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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\TUNED MASS DAMPERS - PART II}





\markboth{Tuned Mass Dampers - Part II}{Tuned Mass Dampers - Part II}






\author{
\begin{minipage}[t]{6 true cm}
{Keith Landry\\
Georgia Southern University\\
Statesboro GA  30458 USA\\
}
\end{minipage}
\ \
\begin{minipage}[t]{6 true cm}
{Brian Winkel, 
Director SIMIODE\\ 
Chardon OH 44024 USA\\
BrianWinkel@SIMIODE.org}
\end{minipage}
}

\makeStitlePDFLaTex



\begin{abstract}
We offer students an opportunity to build mathematical models to mitigate dangerous swaying in structures using structural improvements called Tuned Mass Dampers (TMD).  We model the motion of the original structure as a spring-mass-dashpot system (one degree of freedom) with stiffness replacing the spring constant subject to internal  resistance to sway. We add another, smaller spring-mass-dashpot system whose frequency is tuned to reduce the overall oscillation of the original structure and consider the effects of this two degree of freedom model. We foster analysis of the steady state solution and ask how a TMD responds to a range of driver frequencies, still working in the ideal model in which there is no resistance or damper term in either the original structure's spring model or in the TMD. Then we study the more realistic situation in which both the original mass and the damper mass experience resistance or damping. 
\end{abstract}


\section*{SCENARIO DESCRIPTION}

We refer the reader to Part I of this Modeling Scenario \cite{LandryWinkel2016S} \teacher{(and for teachers \cite{LandryWinkel2016T})} for background  material and rationale for engineers to use Tuned Mass Dampers. A very good explanation of just what a Tuned Mass Damper (TMD) is can be seen in the YouTube video at \cite{PE2016} where the right claim is that TMDs offer an example of elegance in engineering.  Basically, a TMD is a pair of coupled damped harmonic oscillators. The name comes from their use in stabilizing things like buildings  or bridges or machinery which can be forced to oscillate by the wind or other forces.

\section*{Modeling a Tuned Mass Damper}



We will continue (from Part I\cite{LandryWinkel2016S}) to study the oscillation of a mass structure.  We represent the stiffness of the original mass of a structure as depicted in Figure~\ref{fig:5-40-MITTwoMass} using a spring and the energy absorption from that mass using a damper. We  attach a spring and damper (TMD) to the original mass and explore how much this TMD can reduce vibrations due to external forces. Displacement of the mass will be measured by distance from an initial equilibrium position with a single degree-of-freedom. Displacement of the TMD mass will be measured by distance from an initial equilibrium position with a second degree-of-freedom about its own initial equilibrium position. 


Consider the following situation shown in  Figure~\ref{fig:5-40-MITTwoMass}.  A large mass  $m_1$  on a ``spring'' with spring constant $k_1$ and a resistance force or damper proportional to velocity of the mass with damping coefficient $c_1$ is coupled to a smaller mass  $m_2$ by a spring with spring constant  $k_2$ and damper with damping coefficient $c_2$.  We will assume $c_1\neq 0$ and $c_2\neq 0$, i.e. we consider the more realistic (than seen in Part I\cite{LandryWinkel2016S}) situation in which both original structure and damper experience damping.  

\begin{figure}[!h]
\centering
\includegraphics[width=6cm]{5-40-MITTwoMass.png}
\parbox{9.5 true cm}{\caption{\label{fig:5-40-MITTwoMass} Diagram of the TMD for developing full set of two degrees of freedom differential equation model \cite{MITpdf}.}}
\end{figure}

Recall in Part I \cite{LandryWinkel2016S} that we assumed  $c_1 =  0$ and $c_2 = 0$ so we had  no damping of the original structure or our attached damper. Thus our original structure  could experience resonance without a damper attached. Now in Part II  we first study the situation in which  the original structure has resistance to sway or damping, $c_1 \neq  0$, but still no attached TMD. We will see that with damping ($c_1 \neq  0$)  the displacement of the original structure reaches a maximum amplitude; however,  it does not go into resonance. This maximum amplitude could be dangerous and so we want to employ a TMD to reduce this possibly too high an amplitude of the structure's displacement.
Indeed, we want to confirm in this case that by tuning the values of $m_2$ and $k_2$ the maximum amplitude of $m_1$'s oscillations can be lowered considerably.  However, we first offer a review of the undamped structure.  

\subsection*{Structure Without Resistance or Damping - Review from Part I}
In this case with no resistance ($c_1=0$ and $c_2=0$) when we apply an external load or driver force, $f(t)$, we have the following single degree of freedom ($y(t)$) differential equation model for the displacement of our mass from its static equilibrium, i.e.~a spring-mass (no dashpot) system:
\begin{equation}\label{eq:sm}
m y''(t) + k y(t) = f(t)\, , \quad y(0) = y_0\, , \quad  y'(0) = v_0 .
\end{equation}

The natural frequency of mass $m_1$ can be obtained from the solution of the homogeneous portion, (\ref{eq:homo}), of our differential equation (\ref{eq:sm}),

\begin{equation}\label{eq:homo}
m y''(t) + k y(t) = 0,   \quad y(0) = y_0\, , \quad  y'(0) = v_0 .
\end{equation}
From (\ref{eq:hs}) we see that  the natural frequency of the  undriven or homogeneous solution of (\ref{eq:homo})  is $\omega_0=\sqrt{\frac{k}{m}}$ while the complete homogeneous solution to (\ref{eq:sm}) is:

\begin{equation}\label{eq:hs}
y(t) =
 \frac{
v_0\sqrt{m}  \sin
   \left(\sqrt{\frac{k}{m}}
   t \right)+y_0 \sqrt{k}
   \cos
   \left(\sqrt{\frac{k}{m}}
   t\right)
}{\sqrt{k}} \, .  
\end{equation}

Recall what happens if the driver function $f(t)$ has the same frequency as the natural  frequency, $\omega_0$,  of the spring-mass system - RESONANCE and trouble. To see this we divide both sides of (\ref{eq:sm})  by $m$ and rearrange the resulting equation using $\omega_0^2 = \frac{k}{m}$ to obtain (\ref{eq:smdiv}):

\begin{equation}\label{eq:smdiv}
 y''(t) + \omega_0^2 y(t) = \frac{1}{m}f(t)\, , \quad y(0) = y_0\, , \quad  y'(0) = v_0\,  .
\end{equation}

Now if $f(t) = \sin(\omega t)$ and this driver has the same natural  frequency as our mass system, i.e.~$\omega = \omega_0$,  then our solution of (\ref{eq:smdiv}) is seen in (\ref{eq:res}) with a plot offered in Figure~\ref{fig:5-40-ResonancePlot}.

\begin{equation}\label{eq:res}
y(t) = -\frac{t \cos ( \omega_0 t)}{2
   m \omega_0 }+ \frac{\sin ( \omega_0 t)}{2 m \omega_0
   ^2}-\frac{\sin ( \omega_0 t) \cos
   ^2( \omega_0 t)}{2 m \omega_0
   ^2}+\frac{\sin (2  \omega_0 t)
   \cos ( \omega_0 t)}{4 m \omega_0
   ^2}+\frac{v_0 \sin
   ( \omega_0 t)}{\omega_0 }+y_0
   \cos (\omega_0 t )\, .
\end{equation}


\begin{figure}[!h]
\centering
\includegraphics[width=7cm]{5-40-ResonancePlot}
\parbox{11.5 true cm}{\caption{\label{fig:5-40-ResonancePlot} Plot of solution of (\ref{eq:smdiv})  with values, $\omega_0 =  1$, for the natural frequency of the system and the driver as well, and $y_0 =2, v_0= 0$, and $m =1$ . In this case  $f(t) = \sin(\omega t)$ is the driver function.}}

\vspace*{-.5 true cm}
\end{figure}

The key term in this solution is the very first term. All the other terms are of period $\frac{2 \pi}{m}$ with finite amplitudes. The first term $-\frac{t \cos (
   \omega_0 t)}{2 \omega_0 }$ grows without bound and accordingly we refer to this solution phenomenon as {\it resonance}. For the following values, $\omega_0 =  1, y_0 =2, v_0= 0$, and $m =1$ we plot our solution (\ref{eq:res}) over the time interval $[0, 100]$ in  Figure~\ref{fig:5-40-ResonancePlot} and see   the solution  grows indefinitely; this is resonance.










\subsection*{Structure With Resistance or Damping}

In Part I \cite{LandryWinkel2016S} and in the previous section we analyzed the case where there is no resistance or damping in either the original structure or the damper, i.e. $c_1 =c_2=0$, and we found that if we tuned the damper by letting its natural frequency be the same as the natural frequency of the original structure we would stop the resonance and effectively reduce the oscillation of the original structure to 0 when a driving force with the same frequency as the natural frequency of the structure occurred. This means if we prescribe $k_2$ and $m_2$ so that $\sqrt{\frac{k_1}{m_1}} = \sqrt{\frac{k_2}{m_2}}$ in the model (\ref{eq:goveqg}) for the original structure's displacement, $x_1(t)$, and the TMD's displacement, $x_2(t)$ we will have no resonance, indeed, no motion of the original structure when it is driven  by an external force $p_0\cos(\omega t)$ with $\omega = \sqrt{\frac{k_1}{m_1}} = \sqrt{\frac{k_2}{m_2}}$.

\vspace*{-.6 true cm}
\begin{eqnarray}\label{eq:goveqg} \nonumber
m_1 x_1''(t) &=& - k_1 x_1(t)   - k_2 (x_1(t) - x_2(t))  + p_0\cos(\omega t)\\ 
m_2 x_2''(t) &=& - k_2(x_2(t) - x_1(t)) 
\end{eqnarray}

 
We now permit resistance or a damping force on our original mass, $m_1$. So we apply a nonzero resistance or damping force $c_1 x_1'(t)$ to the forces acting on mass $m_1$, but allow no TMD to be applied.  That is,  in this case we have  resistance ($c_1\neq 0$ and $c_2=0$) and no attached damper when we apply an external driver force, $f(t)$. Indeed, we have just the following single degree of freedom ($y(t) = x_1(t)$) differential equation model for the displacement of our mass, $m_1$, from its static equilibrium:
\begin{equation}\label{eq:smr}
m y''(t) +  c y'(t) + k y(t) = f(t)\, , \quad y(0) = y_0\, , \quad  y'(0) = v_0 .
\end{equation}


If we use $f(t) = p_0 \cos(w t)$ as a driver then the closed form solution of (\ref{eq:smr}) is given by (\ref{eq:solr}):

\tiny
\begin{eqnarray}\label{eq:solr}\nonumber
y(t) &=& \frac{m_1 p_0 \omega ^2
   \exp \left(\frac{t
   \sqrt{c_1^2-4 k_1
   m_1}}{m_1}-\frac{t
   \left(\sqrt{c_1^2-4 k_1
   m_1}+c_1\right)}{2
   m_1}\right)}{2
   \left(c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega ^2+m_1^2
   \omega ^4\right)}-
     %
     \frac{c_1
   m_1 p_0 \omega ^2 \exp
   \left(\frac{t \sqrt{c_1^2-4
   k_1
   m_1}}{m_1}-\frac{t
   \left(\sqrt{c_1^2-4 k_1
   m_1}+c_1\right)}{2
   m_1}\right)}{2
   \sqrt{c_1^2-4 k_1
   m_1} \left(c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega ^2+m_1^2
   \omega ^4\right)}\\ \nonumber
    %
   && -\frac{k_1
   p_0 \exp \left(\frac{t
   \sqrt{c_1^2-4 k_1
   m_1}}{m_1}-\frac{t
   \left(\sqrt{c_1^2-4 k_1
   m_1}+c_1\right)}{2
   m_1}\right)}{2
   \left(c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega ^2+m_1^2
   \omega ^4\right)}-
     %
 \frac{c_1
   k_1 p_0 \exp
   \left(\frac{t \sqrt{c_1^2-4
   k_1
   m_1}}{m_1}-
    \frac{t
   \left(\sqrt{c_1^2-4 k_1
   m_1}+c_1\right)}{2
   m_1}\right)}{2
   \sqrt{c_1^2-4 k_1
   m_1} \left(c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega ^2+m_1^2
   \omega ^4\right)} \\ \nonumber
    %
   &&+
   \frac{m_1
   p_0 \omega ^2 e^{-\frac{t
   \left(\sqrt{c_1^2-4 k_1
   m_1}+c_1\right)}{2
   m_1}}}{2 \left(c_1^2
   \omega ^2+k_1^2-2 k_1
   m_1 \omega ^2+m_1^2
   \omega ^4\right)}  
   %
  + \frac{c_1
   m_1 p_0 \omega ^2
   e^{-\frac{t
   \left(\sqrt{c_1^2-4 k_1
   m_1}+c_1\right)}{2
   m_1}}}{2 \sqrt{c_1^2-4
   k_1 m_1}
   \left(c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega ^2+m_1^2
   \omega ^4\right)} 
    %
-\frac{k_1
   p_0 e^{-\frac{t
   \left(\sqrt{c_1^2-4 k_1
   m_1}+c_1\right)}{2
   m_1}}}{2 \left(c_1^2
   \omega ^2+k_1^2-2 k_1
   m_1 \omega ^2+m_1^2
   \omega ^4\right)}\\ \nonumber
    &&+
     \frac{c_1
   k_1 p_0 e^{-\frac{t
   \left(\sqrt{c_1^2-4 k_1
   m_1}+c_1\right)}{2
   m_1}}}{2 \sqrt{c_1^2-4
   k_1 m_1}
   \left(c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega ^2+m_1^2
   \omega ^4\right)}+
     %
     \frac{c_1
   p_0 \omega  \sin (t \omega
   )}{c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega ^2+m_1^2
   \omega ^4}-\frac{m_1
   p_0 \omega ^2 \cos (t \omega
   )}{c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega ^2+m_1^2
   \omega ^4}\\ 
    &&+
     \frac{k_1
   p_0 \cos (t \omega
   )}{c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega ^2+m_1^2
   \omega ^4}\, .
   \end{eqnarray}
   \normalsize
   
   
   We will look for the steady state response of mass $m_1$ to the driver and thus ignore the decaying exponential terms of (\ref{eq:solr}). This means we are interested only in the pure cosine and sine terms.  We use the trigonometric identity  (\ref{eq:trigID}) which is often used by engineers to combine two periodic functions with the same frequency into one phase-shifted periodic function.
   
   \begin{equation}\label{eq:trigID}
   A \cos(x) + B \sin(x) = \sqrt{A^2 + B^2} \sin\left(x + \arctan\left(\frac{A}{B}\right)\right)
   \end{equation}
   for  $B \neq0$ and $-\frac{\pi}{2} <  \arctan\left(\frac{A}{B}\right) <\frac{\pi}{2}$ to determine the resulting amplitude, $ \sqrt{A^2 + B^2} $ of the steady state solution here, i.e. when $x = w t$. First we determine the coefficients of $\cos(\omega t)$ ($A$) and of $\sin(\omega t)$ ($B$) from which we can compute the magnitude, $ \sqrt{A^2 + B^2}$, of the steady state solution.
   
   We find
   
   \begin{equation}\label{eq:A}
   A = \frac{k_1
   p_0}{c_1^2
   \omega ^2+k_1^2-2
   k_1 m_1 \omega
   ^2+m_1^2 \omega
   ^4}-\frac{m_1
   p_0 \omega
   ^2}{c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega
   ^2+m_1^2 \omega ^4}
   \end{equation}
   and
      \begin{equation}\label{eq:B}
   B = \frac{c_1 p_0
   \omega }{c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega
   ^2+m_1^2 \omega ^4}
   \end{equation}
   from which we (with algebraic simplification) can compute the amplitude of the steady state response of mass $m_1$ to the driver function $f(t) = p_0 \cos(\omega t)$, after simplifying: 
   \begin{equation}\label{eq:Amp}
   {\rm Amp}(\omega) = \sqrt{\frac{p_0^2}{
   c_1^2 \omega
   ^2+k_1^2-2 k_1
   m_1 \omega
   ^2+m_1^2 \omega ^4}}
   \end{equation}
    
   From ${\rm Amp}(\omega)$, the amplitude of the steady state response of mass $m_1$, we can determine the frequency, $\omega_M$,  at which the displacement or amplitude of position $x_1(t)$ is greatest. We shall refer to this as the {\it maximum frequency response\/} of mass $m_1$. 
   
   We show a typical maximum frequency response curve in Figure~\ref{fig:5-40-ResponseAMP}.
   
   \begin{figure}[!h]
\centering
\includegraphics[width=7cm]{5-40-ResponseAMP}
\parbox{11.5 true cm}{\caption{\label{fig:5-40-ResponseAMP} Maximum frequency response plot for the steady state  solution of (\ref{eq:smr}) with driver $f(t) = p_0 \cos(\omega t)$  for  $p_0=100$, $y_0 =0$, and  $y'(0) = v_0= 0$, while $m =10$ , $k = 90$, and $c=5$.}}

\vspace*{-.5 true cm}
\end{figure}

   
 
The maximum frequency response of mass $m_1$ can be obtained by differentiating ${\rm Am  p}(\omega)$ with respect to $\omega$ and setting that derivative equal to 0 to find the maximum.  With the numbers offered in Figure~\ref{fig:5-40-ResponseAMP} using this optimization process we find that   the maximum amplitude response of 6.68994 occurs at $\omega =  2.97909$.

So while we do not have resonance, which we can obtain when $c = 0$ and the driver frequency is the same as the structure's natural frequency, we do have a frequency, $\omega =  2.97909$,  which is close to that of the natural frequency of the undamped structure, namely $\omega = 3$,  wherein if the structure possessed damping ($c_1 \neq 0$) there would be a maximum frequency response which could be unacceptable. Hence we seek the characteristics of a TMD which will mitigate this maximum frequency response in the more realistic case.
% first where only the structure has damping and then where both the structure and the damper have damping.



\section*{Large Frequency Amplitude Responses Spell Trouble}

In general, engineers do not like resonance. Neither  do they  like  a large amplitude response to possible   driver frequencies, as it means things could be getting out of control, going beyond tolerance levels, and possibly causing havoc. For suppose (\ref{eq:smr}) were a model of a tall structure of mass $m$, $k$ represented a restorative force or stiffness in the structure, and $c$ represented a resistance or damping in the sway of the structure due to friction or internal resistance to motion in the structure. If $y(t)$ is the horizontal displacement of the corner of the structure from its static equilibrium  due to a force, $f(t)$, from seismic tremors  or wind then a high amplitude response to these forces would mean that the structure would swing broadly and could eventually break.  Thus engineers  design in light of amplitude frequency responses to typical driver frequencies. In particular, if the structure has certain properties, namely, $m_1$, $k_1$, and $c_1$,  then the structure needs to be able to counter (better that than just sustain) the force $f(t) = p_0 \cos(w t)$ and here is where  a TMD can  mitigate the maximum amplitude frequency response. So how do we tune the TMD in this situation?


\subsection*{Activities}



  \subsubsection*{\hspace*{.6 true cm} Activity 1}

Using Newton's Second Law, which says that the mass times the acceleration of that mass is equal to the sum of all external forces acting on the mass, and  Figure~\ref{fig:5-40-MITTwoMass} as a Free Body Diagram source, show that the governing equations for the motions of the structure and the damper  are given by (\ref{eq:goveq}). Hint:  For consistency all downward forces, positions, velocities, and accelerations are positive.


\vspace*{-.6 true cm}
\begin{eqnarray}\label{eq:goveq} \nonumber
m_1 x_1''(t) &=& - k_1 x_1(t) - c_1 x_1'(t)  - k_2 (x_1(t) - x_2(t)) - c_2(x_1'(t) - x_2'(t))  + p_0\cos(\omega t)\\ 
m_2 x_2''(t) &=& - k_2(x_2(t) - x_1(t)) - c_2 (x_2'(t) - x_1'(t))
\end{eqnarray}

In our study here we will now presume an ``ideal'' structure with no resistance or damping coefficient, i.e.~$c_1 \neq 0$, and the same for the damper, i.e.~$c_2 \neq 0$. As we can see there is a driving force applied to the structure of $f(t) = p_0 \cos(\omega t)$. Our interest will be in seeing if we can tune the added mass damper to reduce or eliminate the possibility of large maximum amplitude frequency responses when the initial structure is driven by a force with a frequency equal to that of the natural frequency of the initial mass system.

\subsubsection*{\hspace*{.6 true cm} Activity 2}

\begin{itemize}
\item [a)] Determine the form of (\ref{eq:goveq}) when $c_1 = 5 $  and $c_2 = 0.5$.
\item [b)] In the resulting system use $m_1= 10$ and $k_1= 90$ with $y(0) = 0$ and $y'(0) = 0$, while  $f(t) = 10 \cos(3 t)$ and set $m_2 = 0$ with $k_2 = 0$, i.e.~withdraw the second mass system.  Explain the physical  significance and impact results of the numbers  10, 90, and 3 in the above sentence in terms of the motion of the initial oscillator.
\item [c)] Solve the resulting single differential equation for the motion of mass $m_1$ over a time interval of 20 units and plot the displacement of that mass, $x_1(t)$, over that time interval. Explain what you see.
\item [d)]  In (\ref{eq:goveq}) use the values of  $m_1=  10$ and $k_1= 90$ with $y(0) = 0$ and $y'(0) = 0$ and $f(t) = 10 \cos(3 t)$ and set $m_2 = 1$ with varying values of $k_2$. Keep  $c_1 = 5$  and $c_2 = 0.5$. What do you observe in the maximum amplitude of the initial mass $m_1$ as one changes $k_2$? Defend your observation with data or plot.
\item [e)] From (d) what is the ``best" value of $k_2$?  Be sure you define the word ``best."
\item [f)] For the best value of $k_2$ determine the maximum amplitude displacement for mass $m_1$, $x_1(t)$ over a range of frequencies.
\end{itemize}

\subsubsection*{\hspace*{.6 true cm} Activity 3}

We consider another, comparable configuration as Activity 2 for practice. Notice that mass $m_2$ is only 1\% of mass $m_1$ which is quite realistice in structural design usingTMDs.
\begin{itemize}
\item [a)] Determine the form of (\ref{eq:goveq}) when $c_1 = 5$  and $c_2 = 0.5$.
\item [b)] In the resulting system use $m_1=  10$ and $k_1= 90$ with $y(0) = 0$ and $y'(0) = 0$ while $f(t) = 10 \cos(3 t)$ and set $m_2 = 0.1$ with $k_2 = 0$, i.e.~withdraw the second mass system.  Explain the physical  significance and impact results of the numbers  10 , 90, and 3 in the above sentence in terms of the motion of the initial oscillator.
\item [c)] Solve the resulting single differential equation for the motion of mass $m_1$ over a time interval of 20 units and plot  the displacement of that mass, $x_1(t)$, over that time interval. Explain what you see.
\item [d)]  In (\ref{eq:goveq}) use the values of  $m_1=  10$ and $k_1= 90$ with $y(0) = 0$ and $y'(0) = 0$ and $f(t) = 5 \cos(3 t)$ and set $m_2 = 0.1$ with varying values of $k_2$. Keep  $c_1 = 5$  and $c_2 = 0.5$. What do you observe in the maximum amplitude of the initial mass as one changes $k_2$? Defend your observation with data or plot.
\item [e)] From (d) what is the ``best" value of $k_2$?  Be sure you define the word ``best."
\item [f)] For the best value of $k_2$ determine the maximum amplitude displacement for mass $m_1$, $x_1(t)$ over a range of frequencies.
\end{itemize}

\subsubsection*{\hspace*{.6 true cm} Activity 4}



From the introductory  material for this scenario offered above and the analyses in Activities 2 and 3 offer a description of how to build a TMD to stop the resonance phenomena in the case of    (\ref{eq:goveq1}) where there is  damping, i.e.~$c_1 \neq 0$ and $c_2 \neq 0$, 

\begin{eqnarray}\label{eq:goveq1} \nonumber
m_1 x_1''(t) &=& - k_1 x_1(t) - c_1 x_1'(t)  - k_2 (x_1(t) - x_2(t)) - c_2(x_1'(t) - x_2'(t))  + p_0\cos(\omega t)\\ 
m_2 x_2''(t) &=& - k_2(x_2(t) - x_1(t)) - c_2 (x_2'(t) - x_1'(t))\, .
\end{eqnarray}



\begin{thebibliography}{99}

\bibitem{BG2016} Boston Globe. 2016. The Tuned Mass Damper in the John Hancock Buidling. \url{http://cache.boston.com/news/special/audio\_slideshows/hancock/mass\_tune\_damper/}. Accessed 26 August 2016.

\bibitem{Scary2013} Coomedy vide. 2013.  SCARY Japan earthquake 90 buildings sway.  \url{https://www.youtube.com/} \linebreak {\tt watch?v=iGtRko8y4Fo}. Accessed 18 August 2016. 


\bibitem{DEICON2016}  Dynamics \& Controls.  Tuned Mass Dampers.  \url{http://www.deicon.com/tuned-mass-dampers/}. Accessed 18 August 2016. 

\bibitem{Goldberger1988} Goldberger, P. 1988. Architecture View; A Novel Design And Its Rescue From Near Disaster. {\it New York Times}.  24 April 1988.   \url{http://www.nytimes.com/1988/04/24/arts/architecture-view-a-novel-design-and-its-rescue-from-near-disaster.html?pagewanted=all}. Accessed 26 August 2016.

\bibitem{GoogleTMD2016}  Google. 2016. Tuned Mass Damper.  \url{https://www.google.com/?gws\_rd=ssl\#q=Tuned+Mass+Dampers}. Accessed 18 August 2016.

\bibitem{GCSW2013} Grand Canyon Skywalk. 2013. \url{https://www.youtube.com/watch?v=JOuoEXfS6ZA}. Accessed 27 August 2016.

\bibitem{Irvine2002} Irvine, Tom. 2002. The Citicorp Building Tuned Mass Damper.  {\it Vibration.com Newsletter}. January. pp.~2-6. {\tt http://www.vibrationdata.com/Newsletters/January2002\_NL.pdf}. Accessed 17 August 2016.

\bibitem{Kelzad2007} KelzaD. 2007. Suspension bridge shaken by an earthquake in Chile.  \url{https://www.youtube.com/watch?v=t9kR9T9wZsM}. Accessed 18 August 2016. 

\teacher{
\bibitem{LandryWinkel2008} Landry, K. and B. Winkel. 2008. Peak-Frequency Responses and Tuned Mass Dampers - Exciting Applications of Systems of Differential Equations. {\it The UMAP Journal\/}. 29(1): 15-30.
}

\bibitem{LandryWinkel2016S} Landry, K. and B. Winkel. 2016. 5-40-S-TunedMassDampers-StudentVersion-I . 

\teacher{
\bibitem{LandryWinkel2016T} Landry, K. and B. Winkel. 2016. 5-40-T-TunedMassDampers-TeacherVersion-I . }



\bibitem{Ming-Lai1996} Ming-Lai, Lai. 1996.  Tuned Mass Damper.  Washington DC: United States Patent Office.  \url{www.google.com/patents/US5558191}.Accessed 18 August 2016.

\bibitem{MITpdf} MIT. 2016. Tuned Mass Damper.  \url{http://web.mit.edu/jorloff/www/jmoapplets/secondorder/tunedmassdamper.pdf\#page=3\&zoom=110,35,532}. Accessed 18 August 2016.

\bibitem{Morgenstern1995} Morgenstern, Joe. 1995. The Fifty-Nine Story Crisis. {\it The New Yorker\/}. May 29: 45-53. 

\bibitem{Motioneering2006} Motioneering Inc. 2006. Grand Canyon Skywalk and Custom TMDs. {\it At the Moment: Motion Control News and Views from Motioneering\/}. 5: 1-2.

\bibitem{NISEE2016} National Information Service for Earthquake Engineering (NISEE). 2016.  The Earthquake Engineering Online Archive.   University of California at  Berkeley:  \url{https://nisee.berkeley.edu/}.   Accessed 18 August 2016.
 
\bibitem{Pfisterer2009} Pfisterer Sefag AG. 2009. Product Catalogue - Stickbridge Dampers for effective control of conductor vibarations.  \url{http://www.pfisterer.com/fileadmin/pfisterer/downloads\_en/Stockbridge\_Dampers-PI-EN.pdf}. Accessed 18 August 2016.
 
\bibitem{PE2016} Practical Engineering. 2016. What is a Tuned Mass Damper. \url{https://www.youtube.com/watch?v=f1U4SAgy60c}.

\bibitem{RWDI2016} RWDI:  Consulting Engineers \& Scientists. 2016. Minimising Flopor Vibrations. \url{http://www.rwdimedia.com/uploads/1/1/2/7/11270509/rwdi\_floor\_vibration.pdf }

\bibitem{Taipei101} Taipei 101. Tallest Buildings. \url{ https://www.youtube.com/watch?v=B031ebdzRvc}. Accessed 26 August 2016.

\bibitem{TNB1940} Tacoma Narrows Bridge Collapse ``Gallopin' Gertie''.  2016.   \url{https://www.youtube.com/watch?v=j-zczJXSxnw}.  Accessed 18 August 2016.

\bibitem{Tonev2010} Tonev, Manuil.  2010. Effect of building swaying from the wind.  \url{https://www.youtube.com/watch?v=xIuZQ7nO1pw}.  Accessed 18 August 2016.


\bibitem{Utting2015} Utting, Richard. 2015. Barrel tuners - do they work?  \url{https://www.youtube.com/watch?v=Xq2G2MubmuE}. Accessed 18 August 2016. 


\bibitem{Vanhoenacker2015} Vanoenacker, Mark. 2015. What's that thing hanging from lampposts and powerline?   {\it Slate}.  \url{ http://www.slate.com/blogs/the\_eye/2015/07/31/what\_s\_that\_thing\_stockbridge\_dampers.html}.  Accessed 18 August 2016.

\bibitem{WRSH1973} Wagner, H., V. Ramamurti, R. V. R. Sastry, and K. Hartmann. 1973. Dynamics of stockbridge dampers. {\it Journal of Sound and Vibration\/}. 20(2): 207-220.



\bibitem{YouTubeTMD2016a} YouTube. 2016. Tuned Mass Dampers.  \url{ https://www.youtube.com/results?search\_query=Tuned+Mass+Dampers}. Accessed 18 August 2016. 




\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}



In Part I we  presented background and history concerning Tuned Mass Dampers and then restricted our study to the unrealistic case of no damping $c_1 = 0$ and $c_2 = 0$ in (\ref{eq:goveq}) (Part I) in both the initial mass structure and the TMD. This makes the mathematics more tractable and can give the student the sense of what has to be done to eliminate resonance in a structure.  Here in Part II we address the case in which there is resistance or damping in the original mass and the damper, i.e. $c_1 \neq 0$ and $c_2$.  We offer several opportunities (Activities 2 and 3) for students to play with numbers and render a conjecture in Activity 4 as to the key principle in TMD design in this case,  the natural frequency of the initial mass system and that of the TMD have to be the same, namely $\sqrt{\frac{k_1}{m_1}} = \sqrt{\frac{k_2}{m_2}}$.


We offer a Mathematica notebook 5-40-T-Mma-TMD-TeacherVersion-II and an accompanying  pdf version for those who do not read Mathematica files in which complete analyses and commentary on this material is offered. Here we discuss a number of additional issues one could raise with TMDs. For example, once the optimal parameters are determined to minimize the maximum amplitude response of mass $m_1$, $x1(t)$, one could explore response of the system, i.e.  the maximum amplitude response of mass $m_1$, $x1(t)$, over a range of values of $\omega$, for even though the structure is configured now to respond properly to the designated driver frequency it is worthwhile considering what happens with frequencies near and far from the frequency of original interest. 

\begin{figure}[!h]
\centering
\includegraphics[width=9cm]{5-40-FreqSpan}

\parbox{8 true cm}{\caption{\label{fig:5-40-FreqSpan} Plot of maximum amplitude response of mass $m_1$, $x_1(t)$, over range of driver frequencies, $\omega$.}}
\end{figure}

Figure~\ref{fig:5-40-FreqSpan} displays results of numerical work in which the mass of the TMD is 1\% of the mass of the structure, which is quite realistic. The plot indicates that there is a region over which the system response is satisfactory, i.e. the maximum amplitude response of mass $m_1$, $x1(t)$, is low. Further studies would show that this ``safe'' region would widen if we would devote more mass to the TMD. However, this comes at an economic expense since a larger mass $m_2$ means that more space in the structure would be taken up by the TMD and in the case the structure is a building this means less space for generating income. So there is a trade-off worth considering here.





In a published article, \cite{LandryWinkel2008}, the authors have given more insight into the mathematics of Tuned Mass Dampers.

}


\end{document}



\begin{eqnarray}\label{eq:Act4} \nonumber
m_1 x_1''(t) &=& - k_1 x_1(t) - c_1 x_1'(t)  - k_2 (x_1(t) - x_2(t)) - c_2(x_1'(t) - x_2'(t))  + p_0\cos(\omega t)\\ 
m_2 x_2''(t) &=& - k_2(x_2(t) - x_1(t)) - c_2 (x_2'(t) - x_1'(t))
\end{eqnarray}