Instabilità modulazionale en
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| AI-translated from Italian. Original page: Instabilità_modulazionale |
On this page we examine, as an example, just one particular effect caused by the presence of a non-linear force in a chain of coupled oscillators. It is particularly representative of a class of phenomena called "modulational instability", and is at the same time
- a paradigmatic example
- simple to build and study in the laboratory
- simple to simulate numerically
Non-linear effects in the chain of coupled oscillators
Watch carefully the chain of pendulums coupled by the spring on the support. In the two cases the system is exactly the same; only the initial amplitude of the motion changes
| small initial amplitude | large initial amplitude |
|---|---|
| system in the linear regime | system in the non-linear regime |
Remarks
- The initial condition forces all the pendulums to oscillate in unison at first in both cases. In the first two periods the coupling associated with the spring on the pivot is negligible. However, the non-linear effect of the force of gravity is already visible: the period of the oscillations is slightly longer at larger amplitudes (Galileo's law of isochronism is violated in this case)
- After a few oscillations the effect of the coupling becomes evident: the pendulums also feel the coupling force, and no longer behave as if they were independent: the normal modes of the chain begin to emerge through resonance.
- The question, then, is: where does the energy go? That is, which modes of the chain will be reinforced and which damped?
- Watching the two videos, we can see that the answer to this question depends closely on the importance of the non-linear terms.
Where the energy goes
- In the linear regime (video on the left) the oscillation stays uniform: all the pendulums keep oscillating together, with an amplitude that decreases slowly because of friction. The energy stays in the normal mode into which it was put at the start, the one in which all the pendulums oscillate in phase.
- In the non-linear regime (video on the right) the uniform oscillation breaks up after a few periods: some pendulums almost stop, while a few others, close to each other, keep oscillating with a large amplitude (one even goes past the horizontal). The energy, initially spread uniformly along the chain, spontaneously concentrates in a small region.
- This is the modulational instability: in the non-linear regime the small differences in amplitude between one pendulum and another, unavoidable in reality, are not damped but grow, and the uniform oscillation turns into an oscillation whose amplitude varies ("is modulated") along the chain.
- Why does the energy then stay concentrated in a few pendulums? If the coupling depends only on the difference between the displacements of neighbouring pendulums (as for a spring or a wire connecting them), in the uniform oscillation the coupling is not deformed: this oscillation has the natural frequency of the single pendulum, while in all the other linear normal modes the coupling is deformed and adds a restoring force, which raises their frequency. The pendulums oscillating with a large amplitude, instead, have a longer period, that is a lower frequency, which falls below those of all the modes of the chain: they therefore cannot pass their energy to their neighbours by resonance in the form of waves, and they keep it. These localized oscillations are called discrete breathers[1].
Check your understanding
Problem. The frequency of the oscillations can be measured in the two videos: in the linear regime the pendulums oscillate at about 1.51 Hz, in the first seconds of the non-linear regime (while they still all oscillate together) at about 1.41 Hz. How much longer is the period in the non-linear regime? Using the formula for the period of the pendulum at large amplitudes (page non-linear effects), check that the pendulums were released with an amplitude of about 60°.
Solution. The ratio of the periods is 1.51/1.41 ≈ 1.07: the period is about 7% longer. With = 60° we have sin 30° = 0.5, and the first terms of the series give 1 + 1/4 · 0.25 + 9/64 · 0.0625 ≈ 1.07, in agreement with the measurement.Problem. In the non-linear video, after the uniform oscillation has broken up, the few pendulums that keep oscillating with a large amplitude have a frequency of about 1.40-1.45 Hz, while the others oscillate at about 1.6-1.7 Hz. Remembering that the natural frequency of the single pendulum is about 1.51 Hz, which of the two groups oscillates according to the linear normal modes of the chain? Why can the other group not give away its energy?
Solution. The pendulums at 1.6-1.7 Hz: the linear normal modes of the chain have frequencies higher than that of the single pendulum. The large-amplitude pendulums instead oscillate below all the frequencies of the linear modes: they cannot excite them by resonance, so they keep their energy (they are the discrete breathers).Problem. In the non-linear video the pendulums, all alike, start together with the same amplitude. If the chain were perfectly uniform and the initial condition perfectly identical for all of them, would the oscillation stay uniform? Why does this not happen in reality?
Solution. In principle yes: the uniform oscillation is an exact solution of the equations of motion in the non-linear regime too. In the non-linear regime, however, this solution is unstable: any tiny difference (in the lengths of the pendulums, in friction, in the initial push) grows in time instead of fading. In reality these small differences always exist, and the uniform oscillation breaks up after a few periods.Notes
- ↑ S. Flach, A. V. Gorbach, "Discrete breathers — Advances in theory and applications", Physics Reports 467, 1-116 (2008), doi:10.1016/j.physrep.2008.05.002
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