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%              Integration Engineering Laboratory                    %
%              Department of Mechanical and Aeronautical Engineering %
%              University of California                              %
%              Davis, CA 95616                                       %
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%              May 7, 1994                                           %
% Modified: February 16, 2001 by Harry H. Cheng                      %
% Modified: January  01, 2003 by Geoffrey R. Shiflett                %
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\title{Subordinate Oscillator Arrays: \\ Physical Design and Effects of Error}

\author{John Sterling$^{1,2}$\thanks{Address all correspondence to this author at john.a.sterling1@navy.mil.}, Sai Tej Paruchuri$^3$, Joseph Vignola$^2$,   \\  \textbf{Andrew Kurdila$^3$, Teresa Ryan$^4$}
\affiliation{$^1$Naval Surface Warfare Centers, Carderock Division, Bethesda, MD, USA \\
$^2$Mechanical Engineering Department, Catholic University, Washington, DC, USA
\\
$^3$Mechanical Engineering Department, Virginia Tech, Blacksburg, VA, USA
\\
$^4$Dept. of Engineering, East Carolina University, Greenville, NC, USA\\
}}

% %%% first author
% \author{John Sterling\thanks{Corresponding Author}
%     \affiliation{
% 	Engineer, Carderock Division\\
% 	Naval Surface Warfare Centers \\
% 	Graduate Researcher, Catholic University\\
% 	        Washington, DC 20064\\
% 	Email: john.a.sterling1@navy.mil
%     }	
% }



% \author{Sai Tej Paruchuri
%     \affiliation{Graduate Researcher\\
%     Mechanical Engineering\\
%     Virginia Tech\\
%             Blacksburg, VA 24060\\
%     Email: saitejp@vt.edu
%     }
% }
% %\author{Sai Tej Paruchuri
% %    \affiliation{Graduate Researcher\\
% %    Mechanical Engineering\\
% %    Virginia Tech\\
% %    Email: saitejp@vt.edu
% %    }
% %}
% \author{Pablo Tarazaga, Ph.D
%     \affiliation{Associate Professor\\
%     John R. Jones III Faculty Fellow\\
%     Mechanical Engineering\\
%     Virginia Tech\\
%             Blacksburg, VA 24060\\
%     Email: ptarazag@vt.edu
%     }
% }

 


% \author{Joseph Vignola Ph.D.
% \affiliation{Associate Professor\\
%         Mechanical Engineering\\
%         The Catholic University of America\\
%         Washington, DC 20064\\
%           Email: vignola@cua.edu
%  }}     
% \author{ Andrew Kurdila Ph.D.
%     \affiliation{W. Martin Johnson Professor\\
%         Mechanical Engineering\\
%         Virginia Tech\\
%         Blacksburg, VA 24060\\
%         Email:  kurdila@vt.edu
%     }}
%     \author{ Vijaya V. N. Sriram Malladi, Ph.D
%     \affiliation{Research Scientist\\
%         Mechanical Engineering\\
%         Virginia Tech\\
%         Blacksburg, VA 24060\\
%         Email: sriram@vt.edu
%     }}


%         \author{\textcolor{white}{Teresa Ryan Ph.D.}}
% \author{Teresa Ryan Ph.D.\\
% 		Assistant Professor\\
% 		Dept. of Engineering\\
%         East Carolina University\\
%         Greenville, NC 27834\\
%         Email: ryante@ecu.edu
%         }
%       \author{\textcolor{white}{Teresa Ryan Ph.D.}}


\begin{document}

\maketitle    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
{\it Prior work demonstrates that an attached subordinate oscillator array (SOA) can attenuate vibration of a host structure over a frequency range of interest.   A judicious choice of the distribution of masses and stiffnesses of the attached oscillators can result in a relatively flat frequency responseof for the host structure over a desired band.  This response modification can be a significant improvement over classical dynamic vibration absorbers (DVA) that attenuate a structure's response at one target frequency while increasing the frequency response amplitude at nearby side frequencies. Performance of the SOA can be highly sensitive to the uncertainty or disorder in the mechanical properties of the system.  This paper introduces a novel design strategy that can make use of either 3D-printing or piezoelectric SOAs (PSOAs). This strategy has the potential to address and ameliorate.  It is important to note that the design strategy is simple and effective in that it can be carried out without computational optimization techniques by choosing simple or well-known distributions of properties.  
}
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{nomenclature}
%\entry{SOA}{Subordinate Oscillator Array.}
%\entry{PSOA}{Piezoelectric Subordinate Oscillator Array}
%\end{nomenclature}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

The design of dynamic vibration absorbers (DVAs) is a classical topic of modern vibrations that has been well-studied in many texts \cite{Inman2001,ginsberg_mechanical_2001}.  In this work, the system to which a DVA  is attached is called the host.  The essential and well-known feature of DVA performance for a single degree of freedom (SDOF) host system is that it is possible to exactly cancel the frequency response of the host at its natural frequency, provided that the mechanical properties of both host and DVA are known precisely. Even for multi-degree of freedom (MDOF) systems, this result holds approximately true. Such an exact cancellation, however, comes at a cost: the amplitude of the frequency response of the host structure can be substantially increased at frequencies close to the target frequency. A fundamental drawback to an approach using a traditional DVA is the need for exact knowledge of the mechanical properties of both the host and the DVA. The DVA design can therefore be sensitive to uncertainty or temporal drift in the mechanical properties of the system. 

%A number of strategies have been studied over the years to reduce the drawbacks associated with 
%\annote[tjr]{enhance}{this presumes the reader knows what makes a response 'good' or 'bad' because enhancing is making it better.  Maybe worth one extra sentence?} 
%the response modification achieved by a classical DVA. These strategies include: active tuning or adaptation of DVAs; switching or semi-switching techniques; designs based on subordinate oscillator arrays (SOAs); or vibration attenuation via active structural 
%components. \note[tjr]{each of the previous could get one or more citations?} \note[JS]{our last paper was about three quarters lit review.  something like 50 references.} \note[tr]{ehhh, the novice reader isn't reading that one.  Ok, you refer to it in the next sentence, but those feel pretty exposed without references. } A comprehensive review of comparable \note[tjr]{systems}{strategies?  approaches?} is available \cite{paruchuri_passive_2019}.  We  focus our study on developing {\em direct and simple} design strategies that are robust with respect to property uncertainty or disorder.

An SOA generalizes the idea behind a DVA by expanding it into a distribution of multiple sequential DVAs whose effects overlap. Each oscillator in this array has varied dynamic properties and these properties are what define the distribution.  The mass, stiffness, and damping of each individual oscillator can be different in a given SOA design. It has been shown\cite{Vignola2009}  that SOAs are capable of attenuating the vibration of the host in an approximately uniform way over a band of frequencies. In principle, an SOA provides substantial improvement in the robustness of the performance over a classical DVA because the SOA design can tolerate variations in the input frequency around the target frequency. However, as with the design of DVAs, realizing this goal in a physical mechanical SOA can be problematic. Response reduction over the desired flat frequency band exhibits sensitivity to {\em{unintended}} variation in the mechanical properties of the primary or the attached masses.  Fabrication errors or uncertainty can have detrimental effects on the realized performance of systems fabricated with a given design strategy. Philosophically, this robustness issue can be expressed in terms of the sensitivity of a design to error in the distributions of the oscillators \cite{vignola_optimal_2013,ryan_noise_2012}. 

 In this paper, a set of design strategies that could be applied singly or in combination for a reduction of this sensitivity is introduced. The discussion begins with a step-by-step outline of a method to design mechanical SOAs.  Section \ref{sec: SOA Design Space} summarizes each of the pertinent design variables, showing the effects of each on the  resulting modulated system response if the values are not optimal.  Section \ref{sec:geometric design of mech SOA} presents an algorithm to prescribe a geometry for manufacture.
  %Then, a thorough discussion and analysis of the disorder problem so that there is clear reasoning for the additional damping and piezoelectric treatments is understood.    
  In Section \ref{sec:Disorder}, a thorough analysis of the disorder problem is presented.  This analysis demonstrates the need careful attention to error in the SOA and suggests two methods for reducing the effects.   The analysis introduces metrics to quantify disorder levels and provide some insight into acceptable levels of disorder under differing conditions.
  %Section \ref{sec: Disorder Mitigation Strategies}   presents methods of circumventing the problems caused by disorder. 
  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{SOA Design Space}\label{sec: SOA Design Space}
Since the primary contributions of this paper are definitions of a general design strategy, a careful description of all relevant parameters follows.  The SOA is depicted schematically as a collection of oscillators represented in the lumped parameter system model shown in Fig. \ref{fig:soamodel}.  The physical implementation of that model used in this work is shown in Fig. \ref{fig:soadrawing}.  The design of SOA must specify the number of oscillators and their length, width and height.  However, there is no closed form prescription to solve this design problem. Determination of each of these values requires several steps.
\begin{figure}[b]
\begin{center}
%\includegraphics{SOAmodel.jpg}
\includegraphics[trim=0in 0in 0in 0.15in, clip=true,width = 0.48\textwidth]{SOAmodel.jpg}
\end{center}		
\caption{The SOA has a primary mass and subordinate masses attached.  The frequency band of suppression is defined by the values of mass and stiffness in the subordinate elements.} 
\label{fig:soamodel} 
\end{figure} 

\begin{figure}
\begin{center}
%\includegraphics{SOAmodel.jpg}
\includegraphics[scale=.7]{SOAdrawing.png}
\end{center}		
\caption{A drawing of the physical layout used to realize the system model in Fig. \ref{fig:soamodel} }
\label{fig:soadrawing} 
\end{figure} 


In the design process, Eq.  \ref{eq:netaqdelta} governs the minimum required number of oscillators,  
\begin{equation}
 N=\eta \Delta Q
 \label{eq:netaqdelta}
\end{equation}
where N is the minimum number of oscillators,  $\eta$ is the modal overlap, $\Delta$ is the desired fractional bandwidth, and $Q$ is the Quality Factor.  Note that $n$ will be used to functionally indicate individual oscillators where N is that total number of oscillators.   All of these parameters are used to shape the frequency response function from input to host over a target frequency range. The fractional bandwidth is defined as the bandwidth of the effect as a fraction of the center frequency of the range that we aim to correct.  Modal overlap is a measure of modal density of the SOA.  It is a unitless $ \left(\frac{Hz}{Hz}\right) $ value calculated on an individual basis by the half-power bandwidth divided by the frequency span between the fundamental frequency of the current oscillator to that of the next. Modal overlap is defined in more detail Section \ref{sec: Modal Overlap}.
	
There are several additional factors that affect the design of an SOA.  System properties of the host system or primary oscillator and material properties of the SOA to be designed must be considered.  The relevant system properties are (1) the mass of the primary system and (2) the width and center frequency of the band to be suppressed. The relevant material properties are the Quality Factor, Young's modulus, and density.  Also, if there are  significant environmental changes, such as temperature variation, additional care must be taken to make sure the SOA will perform correctly throughout the temperature range.
	
In the next few sections, we explain each of the parameters in the design space and discuss how they can be interpreted.  The goal is to build an intuitive notion of how each parameter can be used to alter the designed performance of the system.  Section \ref{sec: Modal Overlap} details the definition and calculation of modal overlap, $\eta$.  Section \ref{sec: Fractional Bandwidth} explores the effects of $\Delta$, the fractional bandwidth.  Section \ref{sec: Quality Factor} describes the effects of damping in the SOA system.  Section \ref{sec: Mass Ratio} explores the effects of the mass ratio of the primary element to the subordinate elements, $\alpha$.     The generation of the frequency distribution, $\beta$, using the frequency distribution parameter, $p$, is outlined in Sec \ref{sec:betaandq}.  Section \ref{sec:constraints} gives an explanation of several manufacturing constraints.


%\subsection{SOA Parameters}
%	A calculation has been developed to factor in every factor and iterate design solutions.  Passing all of the variables into a set of functions to be detailed at the end of the chapter generates a simple array of length and width values at a constant thickness.  These parameters have been introduced in section 3.


\subsection{Modal Overlap Parameter, $\eta$}\label{sec: Modal Overlap}
Modal overlap is defined as the half-power bandwidth divided by the difference in frequency successive peaks, $\Delta$f. Modal overlap has been studied in the literature \cite{lyon_statistical_1975,feit_vibration_1996} For the system as a whole, modal overlap can be calculated as

\begin{equation}
\eta  = \frac{N-1}{\Delta  Q_\mathrm{SOA} } 
\label{eq:modal_overlap}
\end{equation}
where $Q_\mathrm{SOA} = Q_n$ is the Quality Factor of the n$^{th}$ resonance, and the frequency separation between adjacent resonances is given by $\Delta/(N-1)$.  Modal overlap, $\eta$, is generally set to 2 when calculating the required number of oscillators\cite{vignola_optimal_2013}.  Any value below two means that there is not enough modal density to properly create the desired effect.  The effect shows up as ripple in the combined response and is visible in Fig. \ref{fig:etahighlow}.  $\eta$ greater than 2 would be shown but they line up precisely in the response.  Raising $\eta$ above 2 has little benefit but is revisited in Sec. \ref{sec:parameterinfluence}. 
\begin{figure}[h]
\begin{center}
\centering
\includegraphics[scale=.75]{etahighlow3.png}
\caption{An example of the effects of $\eta$ on the SOA system.  The ripple effect starts small but ramps up as $\eta$ is decreased.}
\label{fig:etahighlow}
\end{center}
\end{figure}
\clearpage
\subsection{Fractional Bandwidth, $\Delta$}\label{sec: Fractional Bandwidth}
The SOA is designed to suppress a band of frequencies over a bandwidth measured as a fraction of the center frequency.  The center frequency, $F_c$, and fractional bandwidth, $\Delta$ must be chosen ahead of time in order to design an SOA.  The required width of the fractional bandwidth is dependent on the nature of the peak to be suppressed.  For example, a fractional bandwidth of 10\% would suppress magnitudes between 95\% and 105\% of the center frequency  The SOA is effective at suppressing peaks of both high and low quality factor, but the fractional bandwidth needs to contain the entire width to be suppressed. The fractional bandwidth is easily visualized and shown in Figure \ref{fig:fbgraph}.  Looking at the figure, it may be tempting to increase the bandwidth artificially to increase the total magnitude reduction.   However, this can lead to additional design issues such as: increased attention to $Q$ in the SOA material, increased SOA physical size, and violating material constraints listed in Sec. \ref{sec:constraints}.

\begin{figure}[h]
\begin{center}
\centering
\includegraphics[scale=.75]{fbgraphnon2.png}
\caption{This shows 10\% vs 25\% bandwidth.  Note the non-dimensional frequencies where each curve begins to diverge from the SDOF are at their respective fractional bandwidth levels.}
\label{fig:fbgraph}
\end{center}
\end{figure}
\subsection{Quality Factor, $Q$}
\label{sec: Quality Factor}
When considering Eq. \ref{eq:netaqdelta} with respect to SOA design, modal overlap, $\eta$, is generally set to two and fractional bandwidth, $\Delta$, is the bandwidth of the frequency range to be suppressed. Those two constraints leave the Quality Factor, $Q$, as the only material property involved in determination of the number of oscillators, $N$, required for the SOA.  Higher Quality Factor will marginally increase the peak suppression but also increase the number of oscillators required.  Using a high-$Q$ material in the SOA also requires strict attention to the fractional bandwidth that is chosen.  If the fractional bandwidth is too high, the high-$Q$ SOA will create artifacts at the edges of the band whereas a low-$Q$ SOA is less sensitive to this effect.  These effects are visible in Fig. \ref{fig:doubleQ}.
%Additional attention is also required to the Quality Factor of the primary peak.  A high \annote{Quality Factor SOA }{you mean high q primary, don't you?} with \annote{fraction bandwidth} wider than necessary will create side-lobes.  If using materials with \annote{low damping}{either stay consistent with high q or explicitly remind the reader that highq is low damping} like metals, additional care must be taken to choose the correct fractional bandwidth.  These side-lobes are shown in Figure \ref{fig:doubleQ}.

\begin{figure}[h]
\begin{center}
\centering
\includegraphics[scale=.75]{doubleq3.png}
\caption{An example of the effect of choosing fractional bandwidth that is too large.  Note that the low-$Q$ response does not express the edge artifacts but also achieves less peak suppression.  Both curves display a 15\% fractional bandwidth} %soa_response_double_graph_eta.m}
\label{fig:doubleQ}
\end{center}
\end{figure}

\subsection{Mass Distribution, $\alpha$, and Mass Ratio, $\mu$} \label{sec: Mass Ratio}
With the number of oscillators required in the array, $N$, determined, the mass distribution $\alpha$ can be defined.  The required total mass of the SOA elements is expressed as the sum of the elements of the mass distribution $\alpha(n)$.  For the bandpass configuration, the optimal mass ratio between the total mass of the SOA and the host structure, $\mu$, has been determined to be $\frac{\Delta^2}{4}$ \cite{vignola_optimal_2013}.   This requirement dictates that more mass will be required to suppress a wider bandwidth.    

For the example presented in Fig. \ref{fig:massratiogood}, a desired response maximizes the peak reduction while minimizing side artifacts.  Increasing the mass ratio too much gives the SOA a significant mass ratio compared to the host.  This high mass ratio will create resonant peaks of its own to the sides of the primary peak.  The SOA will still achieve vibration suppression, but will introduce side peaks that can offset the benefits of the initial peak reduction. This behavior essentially returns the performance profile of the system to that of a classical DVA.  In contrast, decreasing the mass ratio too far reduces the amount of energy tranferrable into the SOA system, reducing the suppression capabilities of the system.  

\begin{figure}[h]
\begin{center}
\centering
\includegraphics[scale=.75]{massratiohighlow3.png}
\caption{An example of the effects of mass ratio on the SOA-adorned system.  The high mass ratio returns the performance to that of a classical DVA.  The low mass ratio decreases the performance of the system. }
\label{fig:massratiogood}
\end{center}
\end{figure}


%\subsection{Young's modulus, $E$ and Density, $\rho$}
%Achieving the desired frequency and mass of each individual beam obviously requires attention to the physical length, width, and thickness.  While the Young's modulus and density are contained in the base equations for these dimensions, they do not require any special attention.  However, they are still always present and must be accurately determined. \note[JS]{These are just material properties and scalars.  Do they even belong in this section?}

\subsection{Frequency Distribution $\beta$ and Frequency Distribution Parameter $p$}
\label{sec:betaandq}
Many of the properties of an SOA design are defined in terms of a discrete distribution of a particular property over the set of oscillators.  The discrete distribution of frequencies across the fractional bandwidth, $\Delta$, is represented by the set $\beta$.  Equation \ref{eq:peq} defines $\beta$ for the $n^{th}$ oscillator in terms of the number of oscillators, $N$, the fractional bandwidth, $\Delta$, and the frequency distribution parameter, $p$.  As presented, Eq. \ref{eq:peq} generates a $\beta$ that has a non-dimensional center frequency of 1.  The actual frequency distribution is created by multiplying the set $\beta$ by the center frequency of the host structure.  
%distribution multiplied by the center frequency creates the $\beta$ distribution which is the actual set of fundamental frequencies targeted for the individual oscillators.  
The $\beta$ distribution represents the target or as-designed isolated natural frequencies of each of the individual oscillators that comprise the SOA.  

Figure \ref{fig:pvalues} shows the effect of distribution parameter $p$ on the frequency distribution $\beta$ and Fig. \ref{fig:2pgraph} shows the effect of $p$ on the actual SOA response.  Note that $p>1$ packs more oscillator frequencies on the edges of the SOA target band.  Therefore, the SOA would be less effective on the at modifying the response at the host structure's center frequency, diminishing the performance of the SOA as a whole for an SDOF system. Such a distribution may be of use on a host structure with a bimodal frequency response. A $p$ of approximately 0.75 produces a flat response near the center of the target band(color), generally best suited for an SDOF host structure.  As $p$ approaches 0.6 and below, a dimple will form in the flat response(color). By $p=0.25$, the SOA system response becomes essentially equivalent to that of a classical DVA with an equivalent mass that is simply the total SOA mass.
\begin{equation}
{\beta _n} = \left\{ {\begin{array}{*{20}{c}}
{\frac{\Delta }{2}\left( {{{\left( {\frac{{2\left( {n - 1} \right)}}{{N - 1}}} \right)}^p} - 1} \right) + 1\,\,\,\,\,\,\,{\rm{for}}\,\,n \leq \frac{N}{2}}\\
{\frac{\Delta }{2}\left( {1 - {{\left( {\frac{{2\left( {n - 1} \right)}}{{N - 1}}} \right)}^p}} \right) + 1\,\,{\rm{   for}}\,\,n > \frac{N}{2}}
\end{array}} \right.
\label{eq:peq}
\end{equation}

\begin{figure}[h]
\begin{center}
\centering
\includegraphics[scale=.75]{pvalues.png}
\caption{The $p$ value affects the density of the frequency distribution about the center frequency.  A $p$ value of 1 provides a linear distribution of frequencies throughout the bandwidth, $\delta$.}
\label{fig:pvalues}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\centering
\includegraphics[scale=.75]{4pgraph2.png}
\caption{The $p$ value affects the density of the frequency distribution about the center frequency. A p-value of 1 will return a balanced curve with equal density of frequency distribution throughout the target band.  A p-value of 0.75 will offer more of a flat-band effect as it has more elements of the frequency distribution closer to the center frequency \note[JS]{match colors to fig 7}}  
\label{fig:2pgraph}
\end{center}
\end{figure}

%soa_response_double_graph_2pgraph.m
%\subsection{Temperature Range}
%Certain materials are more affected by temperature than others.  In general, the stiffness of a plastic is likely to be affected by environmental temperature than metals.  This stiffness change will be manifested as disorder in the system.  


\subsection{Manufacturing Constraints}
\label{sec:constraints}


%A successful design chooses all of the aforementioned parameters, but constraints on those values are important.  When designing SOA systems

Successful design of an SOA must consider the practical constraints on the ranges of each of the aforementioned physical parameters. There are two categories of constraints in the SOA design methodology.  Geometric constraints involve ensuring that certain design assumptions are held true. 
For example, this formulation relies on beam aspect ratios that satisfy Euler-Bernoulli conditions.    
Material constraints will differ between materials and chosen manufacturing processes, but must be considered thoroughly during SOA design.  For example, steel will have a much smaller practical minimum thickness than 3D-printed materials. As another example, steel has been successfully used at a thickness of 0.01 inches, while the recommended minimum thickness for ABS, a common 3D-printed polymer, is nearly 5 times greater at 0.047 inches (1.2mm).   Typical constraints to consider are: minimum and maximum length, minimum width, minimum thickness, thickness to length ratio, and width to length ratio.  The values of these constraints are not absolute and, especially for 3D-printed materials, must be determined independently for each design situation.
%Each material, especially 3D-Printed material, has specific limits on things like minimum and maximum thickness or overall size. 


 %Limits are set on certain geometric and material properties.  These limitations are different for each material and not all of them are applicable to every material.  For example, there is no nominal maximum thickness for a steel part, but there is for 3D-Printed parts.  These limits exist for a variety of reasons ranging from manufacturing capabilities to part fragility to calculation accuracy.  The limits are set on several properties: minimum and maximum length, minimum width, minimum thickness, thickness aspect to length, and width aspect to length.  

%%Minimum width and length apply to all materials and are based on manufacturing tolerances.  Maximum length is applied to 3D-printed parts because several 3D-printed materials are limited by overall workspace size.  Minimum thickness is another manufacturing issue that varies by material and method.  Metals can be very thin and keep dimensionality (0.01 inch has been used successfully before), but 3D-printed materials can be very sensitive in post-processing to being too thin.  The constraints on aspect ratios act to keep the standard beam assumptions valid.  One aims to keep the length/thickness aspect approximately 10/1 and length/width above 5/1.  The values of these constraints are not absolute and, especially for 3D-printed materials, can be dependent on the experience of the researcher to set these personally.

%\note[JS]{Do I need to make a huge table with the values of material library I've collated?  Make an appendix?  Is this appropriate for a journal?  These tables are not analytic in any way simply tricks of the trade i've found after making hundreds of these}
\section{Physical Design of SOAs}
\label{sec:geometric design of mech SOA}
The preceding section has introduced all of the theoretical parameters required for SOA design. At its heart, the purely mathematical SOA is simply a set of property distributions that can be applied to any physical second order system, such as electrical, acoustic, or mechanical.  The following sections will show the steps for the design a planar mechanical SOA.  Section \ref{sec:mechdesign} will discuss mechanical design of an SOA.
%and Sec. \ref{sec:piezodesign} discusses the addition of piezoelectric components to the mechanical geometry \note[JS]{add something about why}.
\subsection{Design of Mechanical SOA}
\label{sec:mechdesign}
As the manufacture of a mechanical SOA is attempted, one must carefully consider the properties of the primary system that is to be damped.  Specifically, the mass and frequency bandwidth of the target system must be determined accurately.  That frequency band is defined on two parameters, center frequency, $F_c$ and fractional bandwidth, $\Delta$. Additionally, the material from which the SOA is constructed is also important.  The material stiffness and density must be known accurately. %The damping will also have an effect, but that is already accounted for in the calculation of $N$.   


%and also that the frequency distribution parameter, $p$, affects the shape of the curve as well.  

Starting from the classical fundamental beam frequency estimate equation from Kinsler \cite{kinsler_fundamentals_1999}, the uniform rectangular cross-section this becomes
%\begin{equation}
%F_n=\frac{1.875^2}{L} \frac{1}{2\pi} \sqrt{\frac{EI}{\rho A}}  
%\label{eq:kinslerfn}
%\end{equation}
%

%Then rearrange some of the compound variables into individual dimensions.
\begin{equation}
F_n=1.875^2 \frac{1}{2\pi} \sqrt{\frac{Et^2}{12\rho L^4}}  
\label{eq:fnequation}
\end{equation}
where $E$ is the material stiffness, {$t$} is thickness, $\rho$ is material density, and $L$ is beam length.  Equation \ref{eq:fnequation} can be rearranged into an equation with length on the left.  Then, realizing that $\beta(n)F_c$ is equivalent to $F_N$, an expression for the length of each element is generated 
\begin{equation}
L(n)=\frac{1}{2\pi}\sqrt[4]{\frac{Et^2}{12\rho}\left(\frac{1.875^2}{\beta(n)F_c}\right)^2}
\end{equation}

%To simplify manufacturing and calculation, the thickness of the part is chosen iterated to produce aspect ratios near 10:1 but that also correspond to standard fractional measurements for ease of manufacturing.  Because the width of the beam does not have a significant effect on the bending frequency of a beam, a set thickness allows the length to be calculated for each frequency via the rearrangement


Recall from Sec. \ref{sec: Mass Ratio}, it was stated that the optimum mass ratio for the flat-band response, $\mu$,  equal to $\frac{\Delta^2}{4}$. The mass of each oscillator is calculated knowing the total SOA mass and distributing it across the array.  For this system, the mass ratio distribution will have a negative linear distribution parameter, optimized in \cite{vignola_optimal_2013} to be -1.  The lowest frequency element will be the most massive one and the highest frequency will be the least massive.   The mass distribution, $\alpha$, is defined as


\begin{equation}
\alpha(n)=-\left(\frac{\mu}{N}\right)\alpha^*(n)
\end{equation}
where $\alpha(n)$ represents the distributions of the masses of each oscillator, $N$ is the total number of oscillators and $\alpha^*$ is a linear distribution from  1-$\frac{\Delta}{2}$ to 1+$\frac{\Delta}{2}$ with $N^*$ elements. 
%{\color{red}This is an attempt to combine and explain 3 lines of MATLAB code commented below.  It is a difficult passage but I believe it says what it needs to say.}
 %mass_slope = fractional_bandwidth*mass_slope_parameter;
 %alpha_star = linspace(1-mass_slope,1+mass_slope,N)';
 %alpha = alpha_star*mass_ratio/N;
%The $\alpha^*$ parameter is similar to $p$ in its limits, but is not given in the SOA parameters because it is not changed or varied.  For simple systems where the mass of the primary system is just the mass of the beam used to create the center frequency, the mass of the oscillators are small.

Once element masses are defined, geometry determination can proceed using a given material's density and material constraints and iterating the thickness.  Knowing that $$m=\rho L Wt,$$
one can define the width distribution as 
\begin{equation}
    W(n)=\frac{\alpha(n)}{\rho L(n) t_i}.
\end{equation} 
where $t_i$ is the iterated thickness level.  Assuming thickness to be constant for all elements, the length and width distributions are adjusted to follow the prescribed $\alpha(n)$.  As such, there are an infinite number of solutions to the SOA design problem.  This infinite set is bounded by eliminating violations of previously discussed material limits, which, again, will vary depending on material and manufacturing method.

%and that the mass of each individual oscillator has now been defined, the width can be determined by stipulating a constant thickness. As thickness is iterated, the length and width distributions are adjusted to keep a constant mass.  These dimensions and aspects are checked against the previously discussed material limits.
%A flowchart for the design algorithm is shown in Fig. \ref{fig:geobuild}.  The geometry building algorithm requires all of the variables defined earlier in this section as well as the mass of the primary.  It then polls information from a library containing both material properties and manufacturing limitations. These limitations are different for each material and not all of them are applicable to every material.  For example, there is no nominal maximum thickness for a steel part, but there are for many 3D-Printed parts when workspaces are on the order of a square foot or less.  
%\note[JS]{TO AK- You asked that the flowchart be labelled with letters and referred to in the design process.   The design process is actually just contained in one box in the flowchart.   The rest of the flowchart is detailing the dimensional acceptability process which is essentially a whole new thing.  I made the flowchart because I thought it did a better job of explaining it than a few paragraphs would.}
%\begin{figure}[h!]
%\begin{center}
%\centering
%\includegraphics[scale=0.75]{geobuild.png}
%\caption{A flowchart for the geometric design of SOAs}
%\label{fig:geobuild}
%\end{center}
%\end{figure}
%\clearpage
%\subsection{Geometric Design of Piezoelectric SOAs}
%\label{sec:piezodesign}
%Adding a piezoelectric patch to each oscillator complicates the matter.  This geometry operates in the 3-1 mode of the piezoelectric model.  Due to cross-coupling between the force and electric fields, the width now has an effect on the overall stiffness. there is no simple formula to determine the proper length.  However, a similar methodology as described above will work, and in this case COMSOL with MATLAB livelink allows the use of both softwares in tandem to iteratively estimate the combined dimensions required.  At low frequencies a closed-form expression has been developed, as well as a described method of matching geometry to a prescribed frequency\cite{zhao_design_2018}. Equation \ref{eq:piezofn} describes this much more complex expression for low frequencies,

%\begin{equation}
%F_n=\frac{1.875^2 t}{4\pi L^2} \sqrt{\frac{E_p [1+3(1+2A)^2 + 4A^3 B]}{3\psi_p[4(1+A)^2(Aq+1)]}},
%\label{eq:piezofn}
%\end{equation}
%\note[JV]{change rho to something else}
%where $t$ is the total thickness, L is the beam length, $E_p$ is the elastic modulus of the base beam, $A$ is the thickness ratio of the piezoelectric patch to the total thickness, $B$ is is the elastic modulus ratio of the piezoelectric patch to the base plate, $\psi_p$ is the density of the base plate, and $q$ is the density ratio of the piezoelectric patch and the base plate.   While many piezoelectric devices operate at high frequency, the natural frequency of beam-shaped piezoelectric patches used in this application have low-frequency resonances.  This is the starting point for the design.
%From there, COMSOL$^{TM}$ using MATLAB$^{TM}$ livelink can be used to iterate the thickness in the general algorithm without a closed form solution.   A basic flowchart is included in Fig. \ref{fig:livelink}

%\begin{figure}[h!]
%\begin{center}
%\centering
%\includegraphics[scale=0.35]{livelinkh.png}
%\caption{A flowchart for calculating geometries of piezo-attached SOAs}
%\label{fig:livelink}
%\end{center}
%\end{figure}

\section{Disorder}
\label{sec:Disorder}
%One of the most important aspects in the design of a subordinate oscillator array is the sensitivity of performance to error in the distributions of mass and frequency.  This is referred to as disorder.  Although the design of the SOA is precise and methodical, there are several sources of disorder.  These include, but are not limited to: manufacturing tolerances, thermal effects, fatigue, and imprecise knowledge of boundary conditions.  The end result is that even though the initial design may perform well, life cycle and environmental factors must be considered.  In the literature\cite{vignola_shaping_2009}\cite{vignola_optimal_2013}, the precision of the $\alpha$ and $\beta$ distributions has thus far needed to be greater than 99 percent to achieve good results.   Error on the order of 1 part in 1000 is detectable and 1 in 100 has a significant effect on the results.
%{\color{red} look for repeated stuff}

%The system diagram is shown in Fig. \ref{fig:SOAmodel}. In this paper, we will consider only the frequency response of the system.  A pair of figures,  Fig. \ref{fig:band_pass_disorder_2016} and Fig. \ref{fig:band_pass_disorder_levels_2016} show several configurations of the system, and again with added error. 
%As a general example and refresher of the basic properties of an SOA-attached system, Fig. \ref{fig:band_pass_disorder_2016} shows the desired response of an SOA-attached system at different bandwidths.  These curves have 0 disorder.
%Fig. \ref{fig:band_pass_disorder_levels_2016} includes four different distributions of the masses, $m_n$, stiffnesses, $k_n$ with low damping.  In this figure, disorder is randomly introduced into the amplitude of the individual elements of the non-dimensional $\beta$ distribution by adding a value within the range of [-1,1] times the original frequency.  This is to say that at each non-dimensional frequency, the associated amplitude is modified.  However, before that random value is added, it is multiplied by the disorder multiplier that is shown in the legend.  This means that when the disorder multiplier is close to 0, the effect on the $\beta$ is distribution proportionally small.  If the disorder multiplier is set to 1, each individual amplitude could be anywhere from 0 to double the original value.   

One of the most important aspects in the design of a subordinate oscillator array is the sensitivity of performance to error in the distributions of mass and frequency.  The physical realization of an SOA design will have unavoidable error, deviation, or drift from the specified design parameters. There are several sources of such unanticipated disorder.  These include, but are not limited to variation due to: manufacturing tolerances, corrosion, fatigue, and imprecise knowledge of boundary conditions.  Even though the initial design may perform well, life cycle and environmental factors must be considered.  As shown in Fig. \ref{fig:band_pass_disorder_levels_2016}, error in the $\alpha$ and $\beta$ distributions on the order of 1 part in 1000 is detectable and 1 in 100 has a significant effect on the results\cite{vignola_shaping_2009,vignola_optimal_2013}.

\begin{figure}[h]
\begin{center}
\includegraphics{band_pass_disorder_levels_2016}
%\includegraphics[trim=0in 0in 0in 0.15in, clip=true,width = 0.48\textwidth]{band_pass_disorder_2016}
\end{center}		
\caption{A flat-band response was designed to have a 10\% bandwidth.  The performance degrades as error in the property distribution increases from $d=.0001$  where the bandpass performance is unaffected to $d=0.1$ where the bandpass is response is not apparent.  } % this plot was generated by bandpass_with_disorder_levels_2016.m
\label{fig:band_pass_disorder_levels_2016} 
\end{figure} 
%the precision of the $\alpha$ and $\beta$ distributions has thus far needed to be greater than 99 percent to achieve good results.   
%{\color{red} look for repeated stuff}

%The system diagram is shown in Fig. \ref{fig:SOAmodel}. In this paper, we will consider only the frequency response of the system.  A pair of figures,  Fig. \ref{fig:band_pass_disorder_2016} and Fig. \ref{fig:band_pass_disorder_levels_2016} show several configurations of the system, and again with added error. 
Figure \ref{fig:band_pass_disorder_levels_2016} includes four different distributions of the masses, $m_n$, and stiffnesses, $k_n$, and constant low damping.  In this figure, disorder is randomly introduced into the $\beta$ distribution.  Disorder is introduced in a statistical manner according to

\begin{equation}
\beta_d(n)=\beta_0(n)+d\epsilon(n),
\label{eq:disordereq}
\end{equation}
where $\beta_d(n)$ is the disordered $\beta$ distribution, $\beta_0$ is the original, zero-disorder distribution, $d$ is the disorder level, and $\epsilon(n)$ is the Gaussian random distribution with a standard deviation of 1.   This formulation may produce values for $\beta_d$ that are negative, setting a maximum disorder such that $d<0.25$ drastically reduces this chance.  At $d=0.25$, the chance of this occurrence is 4 standard deviations from the mean, less than $0.0001$.  Systems with disorder at or above this level will perform so poorly that analysis of the system becomes unimportant.  As the disorder multiplier is reduced to levels that are relevant to SOA system analysis, on the order of $d=0.1$, this chance becomes vanishingly small.  As an error trap, any negative values of $\beta_n$ are eliminated by using the absolute value. 
%mathematically induced into the frequency distribution through a disorder multiplier multiplied by a random number limited by the width of the $\beta$ distribution. In order to calculate the value of each of the isolated natural frequencies, a random number is added or subtracted from each ideal isolated natural frequency value in the range of zero to $\beta_n$.}  From here, it is multiplied by a factor that is given on the x-axis.  This is to say that the far left parts of the x-axis, the possible disorder in the range is small despite the random factor added to the frequency value. {\color{red} Once the x-axis equals 1, the range of the frequency value is [0,2f].} 
Figure \ref{fig:band_pass_disorder_levels_2016} also illustrates how poorly systems with $d>.1$ perform and why disorder levels above that are of little interest.

%\begin{figure}[h]
%\begin{center}
%\includegraphics{band_pass_disorder_2016}
%\includegraphics[trim=0in 0in 0in 0.15in, clip=true,width = 0.48\textwidth]{band_pass_disorder_2016}
%\end{center}		
%\caption{The array of attachments can be designed to make the response of the primary element behave as a bandpass.  This curve is generated with 0 disorder to show optimized performance. } % this plot was generated by bandpass_with_disorder_2016.m
%\label{fig:band_pass_disorder_2016} 
%\end{figure} 
 


%Earlier research indicates that disorder is one of the main confounding factors in SOA applications.  Disorder is a somewhat nebulous term encompassing the state of imperfection in the distributions of the SOA construct.  The effects of these imperfections manifest as (( reduced energy transfer in a time-domain system)) \note[JS]{Maybe talking about a time-domain system isnt even a door worth opening as it has been discussed in Aldo and Teresa's work and isn't really important here?} responses that do not conform to the flat band-rejection response in the frequency domain.  Disorder itself is generally expressed in a statistical sense, an overall error level across the entire distribution.  Disorder This has a nonlinear relationship with the measured effects of disorder, measured in dB as a deviation from the desired response.  Disorder in a statistical sense can be added easily in MATLAB and modelled, and is inescapable in applications due to manufacturing error and environmental factors.
%\section{Introduction disorder starring N=Q*FB*eta}
%{\color{red}With some low Q, i've seen some decent results where $N<Q*F_b*eta$   THIS IS NEW WORK.   Talk about this in detail here.}
%\subsection{Properties of Classical Disorder}

\subsection{Performance metrics}
Two methods have been developed to distill the performance of a complex structure like an SOA into a single number that apply to different situations.
We begin with Fig. \ref{fig:metricgraph} which shows the basic SDOF response of the primary system.  Also shown is an ideal, zero-disorder response as well as a response with 10\% disorder.
%There are two applicable methods of quantifying disorder and in this explanation.
%Fig. \ref{fig:band_pass_disorder_levels_2016}.  
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=.75]{metricgraph.png}
%\includegraphics[trim=0in 0in 0in 0.15in, clip=true,width = 0.48\textwidth]{band_pass_disorder_2016}
\end{center}		
\caption{The difference between the 0-disorder and disordered curves form the basis for the set used to calculate the RMS and MDIB metrics} % this plot was generated by bandpass_with_disorder_2016.m
\label{fig:metricgraph} 
\end{figure} 

As stated before, the response is simulated from the disordered distributions by adding or subtracting randomized amounts to each element in the $\beta$ distribution.  Although there are a discrete number of oscillators, the response of the SOA system is continuous.  This continuous response can be calculated at any arbitrary point to provide better resolution for the metric than simply $N$ oscillators worth of data points.  The difference between the two curves at some number of frequency points that is $>>N$ is calculated, similar to a standard transfer function.  This calculation is shown in Eq. \ref{eq:Geq}. 
\begin{equation}
\label{eq:Geq}
G_M(\Omega_j)=G_0(\Omega_j)-G_d(\Omega_j)
\end{equation}
where $G_M$ is the set of difference between the zero-disorder $G_0$ and the disordered curve $G_d$ and $j$ marks the individual sample in the frequency domain. With $G_M$ created, we can calculate a metric for this individual pair of FRFs.
%\begin{equation}
%G_M(j)=\sum^j_{j=1}(\beta_0(N,j)-\beta_d(N,j))
%\end{equation}
%Both metrics begin with a comparison of the response of the primary system with the SOA attached in the target frequency band and examines the difference in magnitude of the zero-disorder curve. To be clear, this means that at each frequency sample in the target band, the difference between the disordered and 0-disorder curve in the example in Fig.  \ref{fig:metricgraph} is recorded and compiled into a set $G$.
%In this example, the 10\% bandwidth case of Fig. \ref{fig:band_pass_disorder_2016} and disordered curve (any of the curves in Fig. \ref{fig:band_pass_disorder_levels_2016}) at each frequency point and recorded as a set.  

The root-mean-square of the set has been labeled the RMS metric and is calculated as shown in Eq. \ref{eq:RMS}.

\begin{equation}
    \hat{M}_{rms} = \left( \frac{1}{J}\sum^J_{j= 1}\vert G_M(\Omega_j) \vert^2  \right)^{1/2}
\label{eq:RMS}
\end{equation}
%and the maximum difference in the band has been labeled $M_{MDIB}$ and is calculated as
%\begin{equation}
%    M_{rms} = \left( \frac{1}{N}\sum^N_{n= 1}\vert G_{dn} - G_{0n} \vert^2  \right)^{1/2}
%\label{eq:RMS}
%\end{equation}
%\begin{equation}
%    M_{rms} = \left( \frac{1}{\Delta}\int^{\beta_1}_{\beta_N}\left(\vert G_{dn} - G_{0n} \vert^2\right)d\Omega  \right)^{1/2}
%\label{eq:RMS}
%\end{equation}
%
%where $N$ is the number of 
%where $\beta_1$ is the first value in the target band of the SOA, $\beta_N$ is the last value in the target band of the SOA, $\Omega$ is the non-dimensional frequency, $G_0$ and $G_d$ represents the impulse response of the system. 
The greatest absolute value in this set has been labelled the Maximum-Difference-In-Band (MDIB) metric.   This is measured simply in Eq. \ref{eq:MAX} as
\begin{equation}
    \hat{M}_{MDIB}=\underset{j=0\rightarrow J}{MAX}\left(|G_M(\Omega_j)\right|).
    \label{eq:MAX}
\end{equation}
%\underset{}{}
%Referring to Fig. \ref{fig:band_pass_disorder_levels_2016} and Fig. \ref{fig:band_pass_disorder_2016}, the two metrics both begin by taking the difference in amplitude between  For the MDIB metric, the largest absolute value of the set is recorded as the metric.  For the RMS metric, obviously the RMS of the set is taken as the metric.
%Eq 12 and 13 depend can be evaluated for each value of the parameter $\beta$ The curves we are about to see will average Mhat and be able 
%While the RMS will never be greater than the MDIB, these could be used in different scenarios.  The RMS average is used to display the overall performance of the system in overall vibration reduction across the entire band, useful in a general overall analysis and specifically for a fatigue life analysis regime.  The MDIB strategy would be employed if the system was sensitive to any vibration levels in the band, such as in a detection regime.  There are essentially 3 situations that compare the two methods. 

%First, a single high-Q oscillator out of order in an otherwise ordered system would show a low RMS metric, but this artifact would be detected by the MDIB method. 

%Second, a series of moderately disordered oscillators where none is a significant outlier would show similar results between the two methods.  This is the most common situation and is expected under normal manufacturing situations.  The MDIB method would obviously show slightly higher levels but the trends remain identical. 

%Lastly, a system with that is well-built but shifted on the frequency axis is considered.  This case would occur when an SOA was applied to a resonance, but then the resonance moves slightly.  The MDIB metric would consider this a poor system, as there would be an a significant difference at the edges of the target band.  The RMS method would average out the edge artifacts and give a low overall error level due to the fact that the vast majority of in-band levels would match. 

When measuring the performance of an SOA, it is important to consider both metrics each individual application of the SOA system and whether the MDIB or RMS is appropriate.  RMS will be used as the in analysis for the rest of this paper so as not to unnecessarily double calculations.  

These metrics generate a single value that represents the performance of the SOA.  However, as it depends partly on a random distribution, it is prone to inaccuracy and does not lead to any insights on its own.   In order to reduce the variation, the disordered curve $G_d$ is regenerated at the same disorder level and a new $\hat{M}$ is calculated.  This is repeated until the set of $\hat{M}$ has stabilized.  Experience has shown that this occurs after approximately 250 iterations.  In this work, 2500 calculations were made to ensure this stability.  This mean is designated as $M$.
  
%However, for the rest of this paper, the RMS metric will be used.

%{\color{red} Show example of each of the 3 situations.  1 graph Do I need to show figure examples of the situations?  Do I need to detail the effective P distribution that produces these or just show that it is possible?  The descriptions seem easy enough to follow and again, so many figures}

\subsection{Influence of Varying SOA Parameters on Metrics}
\label{sec:parameterinfluence}
Now that we have a single accurate RMS metric $M_{RMS}$, we can begin to vary parameter values.  The following analysis shows the effect of multiple levels of modal overlap and quality factor as disorder increases.  The RMS metric will be shown, but the MDIB, results follow the same trends, only with higher values of the metric. 

In the following figures, the x-axis has a resolution of 24 disorder levels ranged from $10^{-4}$ to $1$.  For each modal overlap, $\eta$, or quality factor, $Q$, $M_{RMS}$ is generated at the given disorder level.  It is important to note that the curves in these figures have 3 regions.  There is a region where $d<0.001$ where the results are defined by the ability of the parameters to generate a response that meets the zero-disorder reference curve.  In other words, this low disorder zone is dominated by the ability of the disordered curve to match the reference curve from Fig. \ref{fig:metricgraph}.  There is a transition zone, $0.001<d<0.1$ where the effects of of disorder and dynamic properties of the system are in conflict, and a zone, $d>0.1$ where the effects of disorder completely take over the results.  In the disorder-dominated zone, there is essentially nothing that can be done to recover the system.

Figure \ref{fig:etagraph} shows an example of the effect of increasing disorder at different levels of $\eta$.  There are two important points to take away from these curves.  The first observation is that for $\eta<2$, there exists levels of $M_{RMS}$ that do not vanish no matter how low the disorder goes.  This effect is due to the intrinsic lack of coverage of the frequency band that having a modal overlap below 2 causes as discussed in Sec. \ref{sec: Modal Overlap}.  
The second observation is that $\eta>2$ has very little effect in the low-disorder regime and only moderate effectiveness in the transition zone. Once modal overlap reaches a value of 2, the requirements for a fully covered frequency band are met and increasing from there is simply adding additional unnecessary oscillators to the system.  The additional oscillators add some resistance to the effects of disorder in the mid-range, but while the disorder is low there is negligible effect.
%does not appreciably lower the $M_{RMS}$


\begin{figure}[h]
\begin{center}
\centering

\includegraphics[scale=1]{etasweepnew.pdf}
\caption{RMS metric showing $\eta$ as a function of averaged random disorder.  Note that even at very low disorder, $\eta<2$ has base levels of error sensitivity. The parameter $\eta>2$ has very little advantage.  By 1\%, the errors begin to line up.}
\label{fig:etagraph}
\end{center}
\end{figure}
Figure \ref{fig:100Qrms} depicts the effect of varying damping levels in the SOA.  Two characteristics are immediately visible.  Much like having $\eta<2$, if the damping is too high, there is static $M_{RMS}$, even in the zero-disorder case, as the highly-damped oscillators are not able to create the reference response. The more highly damped system simply cannot accurately recreate the chosen reference.  On the other hand, if the damping level is too low, $M_{RMS}$ begins to increase more quickly.  This effect is due to the fact that with lower damping, any individual error can have a bigger effect on the system as a whole.   Like with $\eta$ in Fig. \ref{fig:etagraph}, there is a transition point between disorder and damping which determines the dominant factor in the metric.  This region occurs in both figures at the same levels of disorder.  
%What occurs is similar to a Gibb's phenomenon where there is a flat area of inherent disorder in the low-damping curves in the left side of the graph.  

% One must notice that there is a particular $Q$ below which there in which the $M_{RMS}$ curve remains \annote{JS}{Is this the right word to describe the behavior?}concave and also minimizes the static error from having a lower $Q$ than the reference response.  In Fig. \ref{fig:100Qrms}, this is a $Q$ of 50, but this is relative to the $Q_{pri}$ of the host system and will change based on the properties of the host.
 
 One must notice there is a particular $Q$, termed $\hat{Q}$, below which the $M_{RMS}$ curve is only concave up.  When $Q<\hat{Q}$, the static $M_{RMS}$ caused by the inability to precisely recreate the reference response increases.  Conversely, $Q>\hat{Q}$ increases the system's disorder sensitivity in the transition region.  In Fig. \ref{fig:100Qrms}, $\hat{Q}$ is approximately 50.  Numerical simulation indicates that $\hat{Q}$ depends on the primary structure's damping and is generally less than the Quality Factor of the primary structure, which in this case is $Q=100$.
 
 %The value of $\hat{Q}$ is relative to the damping present in the host structure.  
 
 %Conversely, above this $Q$, static error due to inability to precisely recreate the r
 %Above the critical level of damping are curves which trend towards 0 $M_{RMS}$ as disorder goes to 0.  There is a range of $Q$ that minimizes both built-in $M_{RMS}$ and the rate at which $M_{RMS}$ increases with disorder.  This is, however, going to be relative to the $Q_{pri}$ of the system.  For many systems, $Q_{pri}$ is often around 100.  In this case, the best $Q$ is approximately 50.

%%%%%%%%%%%%%%%%DISSERTATION CONTENT
%The interaction between primary and subordinate damping is also complicated and takes several graphs to tell the story.  
%Figures \ref{fig:20Q}, \ref{fig:100Q}, and \ref{fig:250Q} {\color{red} Change the color (copper in MATLAB) of these curves to Teresa's color banding to more easily show progression } depict the effect of damping in the SOA system. Both damping in the primary and subordinate system are considered. $Q_{Pri}$ is set to 20, 100, and 250 respectively. One effect is that as values of the Quality Factor of the SOA increase above that of the primary, MDIB levels increase at nearly all disorder levels but does not affect the RMS levels.  This is because a greater $Q_{Pri}$ increases the potential difference between the 0-disorder primary response and any possible randomly generated difference has a higher difference in amplitude. 

%While initially unexpected, this is due to the fact that the lower damping necessitates many more oscillators and increases the maximum performance of the array. The increased performance also causes the potential for greater difference caused by disorder.  As the disorder levels increase in addition to the decreased damping, the potential for a single oscillator this decreased damping causes the system to be significantly more likely to develop a single oscillator out of tune which strongly influences the MDIB metric.  That is to say, it is much more likely to have a statistical outlier in the sample.
%Despite the scaling differences as the Q increases, one thing appears constant:  a Q of approximately 30-40 falls in the ``pocket" of the graph such that there is low inbuilt error but also a more slowly increasing magnitude. This appears to be the best compromise between performance and error sensitivity based on damping levels.
%In addition, increased damping in the primary system reduces the scaling factor of the MDIB metric but the shapes of the curves remain constant.  In all of these curves, there is a pocket of $Q_{SOA}$ that both minimizes the 0-disorder MDIB and the rate of MDIB increase.  For all of these sets of curves, $Q_{SOA}$ of approximately 30-40 appears to be optimal. \note[JS]{I don't know why, but it shows up however I seem to set up the sim.}

%\begin{figure}[h]
%\begin{center}
%\includegraphics[scale=1]{Disorderengine20qgood.pdf}
%\includegraphics[trim=0in 0in 0in 0.15in, clip=true,width = 0.48\textwidth]{band_pass_disorder_2016}
%\end{center}		
%\caption{This is a set of curves detailing MDIB of several values of $Q_{SOA}$ with $Q_{pri}$ set to 20 }  %this plot was generated by "Soa_response_disorder_engine_Qsweep.m" and the matlab output is disorderengine250q.mat
%\label{fig:20Q} 
%\end{figure} 


%\begin{figure}%[t]
%\begin{center}
%\includegraphics[scale=1]{qsweepmaxqpri100.png}
%\includegraphics[trim=0in 0in 0in 0.15in, clip=true,width = 0.48\textwidth]{band_pass_disorder_2016}
%\end{center}		
%\caption{This is a set of curves detailing MDIB of several values of $Q_{SOA}$ with $Q_{pri}$ set to 100 }  %this plot was generated by "Soa_response_disorder_engine_Qsweep.m" and the matlab output is disorderengine250q.mat
\label{fig:100Q} 
%\end{figure} 
%%%%%%%%%%%%%%%%%%%%%%%DISSERTATION CONTENT%%%%%%%%%%%%%%%%%%%%%%


\begin{figure}%[t]
\begin{center}
\includegraphics[scale=1]{qsweep100soarms.png}
%\includegraphics[trim=0in 0in 0in 0.15in, clip=true,width = 0.48\textwidth]{band_pass_disorder_2016}
\end{center}		
\caption{This is a set of curves detailing RMS of several values of $Q_{SOA}$ with primary damping set to 100 }  %this plot was generated by "Soa_response_disorder_engine_Qsweep.m" and the matlab output is disorderengine250q.mat
\label{fig:100Qrms} 
\end{figure} 

%\begin{figure}%[t]
%\begin{center}
%\includegraphics[scale=1]{Disorderengine250q.pdf}
%\includegraphics[trim=0in 0in 0in 0.15in, clip=true,width = 0.48\textwidth]{band_pass_disorder_2016}
%\end{center}		
%\caption{This is a set of curves detailing MDIB of several values of $Q_{SOA}$ with $Q_{pri}$ set to 250 }  %this plot was generated by "Soa_response_disorder_engine_Qsweep.m" and the matlab output is disorderengine250q.mat
%\label{fig:250Q} 
%\end{figure} 
\clearpage



%\subsection{Bandwidth interactions with Intrinsic Damping}
%Primary systems with a high $Q$ do not need high fractional bandwidth.  Bandwidths much wider than necessary will actually increase the response of the combined system outside where the fractional bandwidth is wider than the width of the primary curve.

%{\color{red} What kind of analysis can I do here.   Is it just worth showing how randomized error in small amounts can make it double humped, shifted lowered hump aside from the statistically averaged error we see in the vignola code examples given in the published papers?}

%\subsection{{\color{blue}reconsider} Effects of Quality Factor on 3-D Printed SOAs}
%Previous work\cite{vignola_shaping_2009} has focused on simplifying the variables required to describe the SOA analytically.  Instead of describing specific length, width, and thickness dimensions for each element, the array is specified according to distributions of frequencies and masses across $N$ oscillators.  In this method, design of effective bandpass arrays is simplified as the frequency distribution combines length and thickness and the mass ratio (defined as a ratio of the sum of masses of the individual elements to that of the primary mass). The mass ratio can be adjusted by changing subordinate cantilever width without an effect on the frequency, as long as the aspect ratio of the subordinate cantilever is maintained to be beam-like and not move to a more plate-like regime.  The frequency distribution minimums, maximums, and frequency spacing define the band and shape of the effect.  The mass distribution contributes to the magnitude of the bandpass response as well as edge effects.  The comprehensive effects of the mass distribution are still under investigation.





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\clearpage
\section{Conclusions}
This work has presented the underlying principles and implementation guidelines for creating a set of attachments that allows the tailoring of the frequency response of a host structure.  The effects of the sensitivity of these attachments to error in the distributions of key properties was analyzed.  Such error may be caused by several factors, such as manufacturing defects, environmental effects, and design methodology problems. %Three ways to mitigate this error sensitivity have been provided.  One or all of these techniques may be used in parallel depending on the system characteristics.

In cases where temporal drift of host structure properties is an issue, a classical vibration absorber shows significant degradation of performance and may even increase response amplitude in nearby frequencies.  The subordinate oscillator array offers improved host structure response modification performance.   If the range of the anticipated frequency drift can be quantified ahead of time, an increased fractional bandwidth can be used to accommodate the drift at the cost of more stringent manufacturing constraints.

Sources of disorder are manifold.  It is not possible to eliminate any of the listed sources of disorder but the magnitude of these effects may be small.  In a normally distributed addition of error to the distributions of SOA parameters, there is a highly nonlinear response.  SOA design parameters can be altered to accommodate sources of disorder.

In cases where the mass of the host structure is large, reasonable manufacturing techniques allow for increasingly tight tolerances which keeps the error levels low, even at low damping levels.  As such, the larger the host structure, the less of an issue disorder of this sort presents.  
%In cases of a system with time-varying frequency response characteristics, the tuning capability presented in the the piezoelectric attachment model allows the SOA to track the moving resonant frequency of the host structure.  This mitigation technique, however, is also the most expensive and difficult option of the three.  
In systems that require less stringent response reduction, constructing an SOA with more highly-damped materials lowers this sensitivity to error.  This sensitivity reduction comes with a moderately decreased ability for the SOA to pull energy from the host structure, but the advantage of reduced sensitivity to disorder may often be worth the compromise.

%The implications of this tuning capability are novel and important for vibration suppression in any system with time-varying frequency characteristics.   A passive SOA can be constructed with additional damping in order to soothe the sensitivity to disorder in a mechanical system and given a higher fractional bandwidth to accomodate center frequency drift.  The increased damping will also decrease the effect of edge artifacts due to the higher bandwidth.  In addition, a more complex PSOA can be constructed that can more precisely account for frequency drift while also offering excellent performance to a given resonance.  

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%\section{Discussions}
%Fill


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\begin{acknowledgment}
This work is supported by Naval Surface Warfare Center, Carderock Division
\end{acknowledgment}

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