- #76

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You can ignore damping in the case of one driven oscillator over a short window, because negligible energy is lost in that short window (by the definition of "short"), but I see your point about the beat frequency. On certain points, I've been confusing the two situations myself. Your point about the beat frequency is having me scratch my head. I mean, you could say that there is no steady state for a single undamped oscillator, its coordinate is cyclic, changing in time. Calling two undamped coupled oscillators a steady state is ok with me, even though there is a beat frequency, its just another frequency that does not die out.I think I am beginning to sort this out in my mind. There are two scenarios involved in this issue and some of my problem has been to confuse the two. There is what happens with a single oscillator, excited from an energy source and there is what happens with isolated coupled oscillators. I can't see how, in either case, it is useful to ignore damping but, in the case of coupled oscillators, it may be OK because there is only a finite amount of energy in the system at the start but there is still energy flow in and out of the two oscillators so each one is either being damped or driven.

For coupled oscillators, one way of looking at it is that energy flows back and forth between the two at the beat frequency. The direction of energy flow is determined by the phase relationship between the two oscillators, which is constantly changing. The notion that you have any 'steady state' condition must be flawed because the whole process involves nothing staying the same - energy is either flowing one way or the other - even if you eliminate the damping. The situation at time t is the result of the past. I can't accept your statement about 10 cycles 'usefully' representing a small enough window to treat the system as undamped because energy is constantly being lost to or gained from the other oscillator and this is a vital part of the co- resonance phenomenon. (This is analogous to treating the Volts on a charging capacitor as being constant, if you take a short enough observation time. Slope is slope, however short an interval its' measured over.)

In the case of a driven oscillator, the phase difference of driver and oscillator at the driving frequency settles down to a constant value and energy continually flows into the oscillator (hence my obsession with the requirement for damping). In this case, whatever happens for the first few cycles of excitation, I don't see how there can be any power flowing into the ω0 mode because the phases are constantly changing wrt the drive and must surely integrate to zero.

I think 10 cycles for a damping period of 1000 cycles is ok. You can represent the behavior of the oscillator very well by assuming it is undamped. You can represent it very well by a period of 1/10 of a cycle too. The fact that you haven't covered a number of cycles is not an issue. The fact that for 10 cycles you may not have covered a number of "beat cycles" is not an issue.

There's three time periods here, the average period of the two oscillators (a good number if they are separated by a frequency difference small compared to their frequencies), the beat period (corresponding to the difference in the two frequencies) and the damping period, or damping time. I think if the damping time is much longer than the average period, then the system can be well represented by two coupled, undamped oscillators, in a time window small compared to the damping period, no matter what the beat period. If the beat period is much longer than the window, you don't see much of it, if it is short, you see a lot of it. Slope is slope, but if you take a short enough window, the percent change in the function itself is negligible, and by assuming zero slope, your percent error will be low.

You last question I can't visualize, and my notes don't give me any help, so I will fire up Mathematica and try to solve two coupled undamped oscillators.