General:
The chaotic pendulum is best viewed with Netscape. The animation that gives you a live update uses a "server push" technique. This works well with Netscape, but not with Internet Explorer. You may have to reload the page to see any changes if you are not using netscape. A snap shot of a Netscape screen is shown below. The velocity, position and Drive data is acquired every few seconds. Data points are acquired with a time interval between points, such that three drive cycles can be viewed. At low frequencies this can take more than 10 seconds. The 1024 point data array is then graphed and the process is repeated after a preset waiting time. The velocity and position values are in arbitrary units. A position of zero corresponds to the pendulum bob hanging straight down. The pendulum bob can go over the top or even wind around several times before it reverses its direction. A change in the sign of the velocity does indicate a reversal of direction, and zero does correspond to the pendulum being at rest. The magnitude of the Drive data is arbitrary and scaled to fit on the graph. The Drive data is shown as a time reference so that you can observe features in the velocity and position data. This is very helpful when looking at a limit cycle or at a sub harmonic response.
Controls:
The user can interact with the pendulum through several controls on the front panel. The controls are described below.
"Phase, Poincare', Power"
This control allows the user to select the type of graph that is displayed. To select the graph type, click on the type of graph desired. Do not try to slide the control. The Poincare' plot scales the graph according to the information generated by the phase space plot. Therefore, it is necessary to first select the phase space plot before you select the Poincare' plot.
The Phase space plot is a graph of the velocity as a function of position. For a periodic system, this is a closed curve and will be elliptical. In a chaotic state, the curve will not be closed and subsequent cycles will not overlap.
A Poincare' section is also a plot of velocity as a function
of time, but only one pair of velocity-position coordinates is used.
The phase space is "strobed" so that the velocity-position values are determined
at a fixed time after the first positive slope of the drive
cycle. When the Poincare section is selected, a thin orange cursor
( vertical line) is shown on the graph of the square wave drive. The
cursor indicates the velocity-position pair that will be plotted (see
"Phase Angle"). When the system is periodic, velocity
at a particular position should simply repeat itself (within experimental
uncertainty). In this case a black dot is observed. In the case
of a limit cycle several dots may paper, indicating the frequency of the
limit cycle. In a chaotic system the velocity-position data points
will not always overlap and a more complex shape is formed. In a random
system, all velocity-position pairs are possible. If you were to wait
long enough, the entire screen would be filled with black dots. In
a chaotic system, however, there are some velocity-position pairs which never
occur, and blank areas are seen on the graph. One can not predict the
order in which the plots are pointed, but can clearly observe which pairs
are not possible. The characteristic shape of this graph can be changed
by altering the timing of the phase space strobe as describe in the
"Phase Angle" section. A Poincare section requires
thousands of points so you must be patient!
Selecting "Power" displays a power spectrum. The power
spectrum is derived from a Fourier transform of the velocity data.
For a periodic system, only only the drive frequency is observed.
"Frequency"
You may adjust the Drive frequency with this control. To set a frequency,
click on one of the scale divisions.
Do not try to drag the
slider. You can set the frequency from 0.2 Hz to 2.0Hz in 0.1 Hz
increments. The slider will move to indicate the frequency after you
have made a selection. The digital display will also indicate the
frequency.
"Phase"
Changing the phase control will
produce different Poincare' sections. The user can vary the time at which
the phase space is "strobed", when generating a Poincare' section.
The phase angle is measured with respect to the positive slope of the square
wave drive signal. With the control set to 0 deg. the cursor (thin
orange line) will fall on the positive slope edge of the first square wave.
180 deg., would correspond to the middle of the square wave.
"Press to alert others"
This control is like
a virtual "Do Not Disturb" sign. Since this is a real experiment, only
one user at a time should be in control. Use this control to signal
others that you have an experiment in progress, so that other users will
not adjust the controls while you are taking data. When you press the
button, the following warning will be seen:
The warning will flash for 30min. or until you press the button again.
The amount of time remaining will be shown near the bottom of the
screen 
Other
users are not locked out, so please respect each
other! After 30min. the display will turn off automatically.
"Run"
You can not control this
switch.
"Save and Clear"
The clear buttons
erase the data displayed on either the phase space plot or the Poincare
section. Users may save the data they acquired during an experiment.
The phase space plot will contain a maximum of 4096 data points. Once
more than 4096 data points are acquired, the oldest data points are eliminated
one at a time. There is no set limit on the number of points in the
Poincare' section. The size of the data points will automatically decrease
once the number of points exceeds 1000. This helps to reveal structure
in the Poincare' section. The data is saved as an Excel spread sheet
format (.xls).
"File Name"
When the user presses a save button, a file name is generated with the following format:
[Graph type- Ph/Po][frequency] [month]-[day] [hour] [min.] [sec.].xls
for example:
Ph0,2Hz 01-10 14 46 39.xls {this is a phase space plot- Ph}
Po0,2Hz 01-05 07 23 40.xls {this is a Poincare' section-Po}
Note that a "," is used to express a decimal number and not a "."
Data may be downloaded at
ftp://physicsonline@pendulum.mercer.edu/
No password is necessary
The physicsonline directory also contains a folder that contains movies
of the actual pendulum. You will be able to se pre-recorded movies
of the pendulum showing periodic and chaotic behavior. You will need
an MPEG player to view the movies ( Real Player or Windows Media Player work
well).
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