COUPLED OSCILLATORS. Two identical pendulums

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Emil Robertson
5 years ago
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1 COUPED OSCIATORS A real physical object can be rearded as a lare nuber of siple oscillators coupled toether (atos and olecules in solids. The question is: how does the couplin affect the behavior of each of the individual oscillators? Two identical pendulus We have two identical pendulus (lenth for which we consider sall oscillations. In order to find what is the siplest otion, we iaine two eperients: If we draw the two asses aside soe distance and release the siultaneously fro rest, they will swin in identical phase with no relative chane in position. The sprin will reain unstretched (or uncopressed and will eert no force on either ass. q q We call this vibration pattern the first ode of vibration of the syste. The other obvious way of startin a syetric oscillation will be to stretch the sprin fro both ends. If we release the asses fro rest siultaneously, we ay notice that: a The sprin now eerts forces durin otion b Fro syetry of otions of A and B, their positions are irror iaes of each other We call this vibration pattern the second ode of vibration of the syste. Note: each pendulu in the one of the odes above oscillates with the sae frequency: the noral oscillation frequency. The two oscillatin patterns are called noral odes. Both are SHM of constant anular frequency and aplitude.

2 General otion as superposition of noral odes We take two coupled pendulus, identical, each startin fro rest. Any otion of the syste, showin no special syetry ay be described as a cobination of the two noral odes of oscillation. q q T T F e F e sinq sinq We assue sall displaceents fro equilibriu:,. Each pendulu swins because of the cobined force of ravity and the strin tension T. The cobined force is: sin q = for ass, but force alon the sprin eerted by ass is F ~ q cosq ~ q ~ / sin q» q = for ass, but force alon the sprin eerted by ass is F ~ q cosq ~ q ~ / Note Displaceents fro equilibriu are very sall, anles q and q are very sall: sinq, ~ q, ; cosq, ~. Displaceents are iven by: θ, θ If we consider, the sprin is stretched by - and the elastic restorin force in the sprin will be F e = k( - The total restorin force on ass is -[ / - k( - ] The total restorin force on ass is: -[ / + k( - ] Equations of otion can be written for each of the asses by usin Newton's second law:

3 d d = - = - + k( - k( - - ( ( The Syetry Method to solve the syste of second order differential equations The two equations are syetric. We add and subtract the: d d ( ( + - = - ( = - Ł + + k ( ł - (3 We obtained equations that look like the SHM. We have obtained independent oscillations in ( + and ( -. We solve for ( +, ( - : + = A cosw p t - = B cosw s t where ω p = and k ω s = + (4 ω p and ω s are called the noral frequencies.

4 In writin the solutions, we need to apply the initial conditions. For this we have to distinuish between the two oscillatin patterns: ( parallel oscillation: q q and ( syetric oscillation: Parallel oscillation: et B = 0, - = 0 The two pendulus are ovin in parallel. The sprin does nothin (as if it didn't eist. They both oscillate with ω p = the natural frequency of free pendulu without couplin. This is the first (lower noral ode of oscillation. Syetric Oscillation: et A = 0, + = 0 The sprin ets epanded/shrunk by twice the oveent of each pendulu. k ω s = + is deterined by both the pendulu and sprin. Each pendulu oscillates with frequency ω s but they are out of phase by π. This is the second (hiher noral ode.

5 General solution of the syste of differential equations is a linear cobination of noral odes: A B (t = cost + cosωst A B (t = cost - cosωst (5 We need a solution that satisfies an initial condition. et's use an eaple: a Initial conditions for the two asses are: (0 = a d & (0 = t = 0 = 0 and & (0 = 0 (0 = 0 A B a = + A B 0 = A = a A = B A = B = a which results if the followin solution: a (t = (cosω pt + cosωst a (t = (cost - cosωst The two noral odes are: + = a cosω t - = a cosω p s t first (lower ode second (hiher ode We define by q = ( + and q = ( - the noral coordinates of the syste

6 If we use: cos α + β α β cos α + cos β = cos cos α + β α β α cos β = sin sin + ωs - ωs (t = a cos t cos t Ł ł Ł ł + ωs - ωs (t = -a sin t sin t Ł ł Ł ł are the two individual oscillations If ω p ωs << + ωs, the sprin constant is very sall and the couplin between the two pendulus is very weak. ω p = ; k ω s = + are the two noral frequencies. We notice that in each noral ode, the individual oscillators oscillates with the sae noral frequency Observation. Up to now, we have studied only coupled oscillations of the sae anular frequency. If the two frequencies are different, we obtain beats: odulations of aplitude produced by two oscillations of slihtly different frequencies.

Vector Spaces in Physics 8/6/2015. Chapter 4. Practical Examples.

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