\section{Structural Analysis}
\label{sec:structural-analysis}
The inital \acl{SMM} \cite{smm} has been set up and analyzed at \acf{IDR} at \acf{UPM}. All figures presented in this chapter are taken from that report.\newline

The main purpose of the structural design is to provide the baseline for dimensioning every hardware component of \acs{AHEPaM} in a way, that it can withstand all mechanical loads induced during the whole mission. The most demanding missions phases are the verification and test phase, and the final launch phase. The main design concept is driven by a couple of factors:

\begin{itemize}
    \item The telescope, including all particle detectors need to be arranged in a stacked arrangement which maintains the relative position of the elements at all times.
    \item The particle detector carriers are potentially made of ceramic material (AlN or $Al_{2}O_{3}$) which can not tolerate large deformations of their bearing seat.
    \item The \acs{BGO} crystals have a large mass of 1 kg per each and are located in the center of the telescope.
    \item The analog electronics signal pathways needs to be minimized
    \item Analog and digital signal side need to be separated and the analog side thoroughly shielded against external noise.
\end{itemize}

From the above list the main objectives of the structural design and thus the focus of the structural analysis has been derived such as:

\begin{enumerate}
    \item The BGO-bracket needs to support the structural loads induced by the \acs{BGO}s.
    \item Since the masses of the \acs{BGO} and the cherenkov sub assembly vary by two orders of magnitude, they will be excited at fundamentally different resonance frequencies. Thus the mechanical elements of the telescope need to be structurally coupled in order to avoid damage of the large SDB..SDD detectors.
    \item The base-frame need to a) support the mechanical loads induced by the \acs{BGO}s  and b) distribute them to the sides of the chassis sides.
    \item The flow of forces shall be along the ebox chassis' outer walls towards the interface feet.
\end{enumerate}

By the time the initial modeling process started, the detailed design of the instrument's cover was not in a representative state. Thus it has been excluded from the presented SMM. It will be included in a future revision.\\
However the overall estimated mass of 350g is low compared to the instrument mass. Additionally a design has been chosen as presented in \cite{ahepam-ddf} that will have a high stiffness. It is therefore expected that a) the instrument cover will have a minor effect on the overall structure and b) that its dynamic behaviour will not be able to excite global modes.\newline Its influence on local modes of the assembly, especially regarding the particle detectors will be part of the above mentioned revision, as their geometric character (large sheet like elements) is similar.

\newpage

\subsection{Model Setup}
\label{sec:smm-detup}
The SMM has been set up using shell elements and has been analyzed using the \textit{NASTRAN} solver. In this initial design state it uses a relatively coarse mesh density focusing on quick iteration cycles rather than high accuracy.\\

The model has been set up following the instrument's FM-status design. That means the following results assume a full arrangement of electronics boards and FM-scale detectors. A tailored DM-status model has not been analyzed.\\

As defined in \cite{ahepam-req} the analyzis applied a static load of 30G \textit{R-AHEPaM-120} and the mission specific random vibration spectrum \textit{R-AHEPaM-160}.

\begin{figure}[h!]
	  \begin{subfigure}{0.5\linewidth}
    	\centering
    	\includegraphics[width=\textwidth]{cau-ath-djf-0007_i1-0/media/smm2_chassis.png}
    	\caption[]{Structural model of \acs{AHEPaM}'s chassis.}
    \label{fig:smm_chassis-mesh}
    \end{subfigure}
	% \hfill
	  \begin{subfigure}{0.5\linewidth}
    	\centering
    	\includegraphics[width=\textwidth]{cau-ath-djf-0007_i1-0/media/smm2_ebox.png}
    	\caption[]{Structural model of \acs{AHEPaM}'s ebox.}
     \label{fig:smm_ebox-mesh}
    \end{subfigure}
	\caption{Shell element meshes of \acs{AHEPaM}'s SMM element.}
    \label{fig:smm_meshes}
\end{figure}

\subsection{Results}
\label{sec:smm-results}

\begin{itemize}
    \item {Complying with \textit{R-AHEPaM-110}, the first global resonance over the whole frequency range up to 2000Hz is $\geq$140 Hz.}
    \item The static load test showed positive margins of safety against yield \textit{MoSy} and ultimate stress \textit{MoSu} for all parts of the instrument. (\textit{R-AHEPaM-120})
\end{itemize}

\subsubsection{Effective Modal Mass}
\label{sec:smm-emm}
An important method of judging the behaviour of a structure is to determine the mass which is excited at a certain frequency, namely the \textit{Effective Modal Mass} (EMM) shown in tables \ref{tab:smm-emm1} to \ref{tab:smm-emm3}. In general low values can be considered as local modes, since they are locally occuring excitation of mostly single parts of the assembly. High values can be considered global modes, since the effect a larger number of parts and thus mass. In the scope of this report a mode is considered 'global' if it shows an EMM$>25\%$. Figures \ref{fig:smm-modes5-12} to \ref{fig:smm-modes68-109} show the dedicated mode shapes. \newline

\newpage

\begin{table}[h!]
	\caption[Effective model masses list.]{Significant effective modal masses up to approx. 260 Hz.}
    \centering
	\includegraphics[width=.85\linewidth]{cau-ath-djf-0007_i1-0/media/smm2_emm1.png}
	\label{tab:smm-emm1}
\end{table}

\begin{table}[h!]
	\caption[Effective model masses list.]{Significant effective modal masses up to approx. 593 Hz.}
    \centering
	\includegraphics[width=.85\linewidth]{cau-ath-djf-0007_i1-0/media/smm2_emm2.png}
	\label{tab:smm-emm2}
\end{table}

\begin{table}[h!]
	\caption[Effective model masses list.]{Significant effective modal masses up to approx. 1100 Hz.}
    \centering
	\includegraphics[width=.85\linewidth]{cau-ath-djf-0007_i1-0/media/smm2_emm3.png}
	\label{tab:smm-emm3}
\end{table}

\begin{figure}[h!]
	  \begin{subfigure}{0.425\linewidth}
    	\centering
    	\includegraphics[width=\textwidth]{cau-ath-djf-0007_i1-0/media/smm2_mode5.png}
    	\caption[]{Mode shape 5.}
    \label{fig:smm-mode5}
    \end{subfigure}
	\hfill
	  \begin{subfigure}{0.425\linewidth}
    	\centering
    	\includegraphics[width=\textwidth]{cau-ath-djf-0007_i1-0/media/smm2_mode12.png}
    	\caption[]{Mode shape 12.}
    \label{fig:smm-mode12}
    \end{subfigure}
	\caption{Global Modes 5 and 12.}
    \label{fig:smm-modes5-12}
\end{figure}

\begin{figure}[h!]
	  \begin{subfigure}{0.425\linewidth}
    	\centering
    	\includegraphics[width=\textwidth]{cau-ath-djf-0007_i1-0/media/smm2_mode22.png}
    	\caption[]{Mode shape 22.}
    \label{fig:smm-modes22}
    \end{subfigure}
	\hfill
	  \begin{subfigure}{0.425\linewidth}
    	\centering
    	\includegraphics[width=\textwidth]{cau-ath-djf-0007_i1-0/media/smm2_mode54.png}
    	\caption[]{Mode shape 54.}
    \label{fig:smm-modes54}
    \end{subfigure}
	\caption{Global Modes 22 and 54.}
    \label{fig:smm-modes22-54}
\end{figure}

\begin{figure}[h!]
	  \begin{subfigure}{0.445\linewidth}
    	\centering
    	\includegraphics[width=\textwidth]{cau-ath-djf-0007_i1-0/media/smm2_mode68.png}
    	\caption[]{Mode shape 68.}
    \label{fig:smm-modes68}
    \end{subfigure}
	\hfill
	  \begin{subfigure}{0.405\linewidth}
    	\centering
    	\includegraphics[width=\textwidth]{cau-ath-djf-0007_i1-0/media/smm2_mode109.png}
    	\caption[]{Mode shape 109.}
    \label{fig:smm-modes109}
    \end{subfigure}
	\caption{Global Modes 68 and 109.}
    \label{fig:smm-modes68-109}
\end{figure}

\newpage

After the modal, the random vibration analysis has been performed by applying the dedicated input spectrum to the model's interface, the feet.\newline
The objective is to find locations throughout the structure which exceed their allowed materials' strength, most prominent the material's yield strength. Table \ref{tab:smm-mos} shows the parts' materials together with the stress, the part has encountered during the analysis. The SMM-report \cite{smm} shows fringe plots for all parts that exceeded their material's threshold. In the scope of this report just an exemplary plot (figure \ref{fig:smm-bgo-fringe}) is shown.

\begin{figure}[h!]
	  \begin{subfigure}{0.65\linewidth}
    	\centering
    	\includegraphics[width=\textwidth]{cau-ath-djf-0007_i1-0/media/smm2_mos.png}
    	\caption[]{Negatvie MoS indicate too high mechanical stresses.}
    \label{tab:smm-mos}
    \end{subfigure}
	%\hfill
	  \begin{subfigure}{0.35\linewidth}
    	\centering
    	\includegraphics[width=\textwidth]{cau-ath-djf-0007_i1-0/media/smm2_fringe-plot-1.png}
    	\caption[]{Exemplary fringe plot of BGO-bracket.}
    \label{fig:smm-bgo-fringe}
    \end{subfigure}
	\caption{Margins of safety derived from random vibration analysis.}
    \label{fig:smm-random}
\end{figure}

\subsection{Discussion}
\label{sec:smm-discussion}
At low frequencies, the global modes occur at the telescope and the bottom plate. They are isolated translational modes, which oscillate along the instruments Y-axis. That is not surprising, since that's the long axis of the instrument chassis' extension and can be considered as the 'soft' axis.\newline
It's important to mention that the presented initial model has been set up with reduced detail level of the included mechanical features. The sheet thickness of the base frame (the interface of the telescope) for example has been implemented with a homogeneous thickness of 4mm, while the underlying CAD-model has been modeled using a sheet thickness of 3mm and stiffening ribs (connecting all boreholes) of 7.5mm. It is assumed that the dynamic behaviour of these features will significantly influence the behaviour and might improve the current results.\newline
More interesting are the higher frequency global modes, because here the boards and the telescope resonate at similar frequencies. In upcoming iterations, with a higher level of detail, it has to be thoroughly investigated if a cross talk is taking place at these elements. If it turns out to be the case, the design will be adapted in order to better separate these modes.\newline
From this first run it can be positively concluded that the chosen coupling of the telescope elements blocks out relative movement in the given frequency range.\newline The analysis also shows negative margins of safety at several parts along the load path of the instrument. While this needs undoubtedly to be improved, it has to be kept in mind that these are findings of the initial preliminary model run. The presented MoS-exceedances can without exceptions be summarized under the premise, that for the given mass of the BGOs a too low plate thickness at the interfaces has been chosen.\newline By increasing the sheets' material thicknesses and/ or increasing the bolt sizes it is expected to bring the current negative values into the necessary positive regime.\\

Although negative aspects have been found by the presented structural analysis it is concluded that the outcome is very positive, since it provides an initial inside view into the structural behaviour of the presented instrument design and significantly helped in locating momentary weak spots which can now be improved. Additionally these spots can relatively easy be modified from an instrument design perspective.\\

