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Re: resonator demo

At 10:58 AM 12/9/00 -0500, 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?

I stand by what I wrote. The examples below are cases where either
a) the system has departed the linear regime, or
b) the driven system has perturbed the drive frequency, in which case my
statement applies to the _actual_ drive frequency, not the intended drive
frequency.

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.

OK.

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.

1) For present purposes it doesn't really matter who is driving whom, but
if that is a topic of interest, see the "cause and effect" thread last
month, e.g. 06:36 PM 10/27/00 -0400.

2) In the present thread, at 01:20 AM 12/8/00 -0500, I wrote:
The canonical example is a damped harmonic oscillator:
.
. ------------|
. handle---spring---mass---dashpot |--- anchor
. ------------|
.

Suppose the mass is rather massive, so that the resonant frequency is
rather low. You can drive this thing non-resonantly. In particular, if
you drive it at some frequency even lower than its (admittedly low)
resonant frequency, you will get a gain of one.

Even if you drive it above its resonant frequency, you can get some action.

Example: When I was seven or eight, we visited the Old Tucson movie
studio. They had an old-time railroad car sitting there. For a joke, my
kid brother and I posed as if we were pushing the railroad car. Somebody
said "that's ridiculous" and we took it as a challenge. So we spent the
next several minutes pushing really hard. It turns out that railroad cars
have really good bearings: damping = zero, restoring force = zero,
resonant frequency = zero. If you push for a long time you can dump a lot
of momentum into one....

Then the question arises, how are you going to STOP the thing?

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.

I don't have access to the specs on that driver, but it is easy to
conjecture that it does not generate a pure-and-simple force. In
particular, if it is operated unloaded (i.e. zero-gram load) does it
exhibit infinite acceleration? I suspect not. I suspect it has some mass
and some spring constant of its own. I further suspect that it has some
parasitic inductance, resistance, and capacitance. Therefore:
a) Considering the driver by itself: the input voltage does not nicely
determine the output force.
b) Considering the driver-plus-pendulum system: This is not an ideal
driven damped oscillator system. It is (at least) two masses and two springs.
c) I will not even discuss the nonlinearities that will occur if the
driver is allowed to "hit the stops" because I think M.E. is smart enough
to make sure this isn't happening.

Analyzing this mass-spring-mass-spring-L-R-C system is certainly possible,
but probably not appropriate for the class in question.

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

Possibly constructive suggestion:

Good force-drivers are hard to come by. Position-drivers are easier. The
analysis is the same (up to an obvious factor of k on the RHS). The last
time I wanted to do this I used a crank attached to a sewing-machine motor
(i.e. a variable-speed motor with a huge gear reduction ratio). There was
No Way any ten-gram load was going to significantly perturb the motion of
that crank.