Re: fallacious filtered white noise

[Andy Dougherty <doughera@LAFAYETTE.EDU>]

On Sat, 9 Dec 2000, Michael Edmiston wrote:

> At one point John Denker wrote "In the linear regime, when driving a
> resonator with an off-resonance signal, it _always_ responds at the
> frequency of the driving force, _never_ at its resonant frequency or any
> other frequency."
>
> John, doesn't this depend upon how strong the driving force is and how
> strongly it's coupled to the oscillating system?

All of this analysis is only valid *after transients have decayed*.
The transient behavior is complex and sometimes difficult to interpret
visually.  There's an analysis in A.P. French's Vibrations and Waves
book.

> However, if I take a very large spring pendulum with mass perhaps even more
> than my mass, I don't think I would be strong enough to make it oscillate at
> a frequency totally determined by me.  In this case it might be difficult to
> decide if I am driving it or it is driving me, except if we assume I am the
> energy source that starts the motion in the first place, then I would be
> inclined to say I am the driver.  But I am not going to be able to make that
> big mass match my frequency unless my frequency matches its natural
> frequency.

In this case, you are not "driving a resonator with an off-resonance
signal".  You are trying very hard to do so, of course, but as you
observe, it is difficult to drive such a system at a precise off-resonance
signal.

> There is an experiment I have students do that is simple to perform, but
> gives complicated results.  I hang a spring pendulum from an electromagnetic
> driver.  The driver is run by a sine-wave oscillator, and the students note
> the behavior of the spring pendulum when the driver frequency is near the
> natural frequency of the spring pendulum.  I use the Pasco-scientific
> SF-9324 driver with a spring pendulum having m = 10g, k = 0.515, T = 0.767
> s, f = 1.304 Hz.
>
> The "textbook" analysis of systems like this start with the differential
> equation x" + b/m x' + k/m x = Fsin(w(d)*t) where b is the damping constant,
> k is the spring constant, the natural angular-frequency is w(0) =
> sqrt(k/m-(b/2m)^2) and the driving angular-frequency is w(d).
>
> All textbooks I have say that when the driving frequency is near the natural
> frequency that the oscillation amplitude will grow to some value, then
> remain steady.  The maximum value of amplitude occurs if w(0) = w(d).
>
> This is not true... at least it is not what we observe experimentally.  What
> we observe is the system oscillates at w(0) with an amplitude that grows and
> falls as the system and driver slowly switch from being in-phase to being
> out-of-phase.  As w(d) approaches w(0) the amplitude oscillations slow down,
> and the system can actually achieve a steady amplitude only when w(0) and
> w(d) match.

Yes.  It looks very much like a beating phenomenon.  This system
has a weak damping, so you are observing a very long transient.  French's
analysis covers this case.

> with it.  So I presume the differential equation is incorrect.  But I wonder
> if it would be possible to design a system for which this differential
> equation would be correct.

It is actually quite difficult in a mechanical system, as you have
correctly observed.

--
    Andy Dougherty              doughera@lafayette.edu
    Dept. of Physics
    Lafayette College, Easton PA 18042