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Re: Midterm Question - Sort of



On Tue, 6 Jul 1999, Doug Craigen wrote:

The thing I've been pondering is that as compared to other "sympathetic
vibration" / "resonance" / ... type examples I can think of, in this
case (or the comparable spring question) the system responds at its own
frequency which is a harmonic of the frequency at which you pump it....
So it *appears* to be multiplication of a frequency. (i.e. if an
optical analog existed it would amount to something such as putting 800
nm in and getting 400 nm out.)

Hey, this same problem confused me in high school! In terms of
coil/capacitor oscillators, how can we pump at some other frequency? The
LC oscillator only has one resonant frequency, so how can 0.5F or 2F have
any effect?

There are two answers. One: the swing-pumping takes the form of a brief
impulse, not a smooth sine wave. Two: the pusher-person only touches the
swing for a portion of the cycle, therefor swing-pumping is nonlinear.

Suppose we use a constant-K spring to connect the swinging person to the
pusher-person, and suppose we pump the swing with a pure sine wave. In
that case the "pusher" would both push and pull on the swing, and so the
swing would "ignore" any pump frequencies which were higher than 1F (it
might respond slightly at the pump frequency, but this would not affect
its 1F amplitude.) (This assumes that the swing itself is a linear
oscillator; that frequency is independant of amplitude.)

If we pump by using a brief impulse, then the Fourier spectrum of the
pumping waveform contains upper harmonics. If we let go of the swing
after pushing, then this acts to "chop up" the pumping wave and create
more upper-harmonic energy.

In optics there ARE such things as frequency doublers. I remember that
Edmund Scientific once sold blocks of, I think, Ammonium Phosphate for
this application.



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