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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Bad Vibrations: Modeling a Building During an Earthquake\\Part I: No Damping}



\markboth{Building Vibrations}{Building Vibrations}

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

%\author{Full Name - First and  Last\\
%Department Affiliation\\ 
%School or Corporate Affiliation\\
%City ST \teacher{Zip or Mail Code} COUNTRY\\
%\teacher{Email Address}
%}

\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 system. In this activity, students will analyze the motion of the roof caused by ground movement. Here we consider an undamped system. Pure resonance is addressed.

Part II \cite{Part2} of this modeling scenario involves damping of a one-story building's vibrations (and in particular, finding the damping the building should have to avoid collapsing). 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 Scenarios \cite{W1} and \cite{W2}.
\end{abstract}

\section*{SCENARIO DESCRIPTION}

An engineer is responsible for designing something that functions properly, not only in the ideal situation, but also in nonideal situations that may occur. One cannot plan for every possibility, but likely scenarios should be taken into account. Vibrations occur in almost every object, not only in mass-spring systems. Vibrations are the reason a loud sound of the correct pitch can cause glass to shatter. The frequency of this pitch is referred to as the \textit{resonant frequency} of the glass. %Therefore, vibrations should be considered during the design process.

Vibrations are important to take into account when designing a space ship. Both the Saturn and Ares rocket experience large vibrations during the first stage when solid fuel was burned due to small disturbances in the flow of the fuel. If these vibrations are at or near the resonance frequency of the spacecraft, the result could be disastrous. During an unmanned Apollo 6 flight, large vibrations caused the main engine to shut down \cite{MIT}. 

Ground vibrations need to be considered in many situations. Most of these vibrations are too small to be noticed by us (without a seismograph or similar device), but sometimes they are large enough to be felt. Buildings must be designed with these vibrations in mind, especially in places that are more likely to experience earthquakes. Our goal is to analyze the movement of a building during an earthquake. Engineers must ensure the movement is small enough that there will not be structural damage to the building or discomfort for the occupants due to motion of the structure. 

In our analysis, we are going to consider \textit{simple} structures that can be idealized as a single point mass. In particular, we are going to analyze the roof of a one-story building.  Assuming that the roof is uniform, we can approximate the roof by a single point mass in the center of the building; see Figure \ref{fig: Roof1}. The roof itself is very stiff and will resist motion. It is horizontal movement of the supporting column that will cause the roof to move.

Your goal is to find the resonance frequency of a building. Similar to a glass, this is the frequency that you do not want the ground to be vibrating at, as it will cause the building to move too far and (eventually) collapse or, in the case of glass, shatter.
%Your goal 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.\\


\begin{figure}[h]
\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}
\vspace*{-.5 true cm}
\end{figure}


\section*{ACTIVITY 1 -- NO VIBRATIONS}


First we consider an (idealized) one-story building, while ignoring damping effects.  We look at the simplified scenario where the roof is supported by a single supporting column. The displacement of the center of mass of the roof, $x(t)$ (see Figure \ref{fig: Roof1}), can be modeled as
\begin{equation}
\label{eq:DE1}
mx'' + kx = 0, \qquad x(0)=x_0, \,\,\, x'(0)=x_1
\end{equation}
where $m$ is the total mass of the roof and $k$ is the stiffness of the supporting column. It is known that 
\[k \mbox{ is proportional to } EI/h^3\]
where $E$ is the elastic modulus of the supporting column, $I$ is the moment of inertia of the column about the bending axis, and $h$ is the length of the column \cite{Cho95}.\\


\noindent\textbf{Problem 1.} What units should the constant $k$ have for the differential equation \eqref{eq:DE1} 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.} If any type of force (such as an earthquake) acts on the system at  $t=0$, the initial values would be nonzero. Find the (analytic) solution for $x(t)$ in this case. Explain why this solution does or does not make sense physically (in the long term).\\

\noindent\textbf{Problem 3.} Suppose that the mass of the roof is increased (for example, by installing a different type of roof or modeling a larger roof/building). Is the displacement more or less (or the same) in this case? Justify your conclusion.\\

\noindent\textbf{Problem 4.} Suppose that the height of the building (and therefore the length of the supporting column) is increased. Is the displacement more or less (or the same) in this case? Justify your conclusion.\\

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

Let $x(t)$ be the displacement of a mass (of size $m$). The \textbf{kinetic energy} of the mass (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 5.} Show that, in this case, the total energy of the system is conserved; in other words, the energy is constant.



\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, supporting column); see Figure \ref{fig: earthquake}.

\begin{center}
\begin{figure}
\centering
\includegraphics[scale=.5]{Roof3}
\caption{Displacement of a roof caused by an earthquake.}
\label{fig: earthquake}
\end{figure}
\end{center}


\noindent\textbf{Problem 6.} Explain why, during an earthquake, it is (physically) reasonable to assume that
\[x_G''(t) = -\dfrac{A}{m}\sin(\omega t).\]
Even though this is reasonable, explain why this assumption/model is not ideal.\\


\noindent\textbf{Problem 7.} Assuming that there is no damping, we have
\begin{equation}
\label{eq: DE3}
mx_T'' + kx = 0.
\end{equation}
Both the ground and the roof/building are accelerating. However, only the roof (not the ground) is affected by the stiffness of the supporting column. Find  a differential equation for $x(t)$ (in terms of $x_G(t)$).\\

\noindent\textbf{Problem 8.} Using the function $x_G(t)$ above (and assuming that the building was not initially moving), solve the associated initial-value problem for $x(t)$.\\

\noindent\textbf{Problem 9.} The roof will collapse if it moves too much. Find one value of $\omega$ that will definitely cause too much motion (in the long term) and cause the building to collapse (eventually). Describe physically and mathematically what is occurring in this case. \emph{Hint: Think of resonance and carefully consider all possible solutions to Problem 8.}\\

\vspace*{-.3 true cm}
\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 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. 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. What is the Deal with Rocket Vibrations? {\it 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{Part2} Weyand, T. 2021. 3-072-S-EarthQuake-Part II. 
\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}

This is Part I of a two part activity. Part II \cite{Part2} looks at different types of damping that may occur in buildings to help prevent vibrations. These projects are written assuming this one will be 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 Part II, be sure to include the background material, figures, and definitions of energy from Part I as an introduction to the project.
\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 II \cite{Part2} (which involves damping) also has multiple parts that can be incorporated into class at the appropriate time.

Students must know how to solve constant-coefficient, linear, nonhomogeneous (the method of undetermined coefficients will suffice)  second-order differential equations in order to complete this activity. It was written assuming that students are familiar with mass-spring-damper systems (including the phenomenon of pure resonance). More background should be added if this is not the case for your students. 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.\\  

\noindent\textbf{\underline{Possible Modifications}}
\begin{itemize}
\item Pure resonance is needed to answer Problem 9. Additional information (and questions) can be added here to teach students about this topic if it has not been covered in class.
\item To shorten this project, the question regarding energy (Problem 5) can be cut without loosing the main point of the project.
\end{itemize}

This project can be completed with pencil and paper. You could easily incorporate a CAS (computer algebra system) by allowing students to solve the differential equations with a CAS to save time (if solution techniques is not one of your main goals) or asking for plots of the solutions.\\

%You will need to clarify which solutions (if any) you want found by hand and where students may use a CAS. For questions that ask students to solve an initial-value problem, I ask them to determine (with justification) what the general solution form is (as is done in Problem 17). They may then use a CAS to find the explicit solution of the initial-value problem.  \\

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

\noindent\textbf{Problem 1.} Notice that first term in the equation has units of
\[ \mbox{mass} \times \mbox{acceleration} = \mbox{mass} \times \dfrac{\mbox{distance}}{\mbox{time}^2}.\]
To make sense, the second term should also have these units. Hence we have
\[\mbox{units of } k \times \mbox{distance} = \mbox{mass} \times \dfrac{\mbox{distance}}{\mbox{time}^2} = \dfrac{\mbox{mass}}{\mbox{time}^2} \times \mbox{distance},\]
which means that the units of $k$ are
\[\dfrac{\mbox{mass}}{\mbox{time}^2}.\]


\noindent\textbf{Problem 2.} 
\[x(t) = \dfrac{x_1\sqrt{m}}{\sqrt{k}}\sin\left(\sqrt{\dfrac{k}{m}}t\right) + x_0\cos\left(\sqrt{\dfrac{k}{m}}t\right)\]
This solution does not make sense in the long term. It says that once the roof starts moving, it will continue moving forever. We know that this is not what happens. At some point, the roof will stop moving.\\

\noindent\textbf{Problem 3.} The amplitude (or, in other words, the maximum displacement) of the vibrations is
\[A=\sqrt{\dfrac{x_1^2m}{k} + x_0^2}\]
(which is the square root of the sum of the coefficients of sine and cosine squared). Using trigonometric identities, we can rewrite the solution as
\[x(t) = \sqrt{\dfrac{x_1^2m}{k} + x_0^2}\sin\left(\dfrac{k}{m}t + \theta\right),\]
where $\theta = \tan^{-1}\left(\dfrac{x_0\sqrt{k}}{x_1\sqrt{m}}\right)$ is the \emph{phase angle}, which also demonstrates that $A$ is the amplitude of the oscillation. If the mass is increased, this means that the amplitude is also increased.\\

\noindent\textbf{Problem 4.} Here we refer back to the amplitude found in Problem 3. If the height is increased, then $k$ (the stiffness) is decreased (assuming everything else remains the same) since $k$ is proportional to $EI/h^3$. If $k$ decreases, the amplitude will increase. This makes sense physically; something taller of the same characteristics will wobble more.\\


\noindent \textbf{Problem 5.} \\
\underline{Brute Force:} Here we have $x(t)$ as in Problem 2 so the velocity is
\[x'(t) = x_1\cos\left(\sqrt{\frac{k}{m}}t\right) - \sqrt{\dfrac{k}{m}}x_0\sin\left(\sqrt{\frac{k}{m}}t\right).\]
The kinetic energy is 
\begin{align*}
E_k(t) &= \dfrac{m}{2}\left[x_1^2\cos^2\left(\sqrt{\frac{k}{m}}t\right) + \frac{kx_0^2}{m}\sin^2\left(\sqrt{\frac{k}{m}}t\right) - 2\dfrac{x_1x_0\sqrt{k}}{\sqrt{m}}\cos\left(\sqrt{\frac{k}{m}}t\right)\sin\left(\sqrt{\frac{k}{m}}t\right)\right]\\
&= \frac{1}{2}\left[x_1^2m\cos^2\left(\sqrt{\frac{k}{m}}t\right) + x_0^2k\sin^2\left(\sqrt{\frac{k}{m}}t\right) - 2x_1x_0\sqrt{mk}\sin\left(\sqrt{\frac{k}{m}}t\right)\cos\left(\sqrt{\frac{k}{m}}t\right) \right]
\end{align*}
and the potential energy is
\begin{align*}
E_p(t) &= \dfrac{k}{2}\left[\dfrac{x_1^2m}{k}\sin^2\left(\sqrt{\frac{k}{m}}t\right) + x_0^2\cos^2\left(\sqrt{\frac{k}{m}}t\right) + 2\frac{x_1x_0\sqrt{m}}{\sqrt{k}}\sin\left(\sqrt{\frac{k}{m}}t\right)\cos\left(\sqrt{\frac{k}{m}}t\right)\right]\\
&= \dfrac{1}{2}\left[x_1^2m\sin^2\left(\sqrt{\frac{k}{m}}t\right)+ x_0^2k\cos^2\left(\sqrt{\frac{k}{m}}t\right) + 2x_1x_0\sqrt{mk}\sin\left(\sqrt{\frac{k}{m}}t\right)\cos\left(\sqrt{\frac{k}{m}}t\right) \right].
\end{align*}

Hence the total energy is 
\[E_T(t) = x_1^2m\left(\cos^2\left(\sqrt{\frac{k}{m}}t\right) + \sin^2\left(\sqrt{\frac{k}{m}}t\right)\right) + x_0^2k\left(\sin^2\left(\sqrt{\frac{k}{m}}t\right)+\cos^2\left(\sqrt{\frac{k}{m}}t\right)\right) = x_1^2m+x_0^2k,\]
which is independent of time, and therefore constant.\\


\noindent\underline{Via the derivative:} Students can also show the energy is constant by observing that
\[E_T'(t) = mx'(t)x''(t) + kx(t)x'(t) = x'(t)(mx''(t)+kx(t)) = x'(t)\times 0 = 0\]
since $x(t)$ solves the homogeneous differential equation \eqref{eq:DE1}. Since the derivative is zero (for all time), the energy is constant.\\


\noindent\textbf{Problem 6.} During an earthquake, the ground does not just move once. One shift deep in the ground causes vibrations at the surface. Hence, something like a sine or cosine function makes sense to represent ground movement; notice that sine has a second derivative of negative sine, as shown here. 
However, this model is not realistic in the long term. This says that the oscillations continue forever. It would be more realistic with some sort of damping term like $e^{at}\sin(bt)$ with $a<0$.\\

\noindent\textbf{Problem 7.}
\[mx'' + kx = A\sin(\omega t)\]

\noindent\textbf{Problem 8.} Assuming that $x(0)=x'(0)=0$, we have
\[x(t) = -\dfrac{A\omega\sqrt{m}}{\sqrt{k}(k-m\omega^2)}\sin\left(\sqrt{\dfrac{k}{m}}t\right) + \dfrac{A}{ k-m\omega^2}\sin(\omega t).\]

\noindent\textbf{Problem 9.} The building will (eventually) collapse if $\sqrt{\dfrac{k}{m}} = \omega$. In this case, the solution is of the form 
\[x(t) = \dfrac{A}{2\sqrt{k^3m}}\left(\sqrt{km}\sin\left(\sqrt{\dfrac{k}{m}}t\right) - tk\cos\left(\sqrt{\dfrac{k}{m}}t)\right)\right).\]
The $t$ in front of the cosine term will result in unbounded oscillation, which will (eventually) cause the building to collapse. In particular, the amplitude of this function is
\[\sqrt{\dfrac{A^2km + k^2t^2}{4k^3m}},\]
which will increase for all time. This means that eventually, the roof will move too far, causing the building to collapse. This is \textbf{pure resonance}. Notice that in this case, the solution form from Problem 8 is not valid, as the denominator is zero. \\



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

I expect that students may struggle with the following parts of this project.
\begin{itemize}    
\item The question related to energy (Problems 5). This computation is challenging if students try to show $E(t)=c$ by plugging the solution $x(t)$ into the definition of $E(t)$. This problem is 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 this question, you may want to provide them a hint to consider $E'(t)$.
\item Thinking of pure resonance in Problem 9. You may want to provide a more specific hint if resonance was only touched on, but not a main topic in class.
\item Thinking in terms of amplitude. The project continually mentions large displacements of something vibrating without explicitly mentioning amplitude. You may want to explicitly guide students to think in this way.
\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{Part2} Weyand, T. 2021. 3-072-S-EarthQuake-Part II. 

\end{thebibliography}
}





\end{document}
