P441 Analytical Mechanics  I. Coupled Oscillators. c Alex R. Dzierba


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1 Lecure 3 Mondy  Deceber 5, 005 Wrien or ls upded: Deceber 3, 005 P44 Anlyicl Mechnics  I oupled Oscillors c Alex R. Dzierb oupled oscillors  rix echnique In Figure we show n exple of wo coupled oscillors, wo pendul, ech of lengh nd ss, coupled by ssless spring of spring consn k. θ θ k Figure : oupled oscillors. We will use he convenion h if θ increses o he righ of equilibriu i is posiive nd oherwise negive. The kineic energy T, grviionl poenil energy U g nd poenil energy due o he spring U s of his syse re given by: T = θ + θ U g = g θ + θ U s = k θ θ The Lgrngin for his syse is L = T U g U s nd Lgrnge s equions yields: θ + g θ + k θ θ = 0 θ + g θ + k θ θ = 0 Defining η k/g nd ω0 = g/ he bove cn be wrien in his copc rix for:
2 θ = ω + η η θ 0 η + η θ θ If we ry soluion of he for: θ Θ = e θ Θ iω hen wih his ddiionl definiion λ ω /ω 0 we hve: + η η η + η Θ Θ Θ = λ Θ 3 bu his looks like solving n eigenvlue/eigenvecor equion: AΘ = λθ. I follows h A λθ = 0. For here o be nonrivil eigenvecors, he deerinn of he rix uliplying Θ hs o be zero o find he eigenvlues we solve for he chrcerisic equion: + η λ η η + η λ = + η λ η = 0 4 nd he soluions o his qudric re λ = corresponding o ω = ω 0 nd λ = + η corresponding o ω = ω 0 + η. The eigenvecors of he rix A re found by using equion 3 nd firs using one of he eigenvlues nd hen he oher. In ech cse one ges he rio Θ /Θ. The overll norlizion is fixed by requiring h he gniude of he eigenvecor be uniy. The eigenvecors re: nd corresponding o he eigenvlues nd + η. Noice h he do produc of he wo eigenvecors is zero hey re orhonorl eigenvecors. If one consrucs rix whose coluns re he eigenvecors, he resuling rix cn be used o do siilriy rnsforion of he rix A. This digonlizes he rix A nd he digonl eleens re he eigenvlues. The rix h does his rick is: S = or explicily: S AS = + η η η + η = η
3 The eigenvecors correspond o Θ +Θ / for he syeric ode wih λ = nd Θ +Θ / wih λ = + η. This wh we found in he ls lecure. These re he cobinions h decouple he equions of oion. In he syeric ode he pendul hve he se pliude nd re in phse he coupling spring does nohing. We could siply reove is. Ech pendulu oscilles wih he se frequency wih λ = giving us ω S = ω 0. In he nisyeric ode, boh pendul gin oscille wih he se frequency, bu now he nisyeric frequency wih λ = + η giving us ω A = ω 0 + k/ obinions of norl odes The norl odes re he eigenvecors which re orhogonl o ech oher nd he os generl oion cn be expressed s liner cobinion of he norl odes. We sw h when we sr boh pendul fro res, wih one pendulu is equilibriu posiion θ 0 = 0 nd he oher displce fro equilibriu θ 0 = θ 0 hen he soluions re: θ = θ 0 cosω S + cosω A θ = θ 0 cosω S cosω A nd using rig relions you should be filir wih: θ θ ωs ω A = θ 0 cos ωs ω A = θ 0 sin ωs + ω A cos ωs + ω A sin = [θ 0 cosω b ] cosω v = [θ 0 sinω b ] sinω v Ech pendulu execues hronic oion wih frequency ω v h is he verge of he syeric nd nisyeric frequencies. Bu in ddiion he pliude is oduled by sinusoidl funcion of ie wih be frequency ω b h is he difference of he syeric nd nisyeric frequencies. If he spring is wek he be frequency is slow. Evenully he firs pendulu s pliude dies o zero bu h s when he oher pendulu is full pliude. The pliude odulion for he wo pendul re 90 ou of phse wih ech oher. See he ie dependence of θ nd θ in Figure. Energy is rnsferred fro one pendulu o he oher nd of course, ol energy is conserved. Anoher coupled oscillor syse Figure 3 shows noher coupled oscillor syse. Two sses ove on horizonl surfce wihou fricion. We ssue h he spring connecing he wo sses hs spring consn k h is sller h he consn for he springs connecing he sses o he wlls. onvince yourself h you know how o inerpre he syeric nd nisyeric odes for his syse. Also, you cn borrow he resuls of he erlier discussion wih his siple replceen: g/ k 0 / 3
4 θ θ ω S + ω A / ω S ω A / Figure : The ngulr displceen s funcion of ie for our wo coupled pendul when he syse is sred fro res wih θ 0 = θ 0 = nd θ 0 = 0. In his exple ω S = ω 0 = 3 s nd ω A = 3.37 s. The fs oscillion of ech pendulu fro equilibriu hs n ngulr frequency h is is he verge of ω S nd ω A while he pliude odulion hppens wih n ngulr frequency h is hlf he difference of ω S nd ω A. k 0 k k 0 Figure 3: Anoher coupled oscillor syse. Ye noher coupled oscillor syse Figure 4 shows ye noher coupled oscillor syse. We hve hree cpciors insed of springs nd wo inducors insed of sses. The volge drop cross one of he inducors due o he ie re chnge of curren is equl o he difference of he volges cross he cpciors ched o he ends of he inducor. Thus we hve: L di d L di b d = Q Q = Q Q 3 4
5 I I b L L Q Q Q 3 Figure 4: Anoher coupled oscillor syse. Differenie hese equions wih respec o ie nd you ge: Ï = L I + L I b I Ï b = L I b L I b I Noe h if I = I b syeric ode hen he bove equions ell us h ω S = / L. The iddle cpcior never ges ny chrge you could reove i. opre his o reoving he iddle spring in eiher Figure or 3. The wo inducors in series give you L nd he wo cpciors in series give you / nd he equivlen L circui hs ω = ω S = / L. If I = I b nisyeric ode hen he bove equions ell us h ω S = 3/ L. This ode hs higher frequency. opre wih he cse of hree idenicl springs in Figure 3. Ye noher coupled oscillor syse Figure 5 shows ye noher coupled oscillor syse he subjec of he nex lecure. y x θ θ Figure 5: Anoher coupled oscillor syse. 5
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