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Re: [Phys-l] Queston: simple pendulum lengthen suspension.



On 03/17/2007 05:41 PM, Bernard Cleyet wrote:
What happens to the amplitude if one suddenly lengthens the suspension at equilibrium position or at max. displacement?

That question is just part of a larger topic. Let me address
the whole topic, since it is just as easy. It illustrates the
grandeur and unity of physics: A playground swingset is related
to the foundations of quantum mechanics and to electronic
measuring instruments.



The length of a pendulum is considered a /parameter/. The same
analysis works for other oscillators (including electrical LC
oscillators).

By modulating ("pumping") the length (or some similar parameter)
you can make a /parametric amplifier/.

You don't really want to do it "suddenly" because that would
break something. It is conventional and sensible to simplify
the problem by considering pump waveforms that are not too
sudden and not too much of a perturbation (percentage-wise).
For concreteness, let the length be
L = L_0 + A_p sin(omega_p t)
where the pump amplitude A_p is small compared to the average
length L_0.

There are various types of parametric amplifiers. The type that
is closest to the spirit of the original question is the /degenerate/
parametric amplifier, or DPA for short. (It is called degenerate
because the signal frequency and the idler frequency are the same,
but if that doesn't mean anything to you, don't worry about it.)

The DPA is the simplest type of parametric amplifier to visualize,
although not necessarily the simplest to use or analyze. In normal
operation the thing is operated on-resonance, i.e. the signal
frequency is at or near the natural resonant frequency.

The pump frequency should be at or near *twice* the natural resonant
frequency. To make a normal amplifier, we shorten the pendulum at
the middle of each half-cycle, doing work against centrifugal force.
At the end of each half-cycle, we lengthen the pendulum. The basic
idea is shown in the figure:
http://www.av8n.com/physics/img48/dpa.png

If the pump amplitude A_p is constant, independent of signal
amplitude, then the /effect/ the pumping has on the signal is
proportional to how much signal was already there. (This is easy
to understand in terms of work done against centrifugal force.)
As a consequence, the effect of the pump can be modeled (to first
order) as a /negative resistance/ and in cases where the negative
resistance dominates other resistances, the signal will grow
exponentially, diverging away from equilibrium. (This is the
mirror-image of the usual damping scenario in which a system
decays exponentially toward equilibrium.)

We are using a pump frequency that is twice the resonant frequency;
we are shortening the pendulum at the middle of each /half/ cycle.
This is definitely different from the usual method that kids
use for pumping a swing, wiggling their legs at 1.0 times the
resonant frequency.

It is straightforward to build a tabletop model of a DPA, using
a low-speed motor with a crank or cam, plus a bob on a string.
(Keep the signal amplitude small or all sorts of nonidealities
will show up. Anharmonicity et cetera.)

It is also possible to demonstrate it using a playground swingset,
by standing on the seat and using your leg muscles to raise and
lower your center of mass appropriately ... but you have to be
careful, because it works too well: it is quite possible to
put so much energy into the system that you exceed the ±90 degree
limits or otherwise get into trouble. A swingset with long ropes
(10m or preferably 15m or more) will permit an effective demo
with relatively less risk.

The foregoing considered the "normal" amplifier case where the
signal of interest was in phase with the pump. Now let's switch
to considering the case where the signal is 90 degrees out of
phase (relative to "normal"). In this case, we are lengthening
the pendulum at the middle of each half-cycle, doing negative
work against centrifugal force. That means that the effect of
the pump can be modeled as a /positive/ resistance.

In the general case, any signal can be decomposed into an in-phase
component and a out-of-phase ("quadrature") component. These are
the /phasor components/. The ideal DPA will amplify the in-phase
component, multiplying it by a factor of G ... and will by the
same logic deamplify the quadrature component, multiplying it by
a factor of 1/G (for some value of G that depends on details).

That is exceedingly interesting and useful, because it means the
DPA preserves area in phase space, in the subspace spanned by
the signal. That is, it performs a Bogoliubov transformation.
http://en.wikipedia.org/wiki/Bogoliubov_transformation

Recall that overall area in phase space *must* be conserved,
so an amplifier (such as a 741) that purports to amplify both
phasor components of a signal must be doing something else
besides ... typically irreversibly mixing in noise from some
other modes. This added noise is required by Liouville's
theorem, and by the second law of thermodynamics, and by the
Heisenberg uncertainty principle (which are three ways of
saying the same thing, if you think about it the right way).
This places a fundamental limit on how accurately you can
measure anything with a non-phase-sensitive instrument: the
so-called SQL (standard quantum limit).

The wonderful thing about the DPA is that (to the extent that
we can neglect parasitic losses) it is a reversible, isentropic
amplifier. One DPA can undo the effect of another, so we know
it isn't adding noise or otherwise degrading the signal.

We can even use a DPA to create "squeezed states" i.e. quantum
states where one phasor component has less noise than the
ground state. That is: colder than cold, blacker than black
(in one component).

Let's be clear: by lengthening a pendulum in just the
right way, you can deamplify one component of the signal
/including its zero-point fluctuations/.

Squeezed states can be used to create a wide class of quantum
nondemolition measuring devices, which can measure (one
phasor component of) a signal very accurately, far beyond
the standard quantum limit. For details on all this, see
http://prola.aps.org/abstract/PRA/v29/i3/p1419_1

Quantum nondemolition measuring devices have been built. It
is impressively spooky to insert a zero-temperature black body
into the beam and see the noise level go /up/.