Two coupled harmonic oscillators We will assume that when the masses are in their equilibrium position, the springs are also in their equilibrium positions The
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Physics 235 Chapter 12 - 1 - Chapter 12 Coupled Oscillations Many important physics systems involved coupled oscillators. Coupled oscillators are oscillators connected in such a way that energy can be transferred between them. The motion of coupled oscillators can be complex, and does not have to be periodic. However, when the oscillators carry out complex motion, we can find a coordinate frame in which each oscillator oscillates with a very well defined frequency. A solid is a good example of a system that can be described in terms of coupled oscillations. The atoms oscillate around their equilibrium positions, and the interaction between the atoms is responsible for the coupling. To start our study of coupled oscillations, we will assume that the forces involved are spring-like forces (the magnitude of the force is proportional to the magnitude of the displacement from equilibrium). Two Coupled Harmonic Oscillators Consider a system of two objects of mass M. The two objects are attached to two springs with spring constants κ (see Figure 1). The interaction force between the masses is represented by a third spring with spring constant κ12, which connects the two masses. Figure 1. Two coupled harmonic oscillators. We will assume that when the masses are in their equilibrium position, the springs are also in their equilibrium positions. The force on the left mass is equal to
F 1 =!"x 1 12 x 2 !x 1 12 x 1 12 x 2 =M x 1The force on the right mass is equal to
F 2 =!"x 2 12 x 1 !x 2 12 x 2 12 x 1 =M x 2The equations of motion are thus
Physics 235 Chapter 12 - 2 -
M x 1 12 x 1 12 x 2 =0 M x 2 12 x 2 12 x 1 =0Since it is reasonable to assume that the resulting motion has an oscillatory behavior, we consider following trial functions: x
1 t =B 1 e i!t x 2 t =B 2 e i!tSubstituting these trial functions into the equations of motion we obtain the following conditions: !+!
12 "M# 2 B 1 12 B 2 =0 12 B 1 12 !M# 2 B 2 =0 These equations only will have a non-trivial solution if !+! 12 "M# 2 12 12 12 "M# 2 =0Note: the trivial solution is B1 = B2 = 0. The requirement for a non-trivial solution requires that the angular frequency of the system is equal to one of the following two characteristic frequencies (the so called eigen frequencies): !
1 "+2" 12 M 2 MFor each of these frequencies, we can now determine the amplitudes B1 and B2. Let us first consider the eigen frequency ω1. For this frequency we obtain the following relations between B1 and B2: !+!
12 "!+2! 12 B 1 12 B 2 12 B 1 12 B 2 12 B 1 +B 2 =0 or B1 = -B2. For the eigen frequency ω2 we obtain the following relations between B1 and B2:Physics 235 Chapter 12 - 3 - !+!
12 B 1 12 B 2 12 B 1 12 B 2 12 B 1 "B 2 =0 or B1 = B2. The most general solution of the coupled harmonic oscillator problem is thus x 1 t =B 1 e +i! 1 t +B 1 e "i! 1 t +B 2 e +i! 2 t +B 2 e "i! 2 t x 2 t =!B 1 e +i" 1 t !B 1 e !i" 1 t +B 2 e +i" 2 t +B 2 e !i" 2 tAnother approach that can be used to solve the coupled harmonic oscillator problem is to carry out a coordinate transformation that decouples the coupled equations. Consider the two equations of motion. If we add them together we get
M x 1 x 2 +!x 1 +x 2 =0If we subtract from each other we get
M x 1 x 2 +"+2" 12 x 1 !x 2 =0Based on these two equations it is clear that in order to decouple the equations of motion we need to introduce the following variables !
1 =x 1 "x 2 2 =x 1 +x 2 The solutions to the decoupled equations of motion are ! 1 t =C 1 e i" 1 t +C 1 e #i" 1 t 2 t =C 2 e i" 2 t +C 2 e #i" 2 twhere the frequencies are the characteristic frequencies discussed before. Once we have these solutions we can determine the positions of the masses as function of time: x
1 t 1 2 2 t 1 t x 2 t 1 2 2 t 1 tPhysics 235 Chapter 12 - 4 - We note that the solution η1 corresponds to an asymmetric motion of the masses, while the solution η2 corresponds to an asymmetric motion of the masses (see Figure 2). Since higher frequencies correspond to higher energies, the asymmetric mode (out of phase) has a higher energy. Figure 2. Normal modes of oscillation. Weak Coupling Coupled oscillations, involving a weak coupling, are important to describe many physical systems. For example, in many solids, the force that tie the atoms to their equilibrium positions are very much stronger than the inter-atomic coupling forces. In the example we discussed in the pervious section, the weak coupling limit requires that κ12 << κ. In this approximation we can show (see text book for details) that our solutions have a high-frequency component that oscillates inside a slowly varying component (see Figure 3). The solutions are thus sinusoidal functions with a slowly varying amplitude. Figure 3. Examples of solution in the weak-coupling limit.
Physics 235 Chapter 12 - 5 - Example: Problem 12.1 Reconsider the problem of two coupled oscillators discussion in Section 12.2 in the event that the three springs all have different force constants. Find the two characteristic frequencies, and compare the magnitudes with the natural frequencies of the two oscillators in the absence of coupling. The equations of motion are
M x 1 1 12 x 1 12 x 2 =0 M x 2 2 12 x 2 12 x 1 =0 (1) We attempt a solution of the form x 1 t =B 1 e i!t x 2 t =B 2 e i!t (2) Substitution of (2) into (1) yields 1 12 "M# 2 B 1 12quotesdbs_dbs20.pdfusesText_26