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Re: A Question About A Simple LRC circuit

I think an appropriate interjection. The Analog computer experiment -- since
dropped. Originally the students were required to do three required expts.
and one of five elective. The elective was dropped and replaced by computer
training in '94.

The intro. and expt. 2. A are relevant.

bc



Experiment I

THE ANALOG COMPUTER EXPERIMENT
Chaos in Nonlinear Differential Equations
INTRODUCTION:
Physicists (and others) have the habit of modeling the unfamiliar with the
familiar. For example, in the previous century electrical circuits were
described in terms of mechanical systems (e.g. a resistor is a dash pot and an
inductor a flywheel). Now you will do the reverse in this experiment. That
this can be done is because such systems are analogous. That is, the
equations that describe their behavior have the same form. We have a great
advantage over our predecessors in that electrical systems are much more
easily "wired up,” and we have sophisticated recording devices.
ADDITIONAL REFERENCES:
1. Ashley, J. A., Introduction to Analog Computation, (Wiley, 1963). This
and the next
instruct analysis of operational amplifier behavior.
2. Jung, Walter G., IC op-Amp Cookbook, (Howard W. Sams, 1974).
3. D. Farmer, J. Crutchfield, H. Froehling, N. Packard, and R. Shaw, Power
Spectra and Mixing Properties of Strange Attractors, Ann. NY. Acad. Sci. 357
453. This paper, written by the founding members of UCSC's dynamical systems
collective, contains a clear description of the Roessler and Lorentz
attractors.
EQUIPMENT:
Besides the previously used Systron-Donner computers, the lab has three modern
(1987) IC based computers, the Comdyna GP-6. Peripherals available include:
the HP 7015B X-Y recorder with remote pen lift (BNC jack underneath), function
generators, Kikusui display oscilloscope, Tektronix model 611 storage
oscilloscopes, attenuator for o’scopes, and the "Collective's" Pulse Box. If
a third group wishes to experiment at the same time, obtain an old manual,
which includes instructions for the S-D computers.
EXPERIMENT:
1. First you must understand the behavior of operational amplifiers
(op-amps).
I-2

A. Wire up a ramp (saw tooth) generator. The simplest starts from zero,
voltage and increases linearly with time. Make one that starts from some
negative voltage, so that when connected to an oscilloscope, the trace will
travel from the left edge of the screen to the right before retracing (using
repeat operation).
B. Model the equation and display on the screen. Make the trace fill
the screen.
2. Do two of the following behavioral investigations:
A. Simple, damped driven and undriven, harmonic oscillators. Record both
position as a function of time and velocity as a function of position (phase
plot). Do not forget to include all the regions of damping behavior and the
mathematical analysis.
B. Investigate the behavior of the driven Duffing or Van der Pol
Oscillator. Note: These exhibit chaotic behavior within a range of driving
parameters and constants. You must find such a set and record the outputs.
Use a function generator (Wavetek) to supply the “driving force” by patching
its output directly to the input resistor of the relevant op-amp.
C. Lorentz, Roessler, or Lotka-Voltera equations. Include plots of limit
cycle and period doubling
D. The lab has an MS-DOS computer interface. The Comdyna manual includes
instructions for this hybrid use. If interested see me.
Warning and Hints
1. Please read and observe the operating instructions for the o'scopes. Do
not damage the kinescope screens.
2. You will note that the storage scopes (Tek. 611) require >1 V. into the
Z - axis to turn on the beam. Use the Pulse Box to supply this. It can also
supply pulses synchronized with the driver to produce Poincaré Sections. It
requires a ramp (saw tooth) input to vary the position (phase) of the
section. Therefore, one must either use a sawtooth to drive the computer or
synchronize two generators. The length and phase of the pulses may be set
using the relevant controls on the front of the Pulse Box. The inverted
output is suitable for the plotter's remote pen lift

3. According to Karl Young the Van der Pol oscillator is unsuitable for P.
sections, but the Duffing yields a “pretty picture of the Veda Attractor.”

bc xii ‘92


"John S. Denker" wrote:

At 05:29 AM 3/6/01 -0500, Robert B Zannelli asked :

what is the mechanical analog
to an electrical circuit. I would think just from intuition that a spring
would serve in a similar manner to a capacitor while mass inertia would be
the equivalent to inductance.

Actually... You can do it either way!

** Scheme 1 ** ** Scheme 2 **
mass ~ inductor mass ~ capacitor
spring ~ capacitor spring ~ inductor
position ~ charge position ~ flux
momentum ~ flux momentum ~ charge
dashpot ~ series resistance dashpot ~ parallel conductance

Note that I said flux and charge, not current and voltage. This was not an
accident. Flux and charge are dynamically conjugate, like position and
momentum. They obey the Heisenberg equation
delta Q delta Phi > hbar/2
whereas voltage and current don't even have the right units for this.

Resistance would correlate to the dissipation
of energy by friction.

Yeah, but be careful, it's 1/resistance in scheme 2.

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

The fainthearted should stop reading now......

a) It turns out that you can do the same thing in plain old mechanics! You
can make exact correspondences that turn the momentum variable into the
position variable and vice versa.

b) In fact there are all sorts of weird things you can choose as your
"coordinate". Once you know the coordinate, the Lagrangian will tell you
what the dynamically-conjugate momentum is.