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Unformatted text preview: Nonlinear Oscillators at Our Fingertips Tanya Leise and Andrew Cohen 1. AN EXPERIMENT WITH COUPLED OSCILLATORS. We begin with an experiment that requires minimal equipment: you just need your left and right fore- fingers. To get warmed up, first point your right forefinger in the natural way (we refer to this position as the “equilibrium position”). Next move your right forefinger back and forth in a motion that could be graphed as shown in Figure 1, where the displace- ment of the fingertip equals the distance from the equilibrium position. 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2 1 1 2 Time in seconds Displacement in inches Figure 1. Oscillation x ( t ) = 2 cos ( 2 π t − π/ 2 ) of a single finger with a frequency ω = 2 π (yielding one cycle per second), amplitude r = 2, and phase angle φ = − π/ 2. Continue oscillating your right forefinger and begin oscillating your left forefinger (it may be helpful to point your fingers at each other and anchor your wrists; only the fingers should move, not your hands). Unless you are a drummer, your fingers will tend to naturally fall into one of two relative motions. The motion that is generally the most comfortable is to move the fingers in unison, as though one were a mirror image of the other. We refer to this as “in-phase” oscillation, and the graphs of the left and right finger motions are identical. The other natural motion is to move the fingers opposite each other, so that they pass each other at the equilibrium point going in opposite directions. We refer to this as “antiphase” oscillation, and the graphs of the left and right finger motions resemble those in Figure 2. 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2 1 1 2 Time in seconds Displacement in inches Figure 2. Antiphase oscillation of the two fingers at a frequency of one cycle per second, where the solid line represents the right finger x R ( t ) = 2 cos ( 2 π t − π/ 2 ) and the dotted line the left finger x L ( t ) = 2 cos ( 2 π t + π/ 2 ) . (The phase difference is the difference between the phase angles: 4 φ = π/ 2 − ( − π/ 2 ) = π .) 14 c ° THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 114 Now we’re ready for the interesting part of this experiment (conceived by Kelso [ 11 ]). Start oscillating your fingers antiphase at a comfortable pace. Gradually speed up the motion, increasing the frequency until something interesting happens. You should experience a sudden “phase transition” from antiphase to in-phase motion. This tran- sition tends to be abrupt; past a certain frequency, most people cannot oscillate their fingers opposite each other, although they can still oscillate them in unison. The graph of this experiment might look something like the caricature in Figure 3. A similar and even more intriguing experiment can be done with two people. Have each person swing one of their legs antiphase with the other person’s leg (each person watches the other person’s leg in order to coordinate the leg motions). Increase the frequency ofother person’s leg in order to coordinate the leg motions)....
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- Spring '11
- limit cycle, frequency increases, nonlinear oscillators