PHYSICAL CHARACTERISTICS OF COUPLED OSCILLATORS

We can see the main physical characteristics of coupled oscillators by observing the motion of two simple pendulums that are coupled together. They can be coupled by attaching their points of suspension to a supporting string as shown in Figure 1.1.
This is a simple experiment that is well worth doing. Both pendulums have the same length l and so their periods of oscillation are equal. The supporting string provides the coupling between the two pendulums. As each pendulum oscillates it pulls on the supporting string and causes the point of suspension of the other

Figure 1.1 Two simple pendulums of length l coupled together by a supporting string. The displacements of the two pendulum masses are considered in the direction perpendicular to the plane of the page.

pendulum to be driven back and forth. The motion of each pendulum affects the other and so their motions cannot be considered in isolation.We consider the motion of the two pendulums in the direction at right angles to the plane of the page. (i) First we displace both pendulum masses by the same amount and in the same direction. When released we observe that the two masses move back and forth in the same directions as each other with the same frequency and the same amplitude. (In this example and for the rest of this chapter we will assume that damping forces can be neglected.) (ii) Next we displace the two masses by the same amount but now in opposite directions. When released the two masses move back and forth in opposite directions. Again they both oscillate with the same frequency as each other but at a frequency that is slightly different from when they move in the same directions. These two distinctly different ways of oscillation are the normal modes of the system. We observe that once the system is put into one or other of these normal modes it stays in that mode and does not evolve into the other. (iii) Now we displace just one mass leaving the other at its equilibrium position. When released the displaced mass moves back and forth but it does so with a steadily decreasing amplitude. At the same time the mass that was initially at rest starts to oscillate and gradually the amplitude of its oscillation increases. Eventually the first mass momentarily stops oscillating having transferred all of its energy to the second mass that now oscillates with the amplitude initially given to the first mass. This process then repeats with the amplitude of the second mass steadily decreasing and that of the first steadily increasing. The cycle continues with the energy repeatedly being transferred between the two masses. This behaviour seems to be strange at first sight and indeed is sometimes used by conjurors to mystify their audience; they might use coconuts as the pendulum masses! However, there is nothing mysterious about the observations. What we are observing is the superposition of the two normal modes described above, as we shall see.