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\documentclass{simiode}
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% With this version of the tex file and the associated SIMIODE.cls file you can produce BOTH a Student Version and a Teacher Version of your modeling scenario or technique narrative.
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\begin{document}

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




\markboth{Spring Costs}{Spring Costs}

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

\makeStitlePDFLaTex


\begin{abstract}
We assume students are familiar with overdamping and underdamping of a spring-mass-dashpot system. Students will apply this knowledge to model the interplay between spring constant, tolerance, and cost. 
\end{abstract}


\section*{SCENARIO DESCRIPTION}



Stiffness of a spring is given by the ``restoration constant'', $k$, in units of Newtons per meter, or N/m. Time is measured in seconds (s).  The greater the stiffness, the more expensive the spring. Let us assume a cost, in dollars, for a 0.2 m long spring of ${\rm Cost}(k) = 13.70 + .01 k^2$. 
 5
Now suppose we are designing the following  spring mass dashpot system:

\begin{equation}\label{eq:Cost}
1.2\cdot y''(t) + 46.4\cdot y'(t) + k\cdot y(t) = 0 \, , \quad y(0) = 0.4 \, , \quad  y'(0) = 0 \, .
\end{equation}

We can assume the initial position of the  spring mass is 0.4 m from the static equilibrium and its initial velocity is 0 m/s in all situations.

\begin{itemize}
\item [i)] Find the values of $k$ which make the system overdamped. 
\item [ii)] Suppose in the case of the overdamped configuration we wish to bring the mass to within 0.1 m of the static equilibrium in the first 0.5 s of motion. We call this 0.1 m the {\it tolerance\/}. What $k$ value would we use to do this? How much would such a spring cost us?  
\item [iii)] What kind of spring performance can we get if we are willing to spend \$600 on the spring while meeting the specifications of (ii)?
\item [iv)] It is conjectured that for a  reduction to x\% of  the tolerance, i.e.  x\% of  tolerance of 0.1 m close to the static equilibrium, there is an x\% increase in price. This means that when $x = 8$ we are comparing costs of a spring which will get us to within 0.1 m of the static equilibrium  in 0.5 s  to a spring that will get us to within $0.01\cdot x = 0.08$ m of the static equilibrium in 0.5 s, a reduction in tolerance to 80\% of original tolerance of 0.1 m.  

Prove or disprove this conjecture, and provide data on costs for springs in terms of $k$ which will get us to within $z/100$ m of the static equilibrium in 0.5 s  for $z = 1,  2, 3, \ldots  , 9$.

Is there a model for this data you can offer?
\item [v)] Finally, can you render a relationship and an approach for finding the minimum cost spring which meets the two criteria (1) drops below distance $a$ meters from the static equilibrium and (2) does so within the time of $b$ seconds after release? Ideally, given the specifications $a$ and $b$ what is the $k$ to produce the cheapest cost  ${\rm Cost}(k) = 13.70 + .01 k^2$ spring and what is that cost?
 
\end{itemize}

\teacher{
\section*{NOTES FOR TEACHERS}
This activity puts the notion of overdamped spring mass dashpot in context of ascertaining the cost for performance. Further, it challenges the students to analyze the cost to improve the spring's performance, measured by cost of the spring, as a function of the spring constant $k$ in order to get the spring mass displacement closer and closer to the static equilibrium in a set time.

We  offer a Mathematica notebook (and accompanying pdf verion) called 3-031-T-Mma-SpringCost-TeacherVersion  with complete analyses.
}





\end{document}
