\experiment{Chaotic Motion}

\newcommand{\SW}{ScienceWorkshop 750 interface\xspace}
\newenvironment{chaoslist}{\begin{compactenum}[\hspace*{.2in}1.]}{\end{compactenum}}

Newton's equations and the extensions supplied by Euler, d'Alembert,
and Lagrange, built a mighty edifice able to explain the motions of 
planets and apples, pendulums and piano wires. The great French
mathematician and astronomer, Pierre-Simon, marquis de Laplace
(1749--1827), sum-med up mechanical philosophy in \emph{A Philosophical
  Essay on Probabilities} (Fifth edition, 1825):
\begin{quotation}
  We may regard the present state of the universe as the effect of its
  past and the cause of its future. An intellect which at a certain
  moment would know all forces that set nature in motion, and all
  positions of all items of which nature is composed, if this
  intellect were also vast enough to submit these data to analysis, it
  would embrace in a single formula the movements of the greatest
  bodies of the universe and those of the tiniest atom; for such an
  intellect nothing would be uncertain and the future just like the
  past would be present before its eyes.
\end{quotation}
Towards the end of the nineteenth century, Henri Poincar\'{e}
(1854--1912) came to appreciate that seemingly simple systems, such as
planets orbiting the Sun, could exhibit quite complicated motions. We
now know that the behavior of many (most?) mechanical systems shows
very strong dependence on initial conditions: systems starting from
nearly the same state evolve into later states that differ
widely. Imperfect knowledge of the initial conditions prevents us from
making valid predictions beyond a limited time interval. Therefore,
Laplace's classical determinism is a delusion.

To be chaotic, a system must be nonlinear and have more than
one degree of freedom. In this experiment you will investigate the
motion of a damped, driven, nonlinear pendulum.

\section{Overview}
\label{sec:chaosoverview}

% Pulley diameters are 10, 29, and 48 mm
% aluminum disk has diameter 9.5 cm and mass 120 g (Pasco website)
% eccentric mass (15 g) (ditto)
% mass is actually 120.26 \pm 0.01 g
% eccentric mass is 14.54 \pm 0.01 g

The pendulum consists of an aluminum disk (9.5-cm diameter, 120 g)
with an eccentric mass (15 g) mounted near the edge. A pulley is
attached to the disk and a string attached to two springs winds around
the pulley. The string is adjusted so that the tension in each spring
is equal when the eccentric mass is rotated to its highest
position (the position of unstable equilibrium). A small magnet can be
positioned near the disk to provide eddy-current damping, which is
proportional to the angular velocity.

For \emph{small} oscillations about either stable equilibrium
position, the natural frequency should be independent of amplitude and
depend only on the rotational inertia of the rotor and the effective
spring constant of the restoring torque (provided by the combination
of springs and gravity on the eccentric mass). 

\subsection{Procedure Overview}

\begin{enumerate}
\item Investigate the symmetry of the potential by releasing the rotor
  from rest near its unstable equilibrium position when the motor arm
  is in a horizontal position, once on each side of vertical. If it is
  asymmetric, carefully adjust the string on the clamp holding the
  string on the right-hand spring to symmetrize.

\item Measure the natural frequency in each potential well with the
  damping magnet far from the disk. Are they the same? Why or why not?

\item Briefly investigate the influence of the damper by recording a
  sequence of releases as you bring the magnet closer to the rotor.

\item Set the drive amplitude to about \val{2.5}{cm} and the magnet
  about 3--4~mm from the rotor. Set the voltage on the DC power
  supply to about \val{3}{V}. You should observe stable oscillation of
  the rotor in one of the potential wells. Is the motion sinusoidal?
  You may wish to perform a fast Fourier transform (FFT) to calculate
  the spectrum.

\item Gradually increase the drive voltage, allowing plenty of time
  for the oscillator to settle after each small change. Eventually the
  oscillator may begin to explore both wells. If it hasn't left the
  well by the time the frequency approaches the natural frequency,
  then either the damping is too great or the drive amplitude is too
  small to generate interesting behavior. Reduce the damping and try
  again.

\item Set the voltage just barely above the chaos threshold and record
  a long run (several minutes), observing the Poincar\'e plot. Alter
  the frequency a small amount and record again.

\item Prepare a few phase plots and Poincar\'e plots for drive
  frequencies just below and just above the chaos threshold.

\item Play and explore!
% By removing the eccentric mass, you can study damping. Does it
% behave like a damped driven SHO? How does the damping depend on
% magnet-disk separation?
% \item Measure the spring constants using the linear system (shouldn't
% require removing anything except the eccentric mass
% \item Computation in Mathematica---I could provide seed code
% \item Is it possible to observe period doubling?
\end{enumerate}



%  The opposite end of one spring is mounted to a rotating arm
% that drives it approximately sinusoidally with an amplitude and
% frequency that you can vary.

% , which is mounted on a Pasco rotary motion 
% sensor, capable of either \dg{1} or \dg{\frac{1}{4}} resolution. Using
% the \SW and DataStudio software, you can
% record angular position and angular velocity at range of frequencies. 

\section{Details}
\label{sec:chaosdetails}

The Pasco software that operates the \SW is called Data Studio. You
will set it up to record angular position, $\theta$, and angular
velocity, $\omega$, as functions of time, as well as the period of the
drive. You can then create live plots to show $\theta$ vs.\ $t$,
$\omega$ vs.\ $t$, a phase plot ($\omega$ vs.\ $\theta$), and a
Poincar\'e plot, which is a phase plot with discrete points plotted
once per cycle of the drive. Of course, you can also export data in
text format to Igor Pro or other analysis packages.


\subsection{Setup}
\label{sec:Chaossetup}

%\setdefaultleftmargin{2.5em}{2.2em}{1.87em}{1.7em}{1em}{1em}

\subsubsection{Hardware configuration}
%
\begin{chaoslist}
  \item Plug the rotary motion sensor into channels one (yellow) and
    two (black) of the \SW.

  \item Plug the photogate into channel 3.

  \item Ensure that the interface's USB cable is connected to the
    computer. Then power on the \SW.

  \item Connect the red and black banana jacks of the DC motor to the
    DC power supply. You may find it useful to connect a digital
    multimeter first, so you can monitor the supply voltage more
    precisely than is possible with the analog meter on the power
    supply. 

  \item Adjust the rotating arm on the drive motor to be about
    \val{2.5}{cm} off axis, and rotate the arm to a horizontal
    position (so that the bottom end of the drive spring is approximately
    in the middle of its range of travel).
\end{chaoslist}
  
\subsubsection{Software configuration}

\begin{chaoslist}
  \item Open Data Studio and create a new activity. You should see a
    picture of the \SW with its ports outlined in yellow. [You will
    only need to go through the following setup procedure once. Once
    you have created an activity with all the appropriate measurements
    and calculations, you can use it for subsequent data files. Just
    Save As a different file, then remove all the data using the
    Experiment menu.]

  \item Click on Port 1 and add a \textbf{Rotary Motion Sensor} from
    the Digital Sensors tab.

  \item Set it up to measure angular position ($\theta$) in degrees,
    angular velocity ($\omega$) in radians/s.

  \item Set the sample rate to \val{50}{Hz}.

  \item Set the Resolution to High (on the Rotary Motion Sensor
    tab). This corresponds to an angular resolution of \dg{0.25}.

  \item Add a photogate to Port 3, and have it measure ``Velocity in
    Gate'' (even though this makes no sense). We will use it instead
    to measure the period and to strobe $\theta$ and $\omega$ for the
    Poincar\'e plot.

  \item Click the \textbf{Setup Timers\ldots} button and label the
    timer Period. Using the Timing Sequence Choices area create a
    sequence of blocked--unblocked--blocked. This will record the
    period each time the drive motor completes a revolution.

  \item Minimize the Experiment Setup window. You may need to come
    back to it later if you decide to change the sample rate or the
    resolution of the rotary motion sensor.
\end{chaoslist}
  
\subsubsection{Calculations}

\begin{chaoslist}
  \item You now need to set up a few calculations to be made
    from the data you will acquire. Click the \textbf{Calculate}
    button.

  \item To calculate the frequency, enter in the Definition box:
    \textsf{f = 1/T}. Click on the Variables popup, select undefined
    variable, which will still be referring to ``x''. It will now
    mention the undefined variation ``T'', which you should tie to the
    Period measurement (Data source).

  \item You now need to set up two additional calculations for the
    Poincar\'e plot, and one for the potential. The values of $\theta$
    and $\omega$ are stored at the sample rate. In addition, we would
    like to record their values once each cycle, using the photogate
    measurements. To manage this, we use a trick. We multiply the
    measurement of Channel 3 by zero and add the value of either
    $\theta$ or $\omega$. Channel 3 is recorded only once per
    revolution of the drive, so including it in the calculation
    produces the desired strobing of $\theta$ and $\omega$.  Create
    these two calculated quantities, then close the Calculate window.
\end{chaoslist}
  
\subsection{Potential Well}
\label{sec:chaosU}

To explore the shape of the potential well, back the damping
magnet all the way out, rotate the eccentric mass to the top, and
release it from rest. If damping is negligible, then
\begin{equation}
  \label{eq:Chaos1}
  E = U(\theta) + \frac{1}{2} I \omega^{2} 
\end{equation}
where $E$ is constant. Therefore, by plotting $-\omega^{2}$ vs.\
$\theta$, you can explore the shape of the potential well. Set up
a calculation for $-\omega^{2}$. Then prepare a graph of $-\omega^{2}$
vs.\ $\theta$:

\begin{chaoslist}
  \item  Drag the Data entry for $-\omega^{2}$ onto the Graph entry in
    the Displays panel.

  \item Mouse over the label for the $x$ axis and popup the menu to
    select $\theta$.

  \item Raise the eccentric mass to the top, click the \textbf{Start}
    button, and release the disk. Stop the measurement after one
    complete oscillation.

  \item Raise the eccentric mass to the top, but make sure it will
    fall on the opposite side, and repeat the procedure. Are the two
    potential wells symmetric? If not, adjust the string to symmetrize.
\end{chaoslist}

  
\subsection{Measurements}

Set the damping magnet to be somewhere between 3 and 5~mm from the
disk and determine the undriven period of the oscillator. Then turn on
the power supply and set the voltage to about 3~V. After an initial
transient, you should see the pendulum oscillate in a periodic
fashion. Record data for a minute or so. Are the oscillations
sinusoidal?

You'll probably want to set up a temporal plot ($\theta$ vs.\ $t$) and
a phase plot ($\omega$ vs.\ $\theta$) to observe the changing behavior
of the oscillator as you \emph{gradually} increase the voltage,
allowing the behavior to settle down after each small change. (You'll
probably also want to turn off individual data points in these two
plots.) Make a rough determination of the voltage/frequency at which
the pendulum begins to explore both potential wells. You should also
set up the Poincar\'e plot at this point and make sure that your
activity file has been saved.

A few last tips about graphs in Data Studio:
\begin{chaoslist}
  \item Graphs have an option called \emph{Full Color}. When selected,
    each visible trace gets colored; when unselected, all traces are
    grayed except the selected trace.

  \item You can use the ``Smart Tool'' to make measurements directly
    on a graph.

  \item You can copy the selected trace to the clipboard. Both
    graphical and textual representations of the data are copied, so
    you can paste directly into a word processor document or into a
    data analysis program. A script is available to permit Igor Pro to
    load the data on the clipboard, preserving the run number from the
    Data Studio activity.
\end{chaoslist}





%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "Phys54Manual"
%%% End: 
