Cp project-1-chaotic-pendulum
- 1. Solving and visualizing the motion of a Harmonic Oscillator Harmonicoscillationiseverywhere inphysics.Itisprobablythe simplestandmostimportantphysical system. Learningthe basicsof harmonicoscillationisaquantumleapto understandinga whole lotof systemsfrommechanics,electronics,andoptics.Thiscomputational projectisafast tract explorationof the propertiesof drivenharmonicoscillation. Simple Harmonic Oscillator The motionof a simple harmonicoscillatorisgovernedbythe equation 𝑚𝑥̈ = −𝑘𝑥 The motionof the oscillator(solutiontothe equationof motion) canbe writtenas 𝑥( 𝑡) = 𝐴 cos(𝜔𝑡 + 𝜙), 𝜔2 = 𝑘/𝑚 where the constants 𝐴,and 𝜙 dependsonthe initial conditions. Usingan ODE solverinMATLAB, usingtwodifferentinitialconditions{𝑥(0) = 0, 𝑥̇(0) = 1} and { 𝑥(0) = 1, 𝑥̇(0) = 0}, do the following 1. Plotthe positionandvelocityasafunctionof time 2. Plotthe potential,kinetic,andtotal energyasafunctionof time 3. Plotthe phase space (i.e.positionvsmomentum),set 𝑚 = 1 4. Describe whathappenstothe plotswhenyouvary 𝜔. To have a betterfeel the problem, trydifferentconditions andparameters. Damped harmonic oscillator Whenthere isfrictionor anylossof energyina harmonicmotion, the motionisdescribedby 𝑥̈ + 2𝛽𝑥̇ + 𝜔0 2 𝑥 = 0 where 𝛽 isthe dampingconstant,and 𝜔0 isthe natural frequency.Usingthe ODEsolverinMATLAB,do the same plotsas before forinitial conditions {𝑥(0) = 0, 𝑥̇(0) = 1},when 𝛽 < 𝜔0, 𝛽 = 𝜔0,and 𝛽 > 𝜔0. Describe the effectof damping inthe three differentregimes. Playwiththe parameterstohave abetter feel of the problem. Driven dampedoscillator Whenthe oscillatorisdrivenbyan external force 𝐹(𝑡),the equationof motioncanbe writtenas 𝑥̈ + 2𝛽𝑥̇ + 𝜔0 2 𝑥 = 𝑓(𝑡) There are infinitelydifferentwaystodrive the system. The mostcommonwaytodo it iswitha wavelike force,say sinusoidalforce of the form 𝑓( 𝑡) = 𝑓𝑑 cos 𝜔 𝑑 𝑡 where 𝑓𝑑 isthe drivingamplitudeand 𝜔 𝑑 isthe drivingfrequency.Now, considerthe case when the oscillatorstartsstationaryat 𝑥 = 1. Solve forthe equation forthe followingparameters 𝑓𝑑 = 100, 𝜔0 = 1 𝛽 = 𝜔0/30, and 𝜔 𝑑 = 𝜔0/5. On equal time axis,plotthe positionandthe drivingforce side byside.Show atleastleast 6cycles of the drivingforce.Youshouldget somethinglike this)
- 2. Include athirdplot forthe positionof the same oscillatorbutwithnodrivingforce. The initial oscillationsare called transientswhichdisappearsafterafew cycles andthe eventual oscillationiscalled an attractor. While keepingotherparametersconstant,describethe effectof the followingonthe transientsand attractor (amplitude andphase): 1. Initial condition(positionandvelocity) 2. Dampingconstant 𝛽 from 𝛽 < 𝜔0 to 𝛽 > 𝜔0 3. Drivingfrequency 𝜔 𝑑 from 𝜔 𝑑 < 𝜔0 to 𝜔 𝑑 > 𝜔0 Showthat the amplitude 𝐴 andphase shift 𝛿 of the attractor follow thisequation ( 𝐴 𝑓𝑑 ) 2 = 1 ( 𝜔0 2 − 𝜔 𝑑 2) + (2𝜔 𝑑 𝛽)2, 𝛿 = arctan ( 2𝛽𝜔 𝑑 𝜔0 2 − 𝜔 𝑑 2 ) To showthis,keepotherparametersconstantandplotthe analytical curve above withthe one from theorytogether.Youshouldgetsomethinglike this for 𝐴 vs 𝜔0. The amplitude ismaximumwhen 𝜔 𝑑 = 𝜔0,whichiscalled resonance.Thatis,the oscillatorabsorbs maximumenergywhenthe natural frequencyof the oscillatoristhe same asthe drivingfrequency. You can try differentdrivingforces,notjustsinusoidal ones. Reference: 1. For more on the analytical side of the harmonicoscillator,consultthe book ClassicalMechanics by John Taylor.Chapter5 is dedicatedforharmonicoscillations. Figuresinthispaperistaken fromthat chapter.