% SIMIODETemplate-TeacherORStudent.tex and simiode.cls for LaTeX2e
% Version 29 July 2015.
%
% This class (simiode.cls) is a modification of that used by
% the journal PRIMUS - Problems, Resources, and Issues in
% Mathematics Undergraduate Studies.
%
% See http://www.tandfonline.com/loi/upri .
%
% The class was developed by  Marc Frantz with suggestions of the
% PRIMUS Editorial Board, especially John Thoo.
%
% This template is to be used with the  class file "simiode.cls".
%
% The latest version of this class and template files
% can be found at
%
%       <http://www.SIMIODE.org> in the Author section.
%
% A "Quick Start Guide" appears below.
%
% This file must be processed with version LaTeX2e or higher.
%
\documentclass{simiode}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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.
%
% 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.
%
%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.
%
% 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.
%
%***********   Toggle the following line to turn the teacher version on (1) or off (0). ***********
%
%
\def\TeacherEdition{0} % Toggle this line's value.
\newcommand{\teacher}[1]{\ifnum\TeacherEdition=1 #1 \fi}
%
%
% The \teacher{ } command wraps any text that will appear only in the teacher version of
% the document. See this file for examples. This has been done for you in the document below.
% Rename the pdf after you compile each final version.
% This option was created  by Eric Sullivan, Carroll College, Helena MT.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}
\usepackage{url}
\usepackage{hyperref}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% NOTE regarding double spacing %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Manuscripts should be submitted doubled spaced.  This is
% accomplished via the \baselinestretch command below.
% However, you may obtain an idea of how your article will
% look in final form, so that you can better plan your
% presentation.  To see your manuscript formatted very
% similarly to the on-line published format of  SIMIODE,
% comment the following line out by inserting a percent sign % (%) at its beginning.

\renewcommand{\baselinestretch}{1.3}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% NOTE regarding macros %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Any macros defined for your paper should be contained in
% the top matter. Likewise, any environment definitions such
% as \newtheorem or \newenvironment should also go in the
% top matter.
%
% Do not \input your macros as separate files within your
% final version because more files make it harder for the
% editors and users to keep track of and/or modify and
% append your paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Top matter is above and
% the beginning of the document below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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\\FORCED VIBRATION AND NO DAMPING}





\markboth{Forced Vibrations and No Damping}{Forced Vibrations and No Damping}






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

% \makeStitle
% replaces: \makeStitle
% Choose only one of the following:
% \makeStitlePDFLaTex  % <--- use this for PDFLaTex (jpg and pdf image files etc.)
   % <--- use this for LaTex (eps image files)

\makeStitlePDFLaTex


\begin{abstract}
We study what can happen when a pure oscillator  (no damper) is driven by a forced vibration function which has the same or close to the same natural frequency as the system it is driving.
\end{abstract}


\section*{SCENARIO DESCRIPTION}




Consider a spring mass dashpot model with mass $m$ kg, resistance $c$ N/(m/s), and spring constant $k$ N/m with a forcing function $f(t)$ in units N    to describe the displacement from static equilibrium, $y(t)$, (\ref{eq:forced}).

\begin{equation}\label{eq:forced}
m\cdot  y''(t) + c\cdot y'(t) + k\cdot  y(t) = f(t)\, , \quad y(0)= y_0 {\quad} {\rm and} \quad y'(0) = v_0\, .
\end{equation}

 In this case we shall deal with an ideal spring and set $c = 0$.  Thus we have the following differential equation  as  (\ref{eq:forced})  becomes (\ref{eq:forcedND}).

\vspace*{.15 true cm}
\begin{equation}\label{eq:forcedND}
m\cdot  y''(t) +  k\cdot  y(t) = F(t)\, , \quad y(0)= y_0 {\quad} {\rm and} \quad y'(0) = v_0\, .
\end{equation}

Recall that (\ref{eq:forcedND}) without a driving function $f(t)$ produces a pure oscillating function with natural frequency $\omega_0 = \sqrt{\frac{k}{m}}$. If now we apply a driver function and our driver is a sinusoidal function, $F(t) = F_0\cos(\omega t)$, we seek to know what the possibilities are for solutions/outputs from (\ref{eq:forcedND}).

For notational convenience let us divide both sides of (\ref{eq:forcedND}) by $m$ and noting that $\omega_0 = \sqrt{\frac{k}{m}}$ we can see that the coefficient of $y(t)$ is $\frac{k}{m} = \left(\sqrt{\frac{k}{m}}\right)^2 = \omega_0^2$. Thus (\ref{eq:forcedND}) now looks like  (\ref{eq:forcedNDT}). Also let us set $y(0)= y_0 = 0$ and $y'(0) = v_0 = 0$ as the driver will move the mass.

\begin{equation}\label{eq:forcedNDT}
y''(t) +  \omega_0^2  y(t) = \frac{F_0}{m} \cos(\omega t)\, , \quad y(0)= 0 {\quad} {\rm and} \quad y'(0) = 0\, .
\end{equation}

We can solve (\ref{eq:forcedNDT}) by hand or, say, use {\it Mathematica\/}'s {\tt DSolve} to solve, obtaining:

\begin{equation}\label{eq:solved}
y(t) = \frac{F_0}{m(w_0^2 - w^2)}(\cos(\omega t) - \cos(\omega_0 t)) \, .
\end{equation}
We can use the trigonometric identity in (\ref{eq:TrigID}) to alter our solution (\ref{eq:solved}) to a more learning friendly format.

\begin{equation}\label{eq:TrigID}
(\cos(\omega t) - \cos(\omega_0 t)) = 2 \sin\left(\frac{\omega_0+\omega}{2}t\right)\sin\left(\frac{\omega_0-\omega}{2}t\right)\, .
\end{equation}

\noindent Thus our solution in  (\ref{eq:forcedNDT}) looks like (\ref{eq:finSol}), 


\begin{equation}\label{eq:finSol}
y(t) = \frac{2 F_0}{m(w_0^2 - w^2)}\sin\left(\frac{\omega_0+\omega}{2}t\right)\sin\left(\frac{\omega_0-\omega}{2}t\right)\, .
\end{equation}

So what might this solution for (\ref{eq:forcedNDT}) look like? Let us examine  the two time dependent terms, ${\rm Term}_1 = \sin\left(\frac{\omega_0+\omega}{2}t\right)$ and ${\rm Term}_2 = \sin\left(\frac{\omega_0-\omega}{2}t\right)$, in the solution. Assuming both $\omega$ and $\omega_0$ are positive numbers, then ${\rm Term}_1$ has a higher frequency  than ${\rm Term}_2$. We note  that ${\rm Term}_1$ has a high frequency $\left(\frac{\omega_0+\omega}{2\cdot 2 \pi}\right)$ Hz while ${\rm Term}_2$ has a low frequency $\left(\frac{\omega_0-\omega}{2\cdot 2 \pi}\right)$ Hz. We say that frequency $\left(\frac{\omega_0+\omega}{2\cdot 2 \pi}\right)$  is {\it modulated at\/} $\left(\frac{\omega_0-\omega}{2\cdot 2 \pi}\right)$, noting that the low frequency term forms an envelope for the higher frequency term. We demonstrate this in Figure~\ref{fig:4-50-Freq}(d).

\begin{figure}[h]

\begin{center}
\begin{minipage}[t]{7 true cm}
\begin{center}
\includegraphics[width=7cm]{4-50-Freq1.eps}
\parbox{6 true cm}{(a) $\omega = 2 \pi\cdot 10$ or 10 Hz and $\omega_0 = 2 \pi\cdot 4$ or 4 Hz  over the interval $[0,10]$.}
\end{center}
\end{minipage}
\ \hspace*{.3 true cm} \
\begin{minipage}[t]{7 true cm}
\begin{center}
\includegraphics[width=7cm]{4-50-Freq2.eps}
\parbox{7 true cm}{(b) $\omega = 2 \pi\cdot 6$ or 6 Hz and $\omega_0 = 2 \pi\cdot 4$ or 4 Hz over the interval $[0,10]$.}
\end{center}
\end{minipage}

\begin{minipage}[t]{7 true cm}
\begin{center}
\includegraphics[width=7cm]{4-50-Freq3.eps}
\parbox{6 true cm}{(c) $\omega = 2 \pi \cdot 4.5$ or 4.5 Hz and $\omega_0 = 2 \pi\cdot  4$ or 4 Hz over the interval $[0,10]$.}
\end{center}
\end{minipage}
\ \hspace*{.3 true cm} \
\begin{minipage}[t]{7 true cm}
\begin{center}
\includegraphics[width=7cm]{4-50-Freq4.eps}
\parbox{7 true cm}{(d) $\omega = 2 \pi\cdot 4.1$ or 4.1 Hz and $\omega_0 = 2 \pi\cdot 4$ or 4 Hz over the interval $[0,40]$.}
\end{center}
\end{minipage}


\parbox{9.9 true cm}{\caption{\label{fig:4-50-Freq} Plots of $y(t) = \frac{2 F_0}{m(w_0^2 - w^2)}\sin\left(\frac{\omega_0+\omega}{2}t\right)\sin\left(\frac{\omega_0-\omega}{2}t\right)$, the solution for differential Equation~(\ref{eq:forcedNDT}), for various values of $\omega$ and $\omega_0$.}}
\end{center}

\end{figure}

In Figure~\ref{fig:4-50-Freq} we note that as the driver frequency, $\omega$, gets closer to the natural frequency, $\omega_0$, that the envelope frequency  of ${\rm Term}_2$, i.e. $\left(\frac{\omega_0-\omega}{2\cdot 2 \pi}\right)$  becomes more pronounced and sinusoidal, while the high frequency term  of ${\rm Term}_1$, i.e.~$\left(\frac{\omega_0+\omega}{2\cdot 2 \pi}\right)$ gets ever higher in frequency. When the driver frequency, $\omega$, gets close to the natural frequency $\omega_0$ we say the system displays {\it beats\/}, a sort of undulation in amplitude (${\rm Term}_2$) of a high frequency tone (${\rm Term}_1$).

\begin{itemize}
\item [a)] We can play the sound of beats in {\it Mathematica\/}.  If the frequencies in our system are in the interval $[20, 20000]$ Hz, then humans can detect the corresponding sound. Consider (\ref{eq:forcedNDT}) with $\omega_0 = 2 \pi \cdot 440$ (440 Hz - the tone of the A note above ``middle" C note on the piano keyboard) and  $\omega = 2 \pi \cdot 480$ (480 Hz).  Using {\tt DSolve} in {\it Mathematica\/}, solve (\ref{eq:forcedNDT}). Then, instead of plotting the obtained solution, use {\it Mathematica\/}'s {\tt Play} command,  which has the same syntax as {\it Plot}, to play the sound of our function solution to   (\ref{eq:forcedNDT}). Here the time interval offered,  $\{t, t_{\rm start}, t_{\rm stop} \}$, refers to the playing time of the sound. Be sure your sound is on, but not too loud and not too long (say, $t_{\rm start} = 0$ s and $t_{\rm start} = 2$ s), lest you wake a slumbering roommate who packed it in earlier, or a classmate sitting next to you, or the librarian in the reference room where you do your homework!  You should recognize this note as the ring tone \cite{TopBitS2010} in the telephone.
\item [b)] Now you try various combinations of $\omega_0$ and $\omega$ (close to each other) to see what beats you can produce. Certain combinations evoke the vibrato or tremolo on a ``weeping" or sad organ background as the heroine  is distressed in ``olde timey'' films. Have you ever heard the annoying signal on the radio generated by $\omega_0 = 2 \pi\cdot 2000$ and $\omega = 2 \pi\cdot 2200$? What is this signal used for?
 
\item [c)]  We massage the solution (\ref{eq:finSol})  to (\ref{eq:forcedNDT}) one more time by dividing and multiplying by $\left(\frac{\omega_0-\omega}{2}\right)$ to obtain (\ref{eq:finSolMod}):


\begin{equation}\label{eq:finSolMod}
y(t) = \frac{ 2 F_0}{m(w_0 + w)}\sin\left(\frac{\omega_0+\omega}{2}t\right)\frac{\sin\left(\frac{\omega_0-\omega}{2}t\right)}{\frac{\omega_0-\omega}{2}} \, .
\end{equation}

Now we were interested in what happened as our driver frequency, $\omega$, got close to our natural frequency, $\omega_0$ and we saw (and heard!) beats in our solution.  Well, what happens as $\omega$ gets very close, arbitrarily close,  $\omega_0$. Indeed, what happens as $\omega \to \omega_0$? In this case the term

$$\frac{\sin\left(\frac{\omega_0-\omega}{2}t\right)}{\frac{\omega_0-\omega}{2}}$$ 
in (\ref{eq:finSolMod}) approaches $t$. We can  obtain this using L'H\^ opital's Rule or using a general {\tt Limit} command in {\it Mathematica\/} to see that {\tt Limit[Sin[a t]/a, a -> 0] = t}.  Thus our solution  (\ref{eq:finSolMod}) to the  (\ref{eq:forcedNDT}) becomes in the limit  (\ref{eq:resonance}) as $\omega \to \omega_0$.

\begin{equation}\label{eq:resonance}
y(t) = \frac{ F_0}{m\cdot \omega_0} t \sin( \omega_0 t) \, .
\end{equation}

Unlike the situation which brought about beats, namely a low frequency envelop modulating a high frequency function, now we have an oscillation with a frequency $\omega_0$ -- that of the system and the driver, since they are the same, but here the solution is  multiplied by $t$. Thus, as time increases this solution gets bigger and bigger, wildly oscillating without end. We call this {\it resonance\/} and for the most part engineers and scientists are not happy when this happens.

\item [d)]Now you try various combinations of $\omega_0$ and $\omega$  to see what resonance you can produce.

\item [e)] Drive (\ref{eq:forcedNDT}) (use $m = 1$ and $F_0=1$) with a driver equal to the natural frequency, say $\omega = \omega_0 = 2 \pi \cdot 440$. Solve the differential equation, plot the solution over $[0, 0.1]$ s and then $[0,4]$ s. What do you see?  Explain what is happening.  Now plot/play the solution in the interval $[0,8]$ s, BUT if you play the solution be sure you are away from those you would not want to disturb!!! Why?
\item [f)]  Given (\ref{eq:forcedND}) or its altered form (\ref{eq:solved}) suppose we had a driver of the form 

\begin{equation}\label{eq:Unknown}
F(t) = F_0 \cos(\alpha t) + G_0 \cos(\beta t)\, ,
\end{equation}
 but did not know what $\alpha$ or $\beta$ actually were. Describe how you could change $w_0$ so you could ``see'' a solution which would tell you, first what $\alpha$ was and then what $\beta$ was.  You may assume it is easy, perhaps as easy as a radio dial, to alter $\omega_0$. Indeed, engineers design and study a simple electrical circuit with resistors, inductors, and capacitors in which the current going through the circuit is modeled as a differential equation and changing the dial (the capacitor) picks out the signals from the ether, much as your changing $\omega_0$ will help you determine $\alpha$ and $\beta$.  In the latter case we call it tuning your radio dial!
\end{itemize}





\teacher{
\section*{NOTES FOR TEACHERS}

We define and illustrate the notions of resonance and beats in driven undamped spring mass or linear second order differential equation. 

We offer explanations and plots in support to show what the phenomenon known as beats means and relate it to sound, in particular the telephone ringer.
We show how the mathematical expression for resonance is obtained through applications of trigonometry and algebra.






\begin{thebibliography}{99}

\bibitem{TopBitS2010} TopBitS.com. 2010. Frequencies of the Telephone Tone. \href{http://www.topbits.com/frequencies-of-the-telephone-tones.html}{http://www.topbits.com/frequencies-of-the-telephone-tones.html}. Accessed 8 June 2010.
\end{thebibliography}
}





\end{document}





%\bibitem{Gause1934} Gause, G.~F. 1971. {\it The Struggle for Existence}. New York:  Dover Publications, Inc. First published in 1934 by The Williams \& Wilkins Company and available completely on the world wide web at \url{http://www.ggause.com/Contgau.htm}. Accessed  31 March 2008.


