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?
If I take a small spring pendulum and grab the mass with my hand and move it
up and down, it moves exactly how my hand tells it to move because the force
of my hand completely overshadows the restoring force and my hand dictates
the momentum of the system. For all practical purposes, the net force on
the mass is the force of my hand. In this scenario I could agree with
John's statement.
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.
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.
I personally have not seen any mathematical analysis of this system in which
the analysis agrees with the experiment. Has anyone else seen such an
analysis? One of the problems here is what I began talking about at the
beginning. The driver (constructed like part of a loudspeaker) is
influenced by the spring pendulum. The oscillator driving the driver stays
sinusoidal, but the motion of the driver mechanism probably does not follow
a perfect sinusoid because the momentum of the spring pendulum interacts
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.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817