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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Bad Vibrations: Modeling a Building During an Earthquake\\Part II: With Damping}





\markboth{Building Vibrations}{Building Vibrations}

\author{Tracy Weyand\\
Rose-Hulman Institute of Technology\\
Terre Haute, IN 47803 USA\\
weyand@rose-hulman.edu
}
\makeStitlePDFLaTex


\begin{abstract}
This modeling scenario considers a one-story building as a simple structure; the roof is modeled as a single point mass (see Figure \ref{fig: Roof1}). Movement of the roof can be modeled similar to a mass-spring-damper system. In this activity, students will analyze motion of the roof under different damping and vibration forces. 

This is a continuation of Part I (no damping) \cite{Part1}; see the Notes for Teachers section for more details. If you are interested in having students analyze a multi-story building as a system of differential equations, see \cite{MarMcd99} for inspiration and SIMIODE Modeling Senarios \cite{W1} and \cite{W2}.
\end{abstract}



\section*{SCENARIO DESCRIPTION}

In Part I \cite{Part1}, we analyzed how ground movement can cause vibrations in a building. Sometimes these resulting vibrations can end up being too big, resulting in discomfort of the people occupying the building or, in the worst case scenario, the building collapsing. To avoid this, damping effects are designed into a structure to attempt to minimize vibrations. Your goal here is to determine how much friction/damping should be designed into a building to keep the roof from moving too far (which would result in the entire building collapsing) when it undergoes minor vibrations from a small earthquake.



\section*{ACTIVITY 1 -- NO VIBRATIONS}


The process by which free vibrations diminish is called \textbf{damping}. Energy of the vibrating system is released by various mechanisms (usually several at once). In a vibrating building, friction at connections and the formation of microcracks are just two possible ways energy is dissipated. It is extremely difficult (and at times impossible) to describe each mechanism individually. 


\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=.85\linewidth]{Roof1}
  \caption{Considering the roof as a point mass at the center}
  \label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=.65\linewidth]{Roof2}
  \caption{Displacement of the roof is measured by $x(t)$}
  \label{fig:sub2}
\end{subfigure}
\caption{Motion of the center of mass of the roof of a one-story building}
\label{fig: Roof1}
\end{figure}



\subsection*{PART A -- Viscous Damping}

Here we idealize the damping forces into a single term:
\[f_D = \gamma x'\]
where $\gamma$ is chosen so that the change of the vibration energy is equal to the change of energy caused by all the damping mechanisms. This is referred to as \textbf{equivalent viscous damping} and $\gamma$ is called the \textbf{viscous damping coefficient}. Considering this damping force also acting on our building, our model now is
\begin{equation}
\label{eq:DE2}
mx'' + \gamma x' + kx = 0, \qquad x(0)=x_0, \,\,\, x'(0)=x_1.
\end{equation}


\noindent\textbf{Problem 1.} What units should the constant $\gamma$ have for the differential equation \eqref{eq:DE2} above to make sense?\\ \textit{Note: you will need to be general here (for example ``time") since explicit measurements (for example ``seconds") were not given yet.}\\

\noindent\textbf{Problem 2.} Find all values of the damping constant $\gamma$ such that the system is underdamped. Recall that, in this case, the system will oscillate, but the amplitude will decrease over time.\\

%\fbox{\begin{minipage}{40em}
%\begin{center}
%{\large{\textbf{Energy}}}
%\end{center}

%The \textbf{kinetic energy} of a mass $m$ (at time $t$) is given by
%\[E_k(t) = \frac{\mbox{mass} \times \mbox{velocity}^2}{2} = \dfrac{1}{2}mv^2(t) = \dfrac{1}{2}m[x'(t)]^2.\]
%The \textbf{potential energy} of a mass $m$ (at time $t$) is defined as
%\[E_p(t) = \frac{\mbox{stiffness}\times\mbox{position}^2}{2} = \dfrac{1}{2}kx^2(t).\]
%The \textbf{total energy} of the mass (at time $t$) is
%\[E(t) = E_k(t) + E_p(t).\]
%\end{minipage}}
%\vspace{1cm}

\noindent\textbf{Problem 3.} Show that the total energy of the system is not conserved in this case (of an underdamped system). What is happening? Justify your conclusion.\\


\subsection*{Part B -- Coulomb Damping}

\noindent Since most structures are underdamped, friction devices are frequently installed in buildings to minimize oscillations. \textbf{Coulomb damping} is a type of (constant) mechanical damping caused by the friction of one object sliding across another. This damping force opposes motion, but \textit{is independent of velocity}. Consider the simple case of a block of wood sitting on a table and attached to a spring. In this system, the Coulomb damping force is the friction between the wood block and the table. See Figure \ref{fig: Coulomb} below.\\

\begin{figure}[h]
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=.8\linewidth]{Coulomb2}
  \caption{Motion in the positive direction ($x'(t)>0$)}
  \label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=.8\linewidth]{Coulomb1}
  \caption{Motion in the negative direction $x'(t)<0$)}
  \label{fig:sub2}
\end{subfigure}
\caption{We need to add arrows to show which direction the Coulomb damping force is acting in each case.}
\label{fig: Coulomb}
\vspace*{-.4 true cm}
\end{figure}


\noindent\textbf{Problem 4.} Explain why the model in \eqref{eq:DE2} is not valid in this situation. 
\vspace*{.1 true cm}

\noindent\textbf{Problem 5.} Assume that the Coulomb damping is a constant $F_0>0$ (independent of velocity) that acts in the opposite direction of motion. Finish drawing arrows in the diagram in Figure \ref{fig: Coulomb} to represent the direction of the Coulomb damping force in each case. Does the Coulomb damping force ($F_0$) act in the same or opposite direction as the spring force ($kx$) in each case? Use this information to write an initial-value problem to model this situation (where the only damping is Coulomb damping), taking the direction of the damping into account. \textit{You should find two differential equations here that depend on which way the motion is occurring.} 
\vspace*{.1 true cm}

\noindent\textbf{Problem 6.} Assume that motion is in the positive direction (i.e. $x'(t)>0$). Solve the initial-value problem from Problem 5.  
\vspace*{.1 true cm}

\noindent\textbf{Problem 7.} Show that the total energy is not conserved in this case. What is happening? Justify your conclusion.


\section*{ACTIVITY 2 -- VIBRATIONS}

If an earthquake occurs, the ground will be displaced some distance $x_G(t)$. Then that total lateral deflection of the roof is
\[x_T(t)=x(t)+x_G(t)\]
where $x(t)$ is the displacement caused by vibrations of the building and $x_G(t)$ is the movement caused by the movement of the ground (and therefore, the supporting column). 

\begin{center}
\begin{figure}[h]
\centering
\includegraphics[scale=.5]{Roof3}
\caption{Displacement of a roof caused by an earthquake.}
\end{figure}
\vspace*{-.4 true cm}
\end{center}

Suppose that your system is experiencing viscous damping. Then our model is
\[mx_T'' + \gamma x' +kx = 0, \qquad x(0)=0, \,\,\, x'(0)=0\]
since only the roof/building (not the ground) experiences the viscous damping. As in Part I \cite{Part1}, we will take
\[x_G''(t) = -\dfrac{A}{m}\sin(\omega t).\]



\noindent\textbf{Problem 8.}  Assuming that the system is underdamped, find the general solution of $x(t)$; you do not need to find any coefficients. Explain why you know this is the correct solution form.
\vspace*{-.1 true cm}

\noindent\textbf{Problem 9.} Assume that the roof of a small building has a mass of 8000 kg and the stiffness of the column is 9000 kg/$\mbox{s}^2$. The viscous damping built into the design is only 30 kg/s. You know that the roof (and building) will collapse if the roof moves more than one meter \textit{from the base of its supporting column}. Since the column, roof, and ground move together from the ground vibration $x_G(t)$, this means that we need $x(t)$ (not $x_T(t)$) to be less than one meter to avoid disaster. If there is a vibration (say, take $A=100$), what values of $\omega$ will cause the building to collapse? \textit{Consider only the long-term vibrations that do not quickly dissipate.}

\section*{MAIN GOAL}

\noindent\textbf{Problem 10.} Assume that the roof of a small building has a mass of 8000 kg and the stiffness of the column is 9000 kg/$\mbox{s}^2$. Assume that the largest earthquake (in recent history) in the area can be modeled by
\[x_G''(t) = -1.25\sin\left(\dfrac{\pi}{2}t\right).\]
What is the smallest value of the viscous damping coefficient $\gamma$ necessary when designing the building to prevent the roof from moving more than 0.5 meters from the base of its supporting column? \textit{Consider only the long-term vibrations that do not quickly dissipate.}\\


\begin{thebibliography}{98}

\bibitem{Cho95} Chopra, A. 1995. {\it Dynamics of Structures: Theory and Applications to Earthquake Engineering\/}. Englewood Cliffs, NJ: Prentice-Hall, Inc.


\bibitem{W1} Landry, K. and Winkel, B. 2016. 5-040-S-TunedMassDampers-Part I. 

\bibitem{W2} Landry, K. and Winkel, B. 2016. 5-040-S-TunedMassDampers-Part II. 

\bibitem{MarMcd99} Marchand, R. and McDevitt, T. 1999. {\it Learning Differential Equations by Exploring Earthquake Induced Structural Vibrations: A Case Study\/}. Int. J. Engng. Ed. Vol. 15, No. 6, pp. 477-485.

\bibitem{MIT} Sauser, B. 2009. {\it What is the Deal with Rocket Vibrations?\/}. MIT Technology Review. Available from \href{https://www.newscientist.com/article/dn13597-nasa-says-spaceships-violent-vibrations-under-control/}{https://www.newscientist.com/article/dn13597-nasa-says-spaceships-violent-vibrations-under-control/}.

\bibitem{Part1} Weyand, T. 2021. 3-072-S-EarthQuake-Part I. 

\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}


This is Part II of a two part activity. Part I \cite{Part1} looks at an undamped building and pure resonance. These projects are written assuming that Part I was completed first. It is possible to combine them or only assign one part.
\begin{itemize}
\item If you combine them, I recommend the following organization:
\begin{itemize}
\item Activity 1 of Part I
\item Activity 1 of Part II 
\item Activity 2 of Part I
\item Activity 2 of Part II
\end{itemize}
\item If you only assign this one (Part II), be sure to include the background material (and energy definitions) from Part I \cite{Part1} as an introduction to the project. You may also want to include Problems 6 and 7 from Part I to have students think about the function proposed for $x_G''(t)$ and the differential equation in terms of $x(t)$ (rather than $x_T(t)$).
\end{itemize}

I intend this activity to be assigned as an out-of-class group project. I give students one week to complete this project, during which time we will be covering mass-spring-damper systems (in thorough detail) in class. Alternatively, this activity could be completed as a series of short in-class activities as new solution techniques of second-order differential equations (or different aspects of mass-spring-damper systems) are addressed in class. This activity is already broken into multiple activities/parts to accommodate this approach. Part I \cite{Part1} (which involves no damping) also has multiple parts that can be incorporated into class at the appropriate time.

Students must know how to solve constant-coefficient, linear, homogeneous (the method of undetermined coefficients will suffice)  second-order differential equations in order to complete this activity. This scenario was written assuming that students are familiar with mass-spring-damper systems (including amplitude, undamped systems, and transient versus steady-state solutions). At the beginning of each term, I provide students a sample of a submitted project to demonstrate what I expect in terms of format, level of detail (in calculations), and sufficient explanations/justifications of both the mathematics and physical situation. I also provide a project-specific rubric for them to see prior to submission so students can make sure they have included everything they are supposed to. A sample rubric is available with the supporting documents of Part I \cite{Part1}.

I expect students to be using a CAS (computer algebra system), such as Maple, when completing this project. If students do not have access to this or using such software is beyond the scope of the class, Problems 9 and 10 should be removed. You will need to clarify which solutions (if any) you want found by hand and where students may use a CAS.\\

\noindent\textbf{\underline{Possible Modifications}}
\begin{itemize}
\item More details could be provided prior to Problem 2 to introduce the concept of underdamped systems. Additional questions could be added here exploring the graph (shape) and solution form of underdamped systems if this topic was not previously covered in class. 
\item \textbf{Students must know the concepts of transient versus steady-state solutions to be able to complete the last two problems.} I did not use that terminology  explicitly in the last question to make sure students were thinking physically/graphically, but you may chose to do so (by asking students to look at the steady-state amplitude). 
\item To shorten this project, the questions regarding energy (Problems 3 and 7) can be cut without loosing the main point of the project.
\item To shorten this project, the questions regarding Coulomb damping (Problems 4-7) can be cut without loosing the main point of the project (since the last few questions only consider viscous damping).
\item This project can be extended by considering two buildings next to one another. Now one must ensure that the total displacement $x_T(t)$ of each building is small enough that that builds could not possibly collide with one another during an earthquake.
\end{itemize}

\noindent\textbf{\underline{Expected Solutions}}\\


\noindent\textbf{Problem 1.} Since each term should have units of force,
\[\mbox{mass} \times \dfrac{\mbox{distance}}{\mbox{time}^2},\]
and $x'$ has units of velocity (distance/time), the constant $\gamma$ should have units of
\[\dfrac{\mbox{mass}}{\mbox{time}}\]
so that the units of $\gamma x'$ are
\[\dfrac{\mbox{mass}}{\mbox{time}}\dfrac{\mbox{distance}}{\mbox{time}} =\mbox{mass}\times \dfrac{\mbox{distance}}{\mbox{time}^2}.\]

\noindent \textbf{Problem 2.} The system will be underdamped if the roots of the characteristic equation
\[mr^2 + \gamma r + k = 0\]
are imaginary, which means that 
\[\gamma^2 - 4mk < 0.\]
Hence the system is underdamped if 
\[0< \gamma < \sqrt{4mk}.\]
The boundary at zero is added (rather than considering negative values of $\gamma$) because physically, it does not make sense to have $\gamma <0$. This would imply negative damping, or in other words, less friction than there is in a perfect vacuum.\\

\noindent\textbf{Problem 3.} We know that
\[E(t) = E_k(t) + E_p(t) = \dfrac{1}{2}[m(x'(t))^2 + kx^2(t)]\]
so we can calculate that
\[E'(t) = mx'(t)x''(t) + kx(t)x'(t) = (mx''(t) + kx(t))x'(t).\]
Since $x(t)$ is a solution to \eqref{eq:DE2}, we know that
\[mx'' + kx(t) = -\gamma x'(t).\]
Therefore, we have that
\[E'(t) = -\gamma (x'(t))^2 < 0.\]
Hence, the total energy is decreasing in this case. This makes sense for this situation; energy is lost due to damping/friction.\\

\noindent\textbf{Problem 4.} Since Coulomb damping is constant (and independent of velocity), the damping force is of the form $F_0$, not $\gamma x'$. This is why \eqref{eq:DE2} is no longer valid.\\

\noindent\textbf{Problem 5.} 
\[mx'' \pm F_0 + kx = 0 \qquad\to\qquad mx'' + kx =  \mp F_0, \qquad x(0)=x_0, \,\, x'(0)=x_1\]
  The sign of $F_0$ (on the right-hand side of the equation) is negative if the motion is in the positive direction and positive if the motion is in the negative direction. When motion is in the positive direction (right), both the spring and friction act in the same direction so they should both have the same sign (on the left-hand side of the equation). When motion is in the negative direction, they have opposite signs.\\ 

\noindent\textbf{Problem 6.} The solution is
\[x(t) = c_1\sin\left(\sqrt{\dfrac{k}{m}}t\right) + c_2\cos\left(\sqrt{\dfrac{k}{m}}t\right) -\dfrac{F_0}{k}\]
where
\[c_1 = \sqrt{\dfrac{m}{k}}\left(x_1+\dfrac{F_0}{k}\right) \qquad\mbox{and}\qquad c_2 = x_0 + \dfrac{F_0}{k}.\] 

\noindent\textbf{Problem 7.} Once again, we have that
\[E'(t) = (mx''(t) + kx(t))x'(t).\]
In this situation, we have $mx'' + kx = -F_0$ so
\[E'(t) = -F_0x'(t) < 0\]
when the displacement is in the positive direction (i.e, $x'(t)>0$). Hence, energy is once again lost from damping/friction.

Notice that if the object is moving in the negative direction, we have
\[E'(t) = F_0x(t) <0\]
since $x'(t)<0$. Therefore, the energy is decreasing no matter which way the roof is moving.\\

\noindent\textbf{Problem 8.} 
\[x(t) = c_1e^{at}\sin(bt) + c_2e^{at}\cos(bt) + B\sin(\omega t) + C\cos(\omega t), \qquad a<0\]
For an underdamped system, the homogeneous solution is a decaying exponential function times a trigonometric function. From the method of undetermined coefficients, we know the form of the particular solution will be sine and cosine (with the same frequency). In particular, we know that
\[a = -\dfrac{\gamma}{2m}<0 \qquad \mbox{and} \qquad b = \dfrac{\sqrt{4mk-\gamma^2}}{2m}.\]

\noindent\textbf{Problem 9.} The associated differential equation here is 
\[8000x''(t) + 30x'(t) + 9000x(t) = 100\sin(\omega t).\]
The solution is of the form
\[x(t) = \dfrac{(-8000\omega^2 + 9000)\sin(\omega t) - 30\omega\cos(\omega t)}{640000\omega^4 - 1439991\omega^2 + 810000} + \mbox{ transient solution}.\]
Therefore the long-term amplitude is
\[\mbox{Amplitude} = \dfrac{\sqrt{(-8000\omega^2+9000)^2 + (-30\omega)^2}}{640000\omega^4 - 1439991\omega^2 + 810000}.\]
We can see from a graph of amplitude versus frequency ($\omega$) that there is a spike near 1. Using a CAS system to solve for when the amplitude is equal to 1, we get 
\[\omega \approx 1.05506 \mbox{  and  } 1.06623.\]
From the graph, it is clear that the amplitude is greater than 1 meter for all values of $\omega$ in between these to values (and nowhere else).

\noindent\textbf{Problem 10.} The associated differential equation is
\[8000x''(t) + \gamma x'(t) + 9000x(t) = 10000\sin\left(\dfrac{\pi}{2}t\right),\]
whose solution is
\begin{align*}
x(t) =& \dfrac{-80000000\pi^2 + 360000000}{324000000+16000000\pi^4 + (\gamma^2-144000000)\pi^2}\sin\left(\dfrac{\pi}{2}t\right)\\
& - \dfrac{20000\pi\gamma}{324000000+16000000\pi^4 + (\gamma^2-144000000)\pi^2}\cos\left(\dfrac{\pi}{2}t\right) + \mbox{transient solution}.
\end{align*}
Therefore the long-term amplitude is
\[\mbox{Amplitude} = \dfrac{\sqrt{(-80000000\pi^2 + 360000000)^2 + (-20000\pi\gamma)^2}}{324000000+16000000\pi^4 + (\gamma^2-144000000)\pi^2}.\]
This graph (of amplitude versus damping $\gamma$) slowly decays as $\gamma$ increases. The amplitude is less than 1/2 if $\gamma$ is more than 10741 kg/s.\\

\noindent\textbf{\underline{Student Challenges}}

I expect that students may struggle with the following parts of this project.
\begin{itemize}
\item The questions related to energy (Problems 3 and 7). It is extremely difficult to show the energy is decreasing in Problems 3 and 7 using brute force. These problems are much easier if students calculate $E'(t)$ (from the definition of $E(t)$) and then use what they know about $x(t)$ (without plugging in the actual function). If you don't want students to spend too long on these questions, you may want to provide them a hint to consider $E'(t)$.
\item Setting up the differential equation (from words) in Problem 5 (particularly getting the sign of the Coulomb force correct). All necessary details are given, but writing a differential equations from words is frequently a challenge. As mentioned above, the section on Coulomb damping (Problems 4-7) can be cut without loosing the main point of the activity.
\item This project is phrased in terms of displacement, not amplitude, to make students think about how to measure the size of the displacement. You may want to define, and then provide the hint of looking at, amplitude if this concept was not covered extensively in class.
\item As addressed earlier, students must understand the difference between a steady-state and transient solution. This knowledge (and a CAS) is necessary for Problems 9 and 10, as these problems will be much more complicated if students look at the entire solution/amplitude (rather than just the steady-state portion).
\end{itemize}

\begin{thebibliography}{99}

\bibitem{Cho95} Chopra, A. 1995. {\it Dynamics of Structures: Theory and Applications to Earthquake Engineering\/}. Englewood Cliffs  NJ: Prentice-Hall, Inc.

\bibitem{W1} Landry, K. and B.~Winkel. 2016. 5-040-S-TunedMassDampers-Part I. 

\bibitem{W2} Landry, K. and B.~Winkel. 2016. 5-040-S-TunedMassDampers-Part II. 

\bibitem{MarMcd99} Marchand, R. and T.~McDevitt, T. 1999. Learning Differential Equations by Exploring Earthquake Induced Structural Vibrations: A Case Study. {\it Int. J. Engng. Ed.} 15(6):  477-485.

\bibitem{MIT} Sauser, B. 2009. {\it What is the Deal with Rocket Vibrations?\/}. MIT Technology Review. Available from \href{https://www.newscientist.com/article/dn13597-nasa-says-spaceships-violent-vibrations-under-control/}{https://www.newscientist.com/article/dn13597-nasa-says-spaceships-violent-vibrations-under-control/}.

\bibitem{Part1} Weyand, T. 2021. 3-072-T-EarthQuake-Part I. 
\end{thebibliography}
}





\end{document}
