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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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% Basically the Student Version contains the title STUDENT VERSION along with the NAME OF SCENARIO, with no abstract, no keywords, no tags, no author identifiers, and no COMMENTS section. The Student Version contains only the STATEMENT of the modeling scenario with appropriate figures, data, instructions, and perhaps a bibliography appropriate for student resourcing, but nothing in the bibliography which would "give away" the nature of the modeling scenario, etc.
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%The Teacher Version contains the material in the Student Version, namely the title NAME OF SCENARIO, now headed with   TEACHER VERSION, but also full author  identification, abstract, key words, tag words, and, following the STATEMENT, a section of COMMENTS which can be discussion of presentation  ideas, "solutions," techniques, etc.
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% Authors can prepare a full Teacher Version using the notions immediately below  to create a pdf for review (but in review please eliminate all information about the author as we use a double-blind referee system. Such information will be restored when accepted and published in SIMIODE at www.simiode.org.
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%\usepackage{amsmath,amssymb}
%\usepackage{color}
%\usepackage{graphicx}
%\usepackage{hyperref}
\usepackage{url}
\usepackage{setspace}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{color}
\usepackage{ulem}
\usepackage{gensymb}
%\usepackage{hyperref}
\usepackage{enumerate}
\usepackage{units}
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\def\kf{\ttfamily\bfseries}
\def\M{{\sl Mathematica\/}}
\def\>>{\textgreater\hspace{-3pt}\textgreater}
\def\yp{{y^\prime}}

%%% Begin Instructor Coefficients

\newcommand{\xo}{3}  %Choose either 3cm, 4cm, or 5cm  (original 3cm)

%Switch a pair of spring and damping constants (e.g. \kone & \bone -> \ktwo & \btwo)

\newcommand{\kone}{220}  % plus or minus 3   (original 220)
\newcommand{\bone}{100} % plus or minus 3  (original 100)

\newcommand{\ktwo}{59} % plus or minus 3  (original 59)
\newcommand{\btwo}{1050} % plus or minus 3  (original 1050)

\newcommand{\kthree}{170} % plus or minus 3  (original 197)
\newcommand{\bthree}{1110} % plus or minus 3   (original 1100)

\newcommand{\kfour}{25}  %plus or minus 3   (original 25)

\newcommand{\amp}{200} %plus or minus 10   (original 200)


%%%End Instructor Coefficients 



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\begin{document}

\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\MILITARY APPLICATIONS OF \\SPRING-MASS SYSTEMS}



\teacher{
    \keywords{second-order differential equations, spring-mass systems, parameter estimation}
    


\tagwords{damping, resonance, beats, Laplace transforms, Dirac Delta function, }
}


\markboth{MILITARY APPLICATIONS OF SPRING-MASS SYSTEMS}{MILITARY APPLICATIONS OF SPRING-MASS SYSTEMS}






\author{Dr. Ivan Dungan and LTC Randy Boucher \\
Department of Mathematical Sciences\\ 
United States Military Academy\\
 \teacher{646 Swift Road}
West Point, NY \teacher{10996} USA\\
\teacher{gregory.dungan@usma.edu and randy.boucher@usma.edu}}



% \makeStitle
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\teacher{
\begin{abstract}
{\color{red}} The project is a collection of five different scenarios regarding the shock system of a trailer; the first three are chronological and the latter two are optional independent scenarios.  In each scenario, students will transform the shock system of a trailer into a second-order differential equation, solve, and interpret the results.  In the first two scenarios, students must compare three different shock absorbers (or shocks) for the trailer.  They must first compare the shocks when being loaded and then while driving on a given terrain.  The third scenario considers a cheaper shock with no damping which leads to a discussion of resonance.  In the fourth scenario, an alternative use of the trailer is being tested which causes a resonance-like phenomenon.  They will use vertical displacement data of the trailer to estimate the damping coefficient of the shock system.  Using the damping coefficient, they can model the trailer and predict the long-term behavior of the trailer.  The final scenario is to model the trailer being pulled over an intense terrain.  They will need to model the terrain using the Dirac Delta function and solve the model using Laplace transforms.  The project has been written from the point of view of a recently commissioned second lieutenant in the U.S. Army, but can be easily adapted to a contracted mathematician or completely stripped of a background story.
\end{abstract}
}

\teacher{

\vspace*{-.1 true cm}
\listkeywords

\vspace*{-.9 true cm}
\listtags
}

\section*{STATEMENT}



You have just completed BOLC (Basic Officer Leadership Course) in the branch of your choice (lucky you) and have PCS'd (Permanant Change of Station) to your first duty station, Joint Base Lewis-McChord, WA.  While sitting with the other new arrivals, a Colonel from the U.S. Army Test and Evaluation Command (ATEC) at Aberdeen Proving Grounds engages the personnel clerk in a quick conversation and addresses the group.
	
``From the looks of things you all will be snow birding until the assignment slate opens up and I need a volunteer.  You must be comfortable with some basic engineering design: mathematical modeling and differential equations. You will be making recommendations on decisions with expensive and vital ramifications."

You eagerly indicate interest. After all, you know about differential equations, but have no idea what snow birding is.  The next day you report to the 51$^{\text{st}}$ Expeditionary Signal Battalion.  The Battalion Commander has assigned you to an initiatives group that has been tasked to perform testing and analysis for ATEC engineers.  Your email has now been set up and you open your first email as a Mathematical Consultant, which is the following memorandum.


\bigskip

GENERAL MATHEMATICAL MODEL:  The sprung mass of a vehicle is the mass of the vehicle minus the suspension and wheels (the unsprung mass).  To simplify the analysis of a suspension system, we assume that the mass of a vehicle, and any forcing function on the suspension, are evenly distributed; in other words, the suspension system can be treated as one shock.  Thus, this simplified suspension system behaves according to the differential equation
$$m\frac{d^2x}{dt^2}+\beta\frac{dx}{dt}+kx=f(t),$$
where $x(t)$ is the displacement from its rest position in meters, $m$ is the mass of the object in $\unit{kg}$, $\beta$ is the damping constant in $\unit{N\cdot s/m}$, $k$ is the spring constant in $\unit{N/m}$, and $f(t)$ is the external force in Newtons (N).

The Rockwell Collins Satellite Transportable Terminal (STT) is a combat-proven, transportable earth terminal designed to establish secure voice, video and data communications virtually anywhere.  Designed to withstand challenging conditions and demanding operations in Iraq and Afghanistan, the U.S. Army has deployed hundreds of STTs for the Joint Network Node (JNN) program.  

\vspace*{.3 true cm}

\begin{center}
\indent \includegraphics[width=7cm]{3-110-HMMWV2.jpg} \hspace{.5 true cm}
\includegraphics[width=7cm]{3-110-STT3.jpg}
\end{center}

These rugged, trailer-based earth terminals deliver up to 56 Mbps (mega bytes per second) throughput - and they can be towed anywhere a HMMWV (High Mobility Multipurpose Wheeled Vehicle) can go.  Rockwell Collins has built more than 800 STT units since the JNN program's inception in 2004. [Rockwell Collins 2009]  Due to recent budget cuts, more economical parts must be used in the production of STTs.  Engineers have tasked you with comparing feasible options for the suspension systems of the STTs.

The suspension system for the STTs needs to be fitted with cost effective shocks that are reliable and protect the satellite while being loaded and transported.  There are three shocks available:  

\begin{center}
    \begin{tabular}{| l | c| c | }
    \hline
   Specifications & Spring Constant $k$ ($\unit{N/m}$) & Damping Coefficient $\beta$ ($\unit{N\cdot s/m}$) \\ \hline
    Air Coasters & \kone & \bone  \\ \hline
    Cloud Riders & \ktwo & \btwo  \\ \hline
    Extreme Shocks & \kthree & \bthree  \\ \hline
    \end{tabular}
\end{center}



\begin{enumerate} 
\item Model the suspension system of a STT trailor while it is loaded with a satellite using these three different shocks.  The trailer of a STT at rest will be loaded with a satellite with a mass of $1800 \unit{~kg}$.  Assuming that the loaded trailer is at equilibrium, the displacement of the shocks before the trailer is loaded is $\xo \unit{~cm}$.

\begin{enumerate}

\item Determine the critically-damped value for each shock i.e. the optimal value to avoid oscillatory motion.  Compare the  damping coefficient of each shock to its critically-damped value.

\vspace{.1in}

\item Solve the spring-mass system for each shock with the given damping value and also with the critically-damped value.  For each shock, plot the solution with the given damping value and the solution with the critically-damped value together.
  
\vspace{.1in}

\item Discuss the model of each shock with regard to the satellite being placed on the trailer.  From your analyses, decide which shock is best for the STT suspension system for loading.  Explain your answer.  
\end{enumerate}


\item \label{ComparisonForcingFunction}  ATEC engineers have specified that if the suspension system bottoms out too many times, the components of the satellite could be damaged, which could cost hundreds of thousands of dollars to repair. Therefore it is important to determine the proper shocks for the suspension system of STTs to avoid any damage to the satellite.  Topographers and engineers have determined that the force on the shocks in the most severe terrain at the highest speed that a HMMWV trailering a STT will attain can be modelled by the function $f(t)=\amp \sin(4t)~\unit{N}$ where $t$ is in seconds.  Assume that $x(0)=0$ and $x'(0)=0$.


\vspace{.1in}
\begin{enumerate}
\item For each shock, solve the associated non-homogeneous equation and plot the solutions.  

\vspace{.1in}
\item Find the maximum/minimum displacement, $x_{max}$ and $x_{min}$ for each suspension system and present these in a table.  Discuss the shock performance in the long-run.

\vspace{.1in}

\item Determine the best shock for the suspension system of the STT and explain your decision from the perspective of the satellite.  

\vspace{.1in}


\item For your recommended shock, solve the associated homogeneous equation.  Identify both the transient solution and the steady-state solution in your solution.  Plot the solutions together to compare.  Explain the effect of the transient solution on the satellite.

\end{enumerate}



\item The preferred option of the budget committee for the suspension system of the STTs are the shocks: Premium Shocks  (Specs: $k = \kfour \unit{~N/m}$; $\beta = 0 \unit{~N\cdot s/m}$).  Present reasonable data persuading our budget team that undamped shocks, although cheap, will not be sufficient.  Consider the same external forcing function from Part \ref{ComparisonForcingFunction}, but with a variable frequency (i.e. $f(t)= \amp \sin(4\omega t)$ where $\omega$ is the degree of roughness of the driving conditions).

\begin{enumerate}

\item \label{CheapShockSolve} Solve the initial value problem for a function with the time, $t$, and frequency parameter, $\omega$, as variables.  Describe the signifigance of $\omega$ and its relation to the speed of the vehicle.

\vspace{.1in}





\item  For what positive value $\omega$ is the above solution invalid?  Characterize the suspension system at this value.

\vspace{.1in}

\item  Create an interactive plot of the solution you found in Part \ref{CheapShockSolve} with respect to the frequency parameter, $\omega$, for the first ten minutes of driving.  Explain the cause of the change of frequency.  Describe any damaging effects the STT might experience while being trailered. 


\end{enumerate}


\item In the field, the trailer of an STT was being tested with an experimental mounting of the Herstal DeFNder FN M3P high powered machine gun in place of the satellite.  
\begin{center}
\indent \includegraphics[width=6cm]{3-110-HerstalDeFNder.png}
\end{center}
While the trailer was at rest and the mounted gun was firing vertically, the vehicle began experiencing pronounced vertical motion.  The soldiers operating the trailer immediately shutdown the gun and returned it to base.   To help understand the phenomenon, some engineers re-fired the weapon and took measurements of the displacement of the trailer from the ground (see \ref{table:RawData} for the reported data).  


\begin{table}   % to make it close to the reference [htbp]
\centering
    \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
    \hline
    t & 0 & 1 &2 &3&4&5&6&7&8 \\ \hline
    x(t)&0&0.0303&0.0581&-0.0045&-0.1188&-0.1363&0.0140&0.1995&0.2022 \\ \hline	
    \end{tabular}
    \label{table:RawData}
\end{table}

\begin{table}  %[htbp]
\centering
    \begin{tabular}{|c|c|c|c|c|c|c|c|c|}
    \hline
    t &9&10&11&12&13&14&15&16 \\ \hline
    x(t) &-0.0261&-0.2696&-0.2557&0.0387&0.3272&0.2971&-0.0496&-0.3709 \\ \hline	
    \end{tabular}
    \caption{Displacement Measurements of Trailer (in meters)}
	\label{table:RawData}
\end{table} 

Since it appeared the structural integrity of the trailer would fail, the engineers shutdown the weapon after $60$ seconds.  Due to the structural restrictions of the trailer, physical experimentation is limited.  The engineers have tasked you with modeling the phenomenon to see if it will stabilize or continue until the point of structural failure.  

The mass of the Herstal DeFNder is $4000 \unit{~kg}$ and the engineers were able to determine the spring constant of the shocks used on the trailer to be $k=5000 \unit{~N/m}$.  Also, our engineers have determined that the amount of force on the trailer produced by the weapon firing is modeled by the function $f(t)=300\cos(t)~\unit{N}$.  

\begin{enumerate}
\item Plot the data of the phenomenon and describe the behavior of the trailer.  
\item Use data and Mathematica to estimate the damping coefficient by minimizing the sum of squared errors (SSE) between the data and the predictions of your model.
\item A suggestion was made that the truck is experiencing resonance or beats.  Is this a reasonable hypothesis?
\item If they do not turn off the vehicle and the shocks can only extend and compress a length of $0.5$ meters, will the structural integrity of the truck be compromised?   
\end{enumerate}




\item Before physical tests are completed by the ATEC, you have been asked to run additional simulations of the STT trailer-based system.  In particular, ATEC senior engineers would like you to consider a scenario for which the trailer traverses a rugged terrain with exposed rocks, an environment the STTs will likely encounter.

A technique for modeling the rugged terrain is to express the force applied to the suspension system by each rock as an impulse. This can be done with a Dirac Delta function of the form, $w_0\delta (t-t_0)$, where $w_0$ is the concentrated load applied at time $t_0$. To model terrain with, $N$, \textit{equally spaced} rocks, we can express the external force, $f(t)$, via several impulse functions, i.e. 
\begin{align} \label{impulsefunction}
f(t) = 
\displaystyle{w_0 \sum_{n=1}^{N} \delta (t-(2n-1) \phi ),}
\end{align}
where $2 \phi$ is the time between impulses and the initial impulse occurs at $t=\phi$.  Your job is to complete an analysis of the STT suspension system for a trailer traversing over rugged terrain. 

\begin{enumerate}
	\item \label{ImpulseSolve} ATEC engineers decided that the \textit{Extreme Shocks} with spring constant $k=170$ $\unit{N/m}$ and damping coefficient $\beta=1110$ $\unit{N\cdot s/m}$ was the best option to prevent damage to the satellites while being loaded and transported. Model the suspension system of a STT trailer while it is loaded with a satellite and traveling over the terrain modeled by \ref{impulsefunction}. For this scenario let $N=13$, $w_0=150$ Newtons and $\phi=3$ seconds. Additionally, the trailer is initially at rest at the equilibrium position (vertically) and has a mass of 1800 kg. 
	\begin{enumerate}
	\item Establish the initial value problem that models the spring-mass system in the scenario described above. Clearly define your independent and dependent variables, parameters, units, and initial conditions.
	
	\item \label{fiveAtwo} Solve \textit{by hand} the IVP established in part \ref{ImpulseSolve} using Laplace transform methods. [Note that the \textit{Mathematica} command {\kf Apart} may be used for partial fraction decomposition if needed. Additionally, the Laplace transform of Equation \ref{impulsefunction} may be accomplished without writing out all the terms in the sum!]
	
	\item Use \textit{Mathematica} to verify your answer for \ref{fiveAtwo} [This may be done visually using graphical techniques.] Additionally, plot the solution and sketch your graph in the space provided below. 
	\end{enumerate}
	\item Find the maximum and minimum displacement, $x_{max}$ and $x_{min}$ for the suspension system. Discuss the performance of the shock system as it traverses the rugged terrain over the long run.

	\item If the shock system can only extend and compress a length of 20 centimeters, will the structural integrity of the truck withstand the rough terrain? What if the time between impulses is decreased, say to $\phi=2$ seconds? What would your guidance be in terms of traversing the STTs in rugged terrain?
	
	
     
\end{enumerate}


\end{enumerate}





%\section*{Submission Version}

%For submission to the manuscript management system, \url{https://simiode.expressacademic.org} authors should submit a Teacher Version with all author identifying information stripped out. This should be a pdf version of the material. 

%Authors may also submit supporting documents such as pdf's of code files, e.g., MatLab, Mathematica, Maple, SAGE, etc. and spreadsheets and data files.  

%Below is a set of References which appears in the Student Version. Note, there is a second set of References one can use for the Teacher Version.

\begin{thebibliography}{98}


\bibitem{FNHerstal2014} FN Herstal. \textit{FN Herstal deFNder FN M3P}.  Virginia, South Carolina, 2016. \url{http://www.fnherstal.com/}.

\bibitem{RockwellCollins2009} Rockwell Collins. \textit{Satellite Transportable Terminal (STT)}.  Cedar Rapids, Iowa, 2009. \url{http://www.rockwellcollins.com/gs}. Accessed 5 December 2016.



\end{thebibliography}

\teacher{
\section*{COMMENTS}
For this project, a Computer Algebra System (CAS) is necessary and the student's will need to be trained in using the CAS well enough to solve differential equations, both analytically and numerically, and plot solutions.  We have chosen to use Mathematica and have included with the project PDF files of solutions for each scenario.  Also, we have included an actual project submission\footnote{Note that in the original project, the weapon in part 4 was mounted to a HMMWV rather than a trailer.}  from a student for parts 1-4.  Note that our students must cite any collaborative work and Mathematica code must be annotated and cited in the text document itself.

The first three scenarios are chronological and the latter two are optional independent scenarios.  Depending on the differential equations book being used, the first three scenarios would be best positioned during or after lessons on spring-mass applications of second-order differential equations (e.g., see \cite{BoyceDiPrima} or \cite{ZillWright}).  The necessary topics in these sections are critical damping and resonance.  The fourth scenario will require a method of parameter estimation.  In the Mathematica solutions, we minimized sum of squared errors which we had previously introduced to the students in a technology lab.  Besides parameter estimation, the students must have an understanding of resonance and beats, and the conditions under which they occur.  In this particular scenario, the data appears to look like resonance which should prompt discussion on the conditions when resonance actually occurs.  If not properly observed, even the model may mislead the student into believing that beats have occured.  The final scenario will require understanding of Laplace transforms and their applications to the Dirac Delta function.

Below we have included example instructions and goals of the project that were provided to the students with the project.

%This is a set of COMMENTS about the scenario, for the teacher only. It might include a ``solution" or a  number of solution strategies. However, the main narrative should be about how to use the scenario in teaching, e.g., what are prerequisites, how long might one devote to it, what resources for students could be expected to be used, what  issues which cause students difficult or ease should be expected to come up, discussion of technology use, pointers to other files, etc.  This set of COMMENTS might have its own  pointers to data sets, supplementary files,  videos, etc. for student and teacher  use, the material for students could be something the teacher might use with discretion. There could be an extended bibliography \cite{Gause1934,Hutchinson1978a}  for additional teacher resources, analyses,  data, etc.

\begin{description}
\item Example Instructions:

This is an INDIVIDUAL assignment. Your primary submission is a hard copy report that should \textbf{not exceed three (3) pages}. To support your primary report, you should include annotated and referenced appendices. 

You should include any additional supporting graphs, equations, and computations in the appendices. An appendix is NOT a place to put printed computer code. If you think that your code is important, you may include it in an appendix, but you should ensure that it includes annotations describing what you are doing. As a backup submission, you may also digitally submit \textit{Mathematica} and \textit{Excel} files that you develop. If you think a computation or equation is important for the reader, then ensure  it is in the report or appendices and not just in your digital file. 

Ensure all work is logical, neat, and organized.  Doing the mathematics correctly is important, but it is also critical to be able to analyze and effectively communicate your mathematical results, as well as reflect on the relevance of your results to the real world.



\item Goals of Project:

\begin{enumerate}
\item Students clearly communicate their mathematical model and solution methods in written form.
\item Students explore aspects of the problem from a different disciplinary perspective to supplement their mathematical modeling.
\item Students design and solve a mathematical model for a real world problem using differential equations.
\item Students confront the ambiguity of problem solving and understand the necessity of making assumptions in formulating a mathematical model and the impact these assumptions have on their solutions.
\item Students apply technology learned in the course to analyze and solve a differential equations model and develop an appreciation for how technology enhances their problem solving capabilities.
\end{enumerate}

\end{description}

\begin{thebibliography}{99}

\bibitem{BoyceDiPrima} Boyce, William E., and Richard C. DiPrima. {\it Elementary Differential Equations and Boundary Value Problems, 10th Ed}. New York: Wiley, 2012. Print.

\bibitem{ZillWright} Wright, Warren G., and Dennis G. Zill. {\it Differential Equations with Boundary-Value Problems, 8th Ed}. Boston: Brooks/Cole, 2013. Print.

\end{thebibliography}
}





\end{document}
