picture update

adv_lab.bib

@@ -36,7 +36,13 @@
   type = {Program}
 }
 
-
+@techreport{ADVLABCHAOS,
+  author = {Burns, Clement},
+  title = {Chaos Lab V3},
+  institution = {Western Michigan University},
+  year = {2017},
+  type = {Lab Guide}
+}
 
 
 

chaos/report/report.tex

@@ -92,13 +92,54 @@ A 2-dimensional phase space is a useful environment in which to identify the cha
     \end{figure}
 
 
-\section{}
-\label{sec:reverbmap}
+\section{Experimental Driven Pendulum}
+\label{sec:experiment}
+An experimental driven pendulum is built in order to determine whether its motion is chaotic. The apparatus is shown in figure \ref{fig:apparatus}. The springs are connected so that they both hold about equal tension while the pendulum is at its top-most position, and so that they can stretch over at least one full turn of the pendulum in either direction. The tension was not measured precisely, nor the weight of the pendulum or the damping of the pendulum. The amplitude of the driving arm was $1.45" \pm 0.1"$. A photogate is used to determine the driving period when necessary.
+
+\begin{figure}
+    \includegraphics[width=6.5in]{appartus.png}
+    \caption{Experimental driven pendulum. This angular motion of the round metal pendulum at the top is expected to exhibit chaotic motion. \cite{ADVLABCHAOS}}
+    \label{fig:apparatus}
+\end{figure}
+
+    \subsection{Resonant Frequency}
+    To determine the natural frequency of the pendulum, it was released from the top position under damped conditions and its frequency of oscillation was measured. The waveform is shown in figure \ref{fig:exp_damped_time}. Fourier transformations of the waveform in figure \ref{fig:exp_damped_fourier} reveal the natural frequency as the frequency where the power spectral density peaks, discounting the low-frequency peak from the flattened waveform due to damping. The natural frequency is estimated to be $0.93\pm0.6 Hz$. 
+    
+    \begin{figure}
+        \includegraphics[width=6.5in]{exp_damped_time.png}
+        \caption{The damped oscillations of the experimental pendulum. The fourier transforms of this function reveal the natural frequency of the pendulum, in figure \ref{fig:exp_damped_fourier}.}
+        \label{fig:exp_damped_time}
+    \end{figure}
+
+
+    \begin{figure}
+        \includegraphics[width=6.5in]{exp_damped_fourier.png}
+        \caption{The fourier frequency-space representation of the damped oscillating. The peak near 0.8 Hz is the natural/resonant frequency of this pendulum.}
+        \label{fig:exp_damped_fourier}
+    \end{figure}
+
+
+
+
+    \subsection{Periodic Motion}
+    To produce periodic motion, the driving arm is run with a driving frequency $1.45 \pm 0.02s$. The observed motions are plotted in phase space in figure \ref{fig:exp_periodic_phase}. The motion still suggests some chaotic behaviour by the variation in the path taken about the critical point, but the poincar\'e section in figure \ref{fig:exp_periodic_poincare} shows that the orbit is converging.
+
+    \begin{figure}
+        \includegraphics[width=6.5in]{exp_periodic_phase.png}
+        \caption{The periodic motion at driving period $1.45 s$is easily recognizable in phase space. It follows a stable orbit around the attractor. This plot still suggests chaotic behaviour, as the }
+        \label{fig:exp_periodic_phase}
+    \end{figure}
+
+    \begin{figure}
+        \includegraphics[width=6.5in]{exp_periodic_poincare.png}
+        \caption{The periodic motion at driving period $1.45 s$is easily recognizable in phase space. It follows a stable orbit around the attractor. This plot still suggests chaotic behaviour, as the }
+        \label{fig:exp_periodic_poincare}
+    \end{figure}
 
 
 \begin{figure}
     \includegraphics[width=6.5in]{chaotic_b_time.png}
-    \caption{Computer-generated model of a driven pendulum with no damping. Arguments: $\theta=x, f=1, I=1, \omega=0.5, \alpha=0, k=\frac{\omega_0^2}{I}=1.5, \phi=0$, 180 time steps.}
+    \caption{Driven oscillating with chaotic motion.}
     \label{fig:chaotic_b_time}
 \end{figure}
 
@@ -112,6 +153,4 @@ A 2-dimensional phase space is a useful environment in which to identify the cha
 \printbibliography
 
 
-
-
 \end{document}