Re: unsympathetic resonances, complex feedback, etc.

[John Denker <jsd@MONMOUTH.COM>]

At 10:07 PM 12/7/00 -0500, Chuck Britton wrote:

>        The 'distinction' that I am referring to is the position held
> by some that 'resonance' requires 'sympathetic driving' of the
> system.

That would be a weird distinction.  Sympathetic driving usually means
transferring energy from one high-Q resonator to another.  But presumably
each of the resonators could exist separately, and could exhibit resonance.

In particular a single LC resonator will resonate quite nicely if you whack
it with a delta function;  no sympathetic anything is required.


========================

Having said what a resonance isn't, perhaps I should be clear about what a
resonance is.  The word traces its origins to acoustics.  It brings to mind
a wave _resounding_ in an underfurnished room, or in an organ pipe.  From
that ancient special usage we derive the current general usage, namely a
peak in the graph of some transfer function versus frequency.  (If there
are multiple peaks we speak of multiple resonances.)

The canonical example is a damped harmonic oscillator:
.
.                              ------------|
.      handle---spring---mass---dashpot    |--- anchor
.                              ------------|
.
where we imagine shaking the handle.  The independent variable X is the
position of the handle, and the dependent variable Y is the position of the
mass.  The gain is G = Y/X.

In this simple case, |G(f)| equals unity at low frequencies (f), has a peak
at the resonant frequency (assuming it's not overdamped), and falls off at
6dB per octave at high frequencies.

BTW this is called a Lorentzian lineshape.


>  Hence the blade of grass between your thumbs and the Tacoma
> Narrows Bridge would not qualify even thought the requisite positive
> feedback is present.

This is getting even weirder.  There is nothing resembling positive
feedback in typical resonators.  Cases in point include LC resonators and
acoustic resonators.

I wouldn't ordinarily recommend it, but it is possible to use feedback
techniques to build systems that exhibit resonance.  In general, the
feedback (i.e. the internal "loop gain" U) can't be classified as positive
or negative.  Actually it is a complex number, or rather a complex function
of frequency.  It swings around in the complex plane, passes abeam the
positive real number 1.0 at resonance, swings around some more, and does
various other things.  Meanwhile G = Y/X is doing other complex things.

The squeal of a badly-engineered mike/speaker system in an auditorium is an
example of this sort of complex feedback.

This topic is covered in upper-division electrical engineering courses such
as circuit analysis and filter synthesis.  This is waaaay over the head of
your typical high-school physics student.

=============

Also:  As we have discussed before, real-world resonators don't have
infinite Q, and it is a bad idea to assume that a resonator will always be
driven at its resonant frequency.