanical[edit] Crystaloscillator Optical[edit] Laser(oscillationofelectromagneticfieldwithfrequencyoforder1015 Hz) OscillatorTodaorself-pulsation(pulsationofoutputpoweroflaseratfrequencies104 Hz–106 Hzinthetransientregime) Quantumoscillatormayrefertoanopticallocaloscillator,aswellastoausualmodelinquantumoptics. Biological[edit] Circadianrhythm Circadianoscillator Lotka–Volterraequation Neuraloscillation Oscillatinggene Segmentationclock Humanoscillation[edit] Neuraloscillation Insulinreleaseoscillations gonadotropinreleasinghormonepulsations Pilot-inducedoscillation Voiceproduction Economicandsocial[edit] Businesscycle Generationgap Malthusianeconomics Newscycle Climateandgeophysics[edit] Atlanticmultidecadaloscillation Chandlerwobble Climateoscillation ElNiño-SouthernOscillation Pacificdecadaloscillation Quasi-biennialoscillation Astrophysics[edit] Neutronstars CyclicModel Quantummechanical[edit] Neutralparticleoscillation,e.g.neutrinooscillations Quantumharmonicoscillator Chemical[edit] Belousov–Zhabotinskyreaction Mercurybeatingheart Briggs–Rauscherreaction Bray–Liebhafskyreaction Computing[edit] CellularAutomataoscillator Seealso[edit] Antiresonance Beat(acoustics) BIBOstability Criticalspeed Cycle(music) Dynamicalsystem Earthquakeengineering Feedback Fouriertransformforcomputingperiodicityinevenlyspaceddata Frequency Hiddenoscillation Least-squaresspectralanalysisforcomputingperiodicityinunevenlyspaceddata Oscillatorphasenoise Periodicfunction Phasenoise Quasiperiodicity Reciprocatingmotion Resonator Rhythm Seasonality Self-oscillation Signalgenerator Squegging Strangeattractor Structuralstability Tunedmassdamper Vibration Vibrator(mechanical) References[edit] ^abTaylor,JohnR.(2005).Classicalmechanics.MillValley,California.ISBN 1-891389-22-X.OCLC 55729992. ^Strogatz,Steven(2003).Sync:TheEmergingScienceofSpontaneousOrder.HyperionPress.pp. 106–109.ISBN 0-786-86844-9. ^"23.7:SmallOscillations".PhysicsLibreTexts.2020-07-01.Retrieved2022-04-21. Externallinks[edit] LookuposcillationinWiktionary,thefreedictionary. 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