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\journal{Journal of Sound and Vibration}
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\begin{document}

\begin{frontmatter}

\title{Teaching Broadband Vibration Attenuation \\ using Fettuccine Pasta}


%% Group authors per affiliation:
\author[rvt]{Sai Tej Paruchuri\corref{cor1}}
\ead{saitejp@vt.edu}
\author[rvt]{Vijaya V. N. Sriram Malladi}
\ead{sriram@vt.edu}
\author[rvt]{Pablo A. Tarazaga}
\ead{ptarazag@vt.edu}
\author[rvt]{Andrew J. Kurdila}
\ead{kurdila@vt.edu}

\cortext[cor1]{Corresponding author}

\address[rvt]{Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 24061}

\section*{Highlights}

\begin{itemize}
    \item Designing an array of linear oscillators, also referred to as Subordinate Oscillator Arrays using dry fettuccine pasta.
    \item Accounting for unknown material properties of structural elements.
    \item Testing multi-DOF structure with Subordinate Oscillator Arrays.
    \item A practical demonstration of vibration isolation using mundane pasta. 
\end{itemize}
\newpage

\begin{abstract}
An array of linear oscillators, also referred to as Subordinate Oscillator Arrays or SOAs, attached to a host structure produce a flat response in the frequency domain. However, such oscillators arrays are highly sensitive to structural disorders. Any uncertainty in structural parameters can induce peaks in the frequency domain. In this paper, we present a systematic approach to design SOAs even with the presence of structural uncertainty. We test our methodology by designing an SOA using fettuccine, a system with a very high uncertainty in its Young's modulus.
\end{abstract}

\begin{keyword}
Subordinate Oscillator Array \sep Fettuccine Pasta \sep Structural Dynamics \sep Education
\end{keyword}

\end{frontmatter}

\linenumbers

\section{Introduction}

An array of linear oscillators can achieve vibration attenuation over a broad range of frequencies \cite{Strasberg1995, Akay2005,Zuo2005,Carcaterra2012,Vignola2009}. These arrays are called Subordinate Oscillator Arrays or SOAs. The isolated natural frequencies of the oscillators in the SOA should be spaced around the natural frequencies of the host structure to achieve broadband attenuation. Various design approaches can accomplish this design condition. The most straightforward method involves the placement of the isolated natural frequencies around the host structure’s natural frequency in a random manner until the necessary attenuation is achieved over the required frequency range. A different approach would involve adding one oscillator at a time into the array corresponding to the highest peak in the desired frequency range and reevaluating the frequency response of the system each time. Naturally, this is a slow approach. However, it is highly efficient, and the SOAs designed using this approach have a much fewer number of oscillators. Vignola et al. \cite{Vignola2009} proposed a method in which discrete distributions are assigned to the natural frequencies of the oscillators. As opposed to choosing individual parameters of oscillators, assigning distributions reduces the number of parameters to select to achieve the necessary frequency distribution. 

The performance of a Subordinate Oscillator Array is highly dependent on the uncertainty in its as well as the host’s structural properties. Vignola et al. studied the effect of disorder on the SOA and showed that the SOAs performance deteriorated with increasing error in the properties.

The design approaches mentioned above do not take uncertainty in structural properties into consideration during the design phase. Arrays with piezoelectric oscillators, studied in \cite{ParuchuriASME2017,ParuchuriIEEE2017}, can adapt their structural properties in the presence of uncertainties. However, such an array of oscillators are complex to design and build when compared to ordinary SOAs. 

In this paper, we present a design approach, adapted from the second method discussed above, that takes the uncertainties of the system into consideration and corrects the model to account for the disorder in the system. We performed experiments on a two degree of freedom system, a model that mimics a two-story building, to test the effectiveness of our approach. We used fettuccine as cantilever beams for oscillators in the array. Considering that fettuccine’s young modulus vary from strand to strand, they make a suitable and inexpensive test subject for our approach. In this following section, we derive the model of the system.
\section{Analytical Model neglecting the mass of posts}

In this section, we are going to present the equations of motion of our system, derived using Lagrange's equations. Figure \ref{lfig_host_structure} shows the host structure, a small scale model of a two-story building, for which we are trying to reduce the amplitude of vibration across two resonant peaks. The host structure under consideration can be modeled as a two degree of freedom system as shown in Figure \ref{lfig_host_structure_model}. 

\paragraph{Kinetic Energy of Host structure} As evident from the experimental setup, the total kinetic energy corresponding to a single story in the structure has contributions from the floor mass as well as the column mass. %If $x_i$ and $m_i$ is the displacement of floor $i$, the kinetic energy of the story can be expressed as 
%
% %\begin{equation}
% \bm{T}_{mass_i} = \frac{1}{2} m_i \dot{x}_i^2(t).
% \end{equation}

The experimental model we considered has two levels. Hence, the total kinetic energy of our setup is
\begin{align}
    T &= T_{mass_1} + T_{mass_2} 
    = \frac{1}{2} m_1 \dot{x}_1^2(t)  + \frac{1}{2} m_2 \dot{x}_2^2(t).
\end{align}

\paragraph{Potential Energy of the Host Structure} The expression for the potential energy of the host structure is much more straightforward to derive and can be expressed using the relation

\begin{align}
U &=  \frac{1}{2} K_1 \left( x_1 - z \right)^2 + \frac{1}{2} K_2 \left( x_2 - x_{1} \right)^2.
\end{align}
where the stiffness of a clamped-guided beam and can be determined using the relation
\begin{equation}
    K_i = \frac{12EI}{L_i^3}.N,
\end{equation}


% \begin{align}
%     U_i &= \frac{1}{2} \underbrace{\left( N k_i \right)}_{K_i} \left( x_i - x_{i-1} \right)^2, \\
%     &= \frac{1}{2} K_i \left( x_i - x_{i-1} \right)^2
% \end{align}
% %
% where $k_i$ is the stiffness of a clamped-guided beam and can be determined using the relation
% \begin{equation}
%     k_i = \frac{12EI}{L_i^3},
% \end{equation}
% %
where $E$ and $I$ are the young's modulus and moment of inertia of the beams respectively. Hence, the total potential energy of the experimental model shown in Figure \ref{} is
%

%

\section{Analytical Model including the mass of posts}
%
Since the mass of the columns in a building is significant, the mass of the springs, which represent the columns in a structure, is taken into account for accurate modeling. 
\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[height=.7\linewidth]{fig_exp_setup.pdf}
  \caption{Host Structure Setup}
  \label{lfig_host_structure}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[height=.7\linewidth]{fig_host_structure_model.pdf}
  \caption{Host Structure Model}
  \label{lfig_host_structure_model}
\end{subfigure}
\caption{Host Structure}
\label{fig_host_structure}
\end{figure}
%

%
The kinetic energy of the columns supporting the $i^{th}$ story of a building can be expressed in the form
%
\begin{equation}
\bm{T}_{beam_i} = N \left(\frac{1}{2} \rho A \int_0^{L_2} \dot{u_i}^2(t) dy \right),
\end{equation}
%
where $\rho$, $A$, $L_i$, $u_i(L_i,t)$ are the density, cross-sectional area, length and displacement along the length of the columns respectively. The term $N$ in the above expression is the number of columns supporting the $i^{th}$ story of the building. As evident from the kinetic energy of the columns expression, all columns are assumed to be identical. The absolute displacement $u_i(L_i,t)$ can be expressed in the form 
%
\begin{align}
u_i &= x_{i-1} + w_i(y,t),
\label{eq_cgbeam_disp}
\end{align}
where $x_{i-1}$ is the absolute displacement of the $(i-1)^{th}$ story of the building and $w_i(y,t)$ is the relative displacement of the columns supporting $i^{th}$ story of the host structure. The columns of our experimental setup, and in buildings, behave as clamped guided beams. The displacement of a clamped guided beam, as shown in Equation \ref{eq_cgbeam_disp}, can be expressed in the form
\begin{equation}
w_i(x,t) = \frac{x_i - x_{i-1}}{L_i^2} \left[ 3y^2 - \frac{2y^3}{L_i} \right].
\end{equation}
%
Therefore, we have the relation
\begin{equation}
u_i - x_{i-1} = \frac{x_i - x_{i-1}}{L_i^2} \left[ 3y^2 - \frac{2y^3}{L_i} \right],
\end{equation}
which can be rearranged as an expression for absolute displacement of the column. It can be expressed as
\begin{align}
    u &= \underbrace{\left[1 - \frac{(3L_i - 2y)y^2}{L_i^3}] \right]}_{\alpha_i} x_{i-1} + \underbrace{ \left[ \frac{(3L_i - 2y)y^2}{L_i^3} \right] }_{\beta_i} x_i, \\
    &= \alpha_i x_{i-1} + \beta_i x_i.
\end{align}
%
Now, the kinetic energy of the columns supporting the $i^{th}$ story of a structure can be expressed in terms of the displacements $x_{i-1}$ and $x_{i}$ in the form
%
\begin{align}
    T_{beam_i} &= N \left( \frac{1}{2} \rho A \int_0^{L_2} (\alpha_i \dot{x}_{i-1} + \beta_i \dot{x}_i )^2 dy \right), \\
    &= \frac{1}{2} \underbrace{ \left[ \rho A \int_0^{L_i} \alpha_i^2 dy \right] }_{m_{ib1}} \dot{x}_{i-1}^2 + \underbrace{ \left[ \rho A \int_0^{L_i} \alpha_i \beta_i dy \right] }_{m_{ib2}} \dot{x}_{i-1} \dot{x}_i + \frac{1}{2} \underbrace{ \left[ \rho A \int_0^{L_i} \beta_i^2 dy \right] }_{m_{ib3}} \dot{x}_i^2, \\
    &= \frac{1}{2} m_{ib1} \dot{x}_{i-1}^2 + m_{ib2} \dot{x}_{i-1} \dot{x}_{i} + \frac{1}{2} m_{ib3} \dot{x}_i^2.
\end{align}
%
The total kinetic energy corresponding to each story in a building is equal to the sum of kinetic energy of the floor and the kinetic energy of the columns. We have
\begin{align}
    T_i &= T_{mass_i} + T_{beam_i} \\
    &= \frac{1}{2} m_i \dot{x}_i^2(t) + \frac{1}{2} m_{ib1} \dot{x}_{i-1}^2 + m_{ib2} \dot{x}_{i-1} \dot{x}_{i} + \frac{1}{2} m_{ib3} \dot{x}_i^2.
\end{align}
%
The experimental model we considered has two levels. Hence, the total kinetic energy of our setup is
\begin{align}
    T &= T_1 + T_2, \\
    &= T_{mass_1} + T_{beam_1} + T_{mass_2} + T_{beam_1}, \\
    &= \frac{1}{2} m_1 \dot{x}_1^2(t) + \frac{1}{2} m_{1b1} \dot{z}^2 + m_{1b2} \dot{z} \dot{x}_{1} + \frac{1}{2} m_{1b3} \dot{x}_1^2 \\
    & \qquad + \frac{1}{2} m_2 \dot{x}_2^2(t) + \frac{1}{2} m_{2b1} \dot{x}_{1}^2 + m_{2b2} \dot{x}_{1} \dot{x}_{2} + \frac{1}{2} m_{2b3} \dot{x}_2^2.
\end{align}
%
In the above expression, $z = x_0$ represents the base excitation of the system. 
%

\paragraph{Equations of Motion of the Host Structure}
%
The expressions for the total kinetic energy and the total potential energy can be substituted into the Lagrange equation to obtain the governing equations of the system. The final form of the equations of motion of the host structure will look like
%
\begin{align}
\begin{bmatrix}
M_{11} & M_{12} \\
M_{12} & M_{22}
\end{bmatrix}
%
\begin{Bmatrix}
\ddot{x}_1 \\
\ddot{x}_2
\end{Bmatrix}
%
+
%
\begin{bmatrix}
K_1 + K_2 & -K_2 \\
-K_2 & K_2
\end{bmatrix}
%
\begin{Bmatrix}
x_1 \\
x_2
\end{Bmatrix}
%
=
%
\underbrace{\begin{bmatrix}
\textcolor{red}{-m_{1b3}} \ddot{z} + K_1 z \\
0
\end{bmatrix}}_{F},
\label{eq_hs_eom_wo_damping}
\end{align}
%
where \textcolor{red}{$M_{11} = m_{1b1} + m_1 + m_{2b1}$, $M_{12} = m_{2b2}$ and $M_{22} = m_{2b3} + m_2$.}
%
\paragraph{Damping} The equation of motion represented by Equation \ref{eq_hs_eom_wo_damping} do not consider damping of the system. Since it is impossible to model the damping in the system directly, the equations of motion are first transferred to modal space. It is much easier to include the modal damping values measured from experiments into the model.  The modal damping matrix has the form
%
\begin{align}
    C_{modal} = \begin{bmatrix}
    2 \zeta_1 \omega_1 & 0 \\
    0 & 2 \zeta_2 \omega_2
    \end{bmatrix}
\end{align}
%
If the Cholesky factorization of the mass matrix is $M = LL^T$, the damping matrix $C$ can be calculated, as discussed in \cite{Inman2001}, using the relation
\begin{equation}
    C = \begin{bmatrix}
    C_{11} & C_{12} \\
    C_{12} & C_{22}
    \end{bmatrix} = L^{-1} P C_{modal} P^T L^{-T}
\end{equation}
%
where $P$ is the eigenvector matrix of $L^{-1}KL^{-T}$.
%
\paragraph{Frequency Response Function of Host Structure} The Laplace transform of the equations of motion will have the form
%
\begin{align}
    \left( M s^2 + C s + K \right) X(s) &= \mathcal{F} Z(s)
\end{align}
%
Therefore, the transfer function from base motion to the displacements $X_1$ and $X_2$ has the form
%
\begin{align}
    \begin{Bmatrix}
    X_1(s)/Z(s) \\
    X_2(s)/Z(s)
    \end{Bmatrix}
    &= \left( M s^2 + C s + K \right)^{-1} \mathcal{F}
\end{align}
%
The frequency response function or the FRF can be obtained by substituting $s = i \omega$ in the above expression. The FRF from the base excitation to the displacements $x_1$ and $x_2$ can be expressed as
%
\begin{align}
    \begin{Bmatrix}
    X_1(\omega)/Z(\omega) \\
    X_2(\omega)/Z(\omega)
    \end{Bmatrix}
    &= \left( (- M \omega^2 + K ) + i C \omega \right)^{-1} \mathcal{F}
\end{align}
%

%
%%
%%%
\paragraph{Equations of Motion of an SOA attached to a host structure} Since the performance of the SOA depends primarily on the placement of the first natural frequency of each oscillator in the SOA, each oscillator can be approximated using a single degree of freedom model. Further, all oscillators in an SOA are tuned to attenuate one particular natural frequency of the host structure. In other words, each SOA is associated with a particular natural frequency of the host structure. Therefore, in cases where multiple resonant peaks are present in the operating range, each peak is associated with an SOA. In case of lumped parameter multi-degree of freedom systems, like the one used in this paper, SOAs can be attached to any of the lumped mass of the structure. 
%
In the derivation of the equations of motion of an SOA attached to a host structure, it is assumed that there is an SOA connected to each primary masses $m_i$ and associated to each natural frequency $\omega_j$ of the host structure. The results discussed in the latter sections will show that the desired performance can be achieved by a fewer number of SOAs. The equations of motion of an SOA connected to mass $i$ will have the form
%
\begin{align}
m_{1j} \ddot{x}_{1j} + c_{1j} \dot{x}_{1j} - c_{1j} \dot{X}_1 + k_{1j} x_{1j} - k_{1j} X_1 &= 0, \\
m_{2j} \ddot{x}_{2j} + c_{2j} \dot{x}_{2j} - c_{2j} \dot{X}_2 + k_{2j} x_{2j} - k_{2j} X_2 &= 0,
\end{align}
%
where $\bm{x}_{1j}$ and $\bm{x}_{2j}$ represent the vector of displacements of the oscillators in the SOAs associated to the $j^{th}$ natural frequency and attached to masses $1, 2$ respectively. In the above equations, $m_{ij}$, $c_{ij}$ and $k_{ij}$ represent the mass, damping and the stiffness matrices of the SOA attached to primary mass $i$ and associated with natural frequency $j$ respectively. The above equations are equivalent to the equations of motion of a single degree of freedom system with base motion. If $ \mathbb{X} = \left \{  \bm{x}_{11}^T, \bm{x}_{12}^T, \bm{x}_{21}^T, \bm{x}_{22}^T \right \}^T $ represents the state vector the final system, the governing equations can be expressed in the form
%
\begin{align}
\mathbb{M} \ddot{\mathbb{X}} + \mathbb{C} \dot{\mathbb{X}} + \mathbb{K} \mathbb{X} = \mathbb{F},
\end{align}
%
where
%
\begin{equation}
    \mathbb{M} = \left[
\begin{array}{cccc|cc}
m_{11} & & & & & \\
& m_{12} & & & & \\
& & m_{21} & & & \\
& & & m_{22} & & \\
\hline
& & & & M_{11} & M_{12} \\
& & & & M_{21} & M_{22}
\end{array}
\right],
\qquad \qquad 
    \mathbb{F} = \begin{Bmatrix}
0 \\
0 \\
0 \\
0 \\
\hline
K_1 \\
0
\end{Bmatrix},
\end{equation}
%
\begin{equation}
    \mathbb{C} = \left[
\begin{array}{cccc|cc}
c_{11} & & & & -c_{11} & 0 \\
& c_{12} & & & -c_{12} & 0 \\
& & c_{21} & & 0 & -c_{21} \\
& & & c_{22} & 0 & -c_{22} \\
\hline
-c_{11} & -c_{12} & 0 & 0 & C_{11} + \sum \left( c_{11n} + c_{12n} \right) & C_{12} \\
0 & 0 & -c_{21} & -c_{22} & C_{21} & C_{22} + \sum \left( c_{11n} + c_{12n} \right) 
\end{array}
\right],
\end{equation}
%
\begin{equation}
    \mathbb{K} = \left[
\begin{array}{cccc|cc}
k_{11} & & & & -k_{11} & 0 \\
& k_{12} & & & -k_{12} & 0 \\
& & k_{21} & & 0 & -k_{21} \\
& & & k_{22} & 0 & -k_{22} \\
\hline
-k_{11} & -k_{12} & 0 & 0 & K_{1} + \sum \left( k_{11n} + k_{12n} \right) & -K_{2} \\
0 & 0 & -k_{21} & -k_{22} & -K_{2} & K_{2} + \sum \left( k_{11n} + k_{12n} \right) 
\end{array}
\right].
\end{equation}
%
%
%%
\paragraph{Frequency Response Function of Host Structure with SOA}
%
The FRF from the base excitation to the displacement $X_1$, $X_2$ of the host structure with SOA can be obtained using the same procedure  discussed above. The frequency response can be obtained using the function
%
\begin{align}
\begin{Bmatrix}
x(\omega)/z(\omega) \\
\hline
X_1(\omega)/z(\omega) \\
X_2(\omega)/z(\omega)
\end{Bmatrix}
&= \left( \left( -M \omega^2 + K \right) + j C \omega \right)^{-1} F 
\end{align}
%
%%
%%%
\section{Experimental Setup}
\textcolor{red}{Sai: I do not have experience in writing experimental sections. I am not really sure about the amount of detail required.}

Figure \ref{} shows a two degree of freedom structure placed on a shaker. The spaghetti array attached to the top and bottom levels of the structure are the SOA corresponding to the first and second natural frequencies of the host structure, respectively. Clay was used as the tip mass of the host structure. 
%
\paragraph{Spaghetti Properties Measurement} As mentioned earlier, we used spaghetti as the oscillator in the SOA and clay as the tip mass of each oscillator. The primary reason for choosing spaghetti is the low cost as well as the ease of changing their length. Before proceeding with the actual SOA experiment, the frequency of few of the spaghetti was obtained. The Young's modulus of these calculated using the relation
\begin{align*}
Relation
\end{align*}
Based on the experimental results, it was noted that Young's modulus of the spaghetti varied from \textcolor{red}{enter values from experiments}, thus making it a suitable material for testing our design methodology. This range was used as the range of possible Young's modulus values for SOA approach which is explained in the next section. The frequency response plot of the single degree of freedom approximation model of spaghetti and the actual response of spaghetti samples can be seen in Figure \ref{}. The frequency response plot of the approximated model was obtained using the relation
\begin{align}
\frac{x(\omega)}{j \omega z(\omega)} &= \frac{ j C \omega + K}{j \omega \left( \left( - M \omega^2 + K \right) + j C \omega \right)} 
\end{align}
and the corresponding plot of actual spaghetti was obtained from experiments.
%
%%
%%%
\section{SOA Design Methodology}
As discussed in Section \ref{}, several SOA design methodologies are available in the literature. These approaches fail to account for the uncertainties in the structure. Since the ratio of each subordinate oscillator to the entire primary structure is of the order of less than 1\%, small structural uncertainties can significantly deteriorate the performance of the SOA \cite{}. Further, uncertainties can arise from improper boundary conditions. For example, if the fixed end of the cantilever oscillator is loose, the effective stiffness of the oscillator will reduce. A methodology to account for the uncertainties in the structure after permanently attaching the subordinate oscillator to the primary structure would be the most effective method to design an SOA. In this section, we propose \textcolor{red}{two simple design approaches}, one that corrects the host structure model and the second to accounts for uncertainties in the SOA.
%
\begin{figure}
\centering
\includegraphics[scale = 0.45]{fig_flow_chart.pdf}
\caption{Flow Chart}
\label{lfig_flow_chart}
\end{figure}
%
\paragraph{Host Structure Model Correction} Vignola et al. discussed the importance of the natural frequency of the host structure as well as the SOA in \cite{} for the successful design of SOAs. Hence, before proceeding to the design of SOAs, it is crucial to correct the host structure's model. In our experiments, the difference in the model and experimental natural frequencies arose from the error in the effective lengths of the columns as well as the modal damping ratios corresponding to the natural frequencies. We define a cost function of the form
\begin{equation}
J_1 = || \omega_{exp, 1} - \omega_{model, 1}(L_1, L_2) ||_2 + || \omega_{exp, 2} - \omega_{model, 2}(L_1, L_2) ||_2.
\end{equation}
%
The column lengths $L_1$ and $L_2$ that minimize the above functions can be considered as the actual lengths of the host structure. Instead of deriving a closed form equation for the minima of Equation \ref{} and solving it for lengths, it is more straightforward to iterate through an array of possible lengths and finding the minima. 
%
After the finding the effective lengths that ensured that the natural frequencies of the model and experiments matched, a new cost function was defined to correct the magnitude of the peaks corresponding to the natural frequencies. The cost function has the form
%
\begin{align}
    J_2 &= \left|\left| \left|\frac{X_1}{Z}(\omega_{exp,1,2}) \right| - \left| \frac{X_1}{Z}(\omega_{model,1,2}) \right| \right|\right|_2 \\
    & \qquad \qquad + \left|\left| \left|\frac{X_2}{Z} (\omega_{exp,1,2}) \right| - \left| \frac{X_2}{Z}(\omega_{model,1,2}) \right| \right|\right|_2.
\end{align}
The modal damping ratios $\epsilon_1$ and $\epsilon_2$ corresponding to the minima of the cost function given above can be considered effective damping ratios and \textcolor{red}{thus used for further design analysis}.
%
\paragraph{SOA Design} \textcolor{red}{Figure \ref{} gives the algorithm.} The design approach involves attaching the substructure without the tip mass to the host structure such that its natural frequency does not align with the host structure's natural frequencies. The natural frequency of the substructure and its modal damping values can be extracted from the frequency response plot of the host structure. In our experiments, it was observed that the uncertainty in the substructure arose primarily due to variable Young's modulus of the material. The Young's modulus of the substructure can be calculated using the relation
\begin{align*}
Young's modulus relation
\end{align*}
The objective of the substructure is to split the highest peak in the frequency response plot into two peaks of comparable magnitudes. In other words, we are interested in finding the tip mass that minimizes the cost function
\begin{align*}
Cost function.
\end{align*}
\textcolor{red}{As evident, the cost function is defined in terms of the difference in magnitudes of the new peaks that were created after adding the substructure.} Similar to the procedure explained above, iteration through a set of permissible tip mass values is a more straightforward approach that finding a closed form expression for the tip mass corresponding to the minima. This procedure is repeated until the desired reduction in magnitude or flat frequency band is achieved. 
%
%%
%%%
\section{Results}
%
In this section, we investigate the effectiveness of the proposed method using simulation and experimental results. 

\paragraph{Model Correction}
Table \ref{} shows the corrected lengths and damping of the host structure obtained after following the methodology explained in Section \ref{}. Figure \ref{} shows the effect of correcting the model. As evident from the figure, the adjusted model accounts for the uncertainties in the structure and matches the experimental data better. 

\paragraph{Spaghetti Property Estimation}
The FRFs of fettuccine of different lengths were obtained experimentally to obtain an estimate of the modulus of elasticity of fettuccine. Figure \ref{} shows the frequency response of fettuccine strands of varying lengths. As evident from the figures, the fettuccine strand has a discernible fundamental mode from which the fundamental natural frequency was determined. The modulus of elasticity of the fettuccine strand was determined using the relation 
\begin{equation}
"first nat freq" and "E" relation
\end{equation}
Calculations showed that the modulus of elasticity varied from $value$ to $value$ and the average was $value$. Since the amount of uncertainty in the area of cross-section, length and moment of inertia is very minimal, any uncertainty can be attributed to the density of the material or the modulus of elasticity or both. After measuring the weights of the spaghetti, major uncertainty in the expected frequency response was attributed to the modulus of elasticity. Table \ref{} shows the final estimates of the structural properties of the fettuccine strands. These values do not represent the actual structural properties of the fettuccine strands. Instead, these values can be used as the initial conditions for design.

\paragraph{Simulation Results}

\paragraph{SOA Results}


%
\begin{center}
    \begin{tabular}{|c|c|c|}
    \hline
    Parameter & Measured values & Corrected Vibrations \\
    \hline
    $L_1$ (m) & $26.5e-2$ & $0.2648$ \\
    $L_2$ (m) & $25.6e-2$ & $0.2562$ \\
    $\zeta_1$ & $0.0011$ & $0.002$ \\
    $\zeta_2$ & $0.0027$ & $0.0035$ \\
    \hline
    \end{tabular}
\end{center}
%
\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[scale = 0.45]{fig_result_1.eps}
  \caption{Before model correction}
  \label{lfig_host_structure_frf_pre_correction}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[scale = 0.45]{fig_result_2.eps}
  \caption{After model correction}
  \label{lfig_host_structure_frf_post_correction}
\end{subfigure}
\caption{Host Structure FRF}
\label{lfig_host_structure_frf}
\end{figure}
%
\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[scale = 0.45]{sim_results_1.eps}
  \caption{Peak 1}
  \label{lfig_soa_sim_frf_1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[scale = 0.45]{sim_results_2.eps}
  \caption{Peak 2}
  \label{lfig_soa_sim_frf_2}
\end{subfigure}
\caption{Simulation Results}
\label{lfig_soa_sim_frf}
\end{figure}
%
\begin{figure}
\centering
\includegraphics[scale = 0.75]{fig_spaghetti_FRF.eps}
\caption{Spaghetti FRF}
\label{lfig_spaghetti_FRF}
\end{figure}
%
\begin{figure}
\centering
\includegraphics[scale = 0.75]{fig_frf_old_soa_frf.eps}
\caption{Old Experiment FRFs}
\label{lfig_frf_old_soa_frf}
\end{figure}
%
\begin{center}
    \begin{tabular}{|c|c|c|c|}
    \hline
    No. & Natural Frequency (Hz) & Predicted Damping Ratio & Predicted Young's Modulus (Pa) \\
    \hline
    1 & 8.462 & 0.01366 & 2.5751e+09 \\
    2 & 8.565 & 0.01032 & 2.6382e+09 \\
    3 & 8.058 & 0.00875 & 2.3943e+09 \\
    4 & 20.676 & 0.0204 & 2.3454e+09 \\
    5 & 20.731 & 0.01848 & 2.3579e+09 \\
    6 & 20.227 & 0.02050 & 2.2446e+09 \\
    \hline
    \end{tabular}
\end{center}
%
\begin{center}
    \begin{tabular}{|c|c|c|c|}
    \hline
    No. & Natural Frequency of Peak (Hz) & Predicted Tip Mass (kg) & Actual Tip Mass (kg) \\
    \hline
    1 & 4.76 & 0.00091446 & 0.00093 \\
    2 & 4.92 & 0.0008194 & 0.00084 \\
    3 & 4.58 & 0.0009664 & 0.00097 \\
    4 & 12.76 & 0.00041633 & 0.00044 \\
    5 & 12.55 & 0.00046008 & 0.00048 \\
    6 & 13.20 & 0.00034724 & 0.00035 \\
    \hline
    \end{tabular}
\end{center}
%
\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[scale = 0.45]{NM_soa_1.eps}
  \caption{Peak 1}
  \label{lfig_soa_nm_frf_1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[scale = 0.45]{NM_soa_2.eps}
  \caption{Peak 2}
  \label{lfig_soa_nm_frf_2}
\end{subfigure}
\caption{Without Mass SOA FRFs}
\label{lfig_soa_nm_frf}
\end{figure}
%
\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[scale = 0.45]{exp_soa_peak_1.eps}
  \caption{Peak 1}
  \label{lfig_exp_soa_peak_1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[scale = 0.45]{exp_soa_peak_2.eps}
  \caption{Peak 2}
  \label{lfig_exp_soa_peak_2}
\end{subfigure}
\caption{Iterative FRFs of SOAs}
\label{lfig_soa_nm_frf}
\end{figure}
%
\begin{figure}
\centering
\includegraphics[scale = 0.75]{exp_final_run.eps}
\caption{Final FRF}
\label{lfig_final_run_frf}
\end{figure}
%
\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[scale = 0.45]{sim_zeta_results_1.eps}
  \caption{Peak 1}
  \label{lfig_soa_zeta_frf_1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[scale = 0.45]{sim_zeta_results_2.eps}
  \caption{Peak 2}
  \label{lfig_soa_zeta_frf_2}
\end{subfigure}
\caption{Damping Results}
\label{lfig_soa_zeta_frf}
\end{figure}
%
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\section{Conclusion}



\section*{References}

\bibliography{j_2_DOF}

\end{document}