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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Spring Inverse Problem}



\markboth{Spring Inverse Problem}{Spring Inverse Problem}

\author{Brian Winkel, Director
SIMIODE\\ 
Chardon OH 44024 USA\\
BrianWinkel@simiode.org
}

\makeStitlePDFLaTex


\begin{abstract}
We are given data on the position of a mass in an oscillating spring mass system and we seek to discover approaches to estimating an unknown parameter.
\end{abstract}


\section*{SCENARIO DESCRIPTION}




This is an example of an  {\it inverse problem\/} in which we are given data and we seek to estimate a parameter in the model so as to fit this data with a model and thus validate the model.

Suppose we have observations (Table~\ref{table:inverseData}) on position, $y(t)$, in cm, vs. time $t$ in s of a small oscillating spring in a dashpot. We know the mass is $m = 1$ g and the spring constant is $k = 20$ dyne/cm. We can measure those. 




\begin{table}[h]\label{table:inverseData}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \hline
$t$ s&  0.0 & 0.1 & 0.2&0.3&0.4&0.5&0.6&0.7&0.8&0.9&1.0 \\ \hline 
$y(t)$ cm & 2.16& 1.77& 1.40 & 1.08 & 0.71 & 0.39 & 0.18 & 0.03 & -0.06 & - 0.01 & -0.10 \\ \hline
\end{tabular}

\parbox{12.2 true cm}{\caption{\label{table:inverseData} Data on the displacement of a mass on the end of a spring.}}
\end{center}

\end{table}

\vspace*{-1 true cm}
\begin{equation}\label{eq:InverseODE}
m\cdot y''(t) + c\cdot y'(t) + k\cdot y(t) = 0 \, , \quad y(0) = 2 \, , \quad y'(0) = 0 \, .
\end{equation}



The differential equation model in (\ref{eq:InverseODE})  discribes the motion of the mass at the end of the spring. We displace the spring 2 cm from its static equilibrium and give it an initial velocity of $y'(0) = 0$, i.e. we just release it.  We seek to find the resistance coefficient $c$ in dyne/(m/s) where these parameters are in (\ref{eq:InverseODE})

Discuss how you would determine $c$. Offer up several ideas. Carry out one approach at least. Confirm your estimate for $c$ in some manner. If you have offered up several approaches compare your results.  How does your model with your $c$ value compare to the data?



\teacher{
\section*{NOTES FOR TEACHERS}

This is an early example of parameter estimation in the setting of an inverse problem, naely, finding a model explanation of some observed data, in this case the position of the mass at the end of an oscillating spring.  We offer a Mathematica notebook and a pdf version of the notebook for those who do not use Mathematics in the Supporting Docs with title 3-026-T-M-SpringInverseProblem in which we offer some suggestions on approaches students might take.

For example 

\begin{enumerate}
\item use Mathematica's Manipulate feature in which we trial and error on $c$ to see how the model fits the data;
\item take differences and plot  $m\cdot v'(t) = -  c\cdot vt) -  k\cdot y(t)$;
\item  use a least squares approach between the data and the solution of (\ref{eq:InverseODE}) with  parameter $c$ as a parameter to be determined; and
\item use pieces of solution formula and graphical estimate period and decay to then determine parameter $c$. This might work if we had data over  longer period of time.
\end{enumerate}

}





\end{document}
