robin/Robin_PHD
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

CH7 (although in CH6 directory)

Browse Source
This commit is contained in:
Robin Clark 2013-01-20 17:47:14 +00:00
parent 28d11d6bce
commit 097da091c8
3 changed files with 21 additions and 13 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

2
submission_thesis/CH6_Evaluation/Makefile
View File

@ -3,7 +3,7 @@
# #
# Place all .dia files here as .png targets # Place all .dia files here as .png targets
# #
DIA = DIA = components_81_euler.png
doc: $(DIA) doc: $(DIA)

BIN
submission_thesis/CH6_Evaluation/components_81_euler.dia Normal file
View File

Binary file not shown.

32
submission_thesis/CH6_Evaluation/copy.tex
View File

@ -123,11 +123,11 @@ The function $fm$ has a component as its domain and the components failure modes
We can represent the number of potential failure modes of a component $c$, to be $ | fm(c) | .$ We can represent the number of potential failure modes of a component $c$, to be $ | fm(c) | .$
\paragraph{Indexing components with the group $G$.} \paragraph{Indexing components with the group $G$.}
If we index all the components in the system under investigation $ c_1, c_2 \ldots c_{|\FG|} $ we can express If we index all the components in the system under investigation $ c_1, c_2 \ldots c_{|G|} $ we can express
the number of checks required to rigorously examine every the number of checks required to rigorously examine every
failure mode against all the other components in the system. failure mode against all the other components in the system.
We can define this as a function, Comparison Complexity, can be represented by a function $CC$, with its domain as $G$, and Comparison Complexity can be represented by a function $CC$, with its domain as $G$, and
its range as the number of checks---or reasoning stages---to perform to satisfy a rigorous FMEA inspection. its range as the number of checks---or reasoning stages---to perform to satisfy a rigorous FMEA inspection.
Where $\mathcal{G}$ represents the set of all {\fgs}, and $ \mathbb{N} $ any natural integer, $CC$ is defined by, Where $\mathcal{G}$ represents the set of all {\fgs}, and $ \mathbb{N} $ any natural integer, $CC$ is defined by,
@ -174,10 +174,10 @@ to identify the hierarchy level.
% \end{equation} % \end{equation}
\subsection{A general formula for counting Comparison Complexity in an FMMD hierarchy} \subsection{A general formula for counting Comparison Complexity in an FMMD hierarchy}
An FMMD Hierarchy will have reducing numbers of functional groups as we progress up the hierarchy. An FMMD Hierarchy will have reducing numbers of {\fgs} as we progress up the hierarchy.
In order to calculate its comparison~complexity we need to apply equation~\ref{eqn:CC} to In order to calculate its comparison~complexity we need to apply equation~\ref{eqn:CC} to
all {\fgs} on each level. all {\fgs} on each level.
We can define an FMMD hierarchy as a set of {\fgs}, $H$.
We define a helper function $g$ with a domain of the level $i$ in an FMMD hierarchy $H$, and a co-domain of a set of {\fgs} (specifically all the {\fgs} on the given level), We define a helper function $g$ with a domain of the level $i$ in an FMMD hierarchy $H$, and a co-domain of a set of {\fgs} (specifically all the {\fgs} on the given level),
defined by defined by
@ -188,7 +188,7 @@ g(H, i) \rightarrow \forall {\FG}^{\xi} \;where\; ({\xi} = {i}) \wedge ({\FG}^{\
\end{equation} \end{equation}
Where $L$ represents the number of levels in the FMMD hierarchy, Where $L$ represents the number of levels in the FMMD hierarchy,
$|g(\xi)|$ represents the number of functional groups on the level $|g(\xi)|$ represents the number of {\fgs} on the level
and $H$ represents an FMMD hierarchy, and $H$ represents an FMMD hierarchy,
we overload the comparison complexity thus: we overload the comparison complexity thus:
%$$ %$$
@ -202,10 +202,11 @@ we overload the comparison complexity thus:
\pagebreak[4] \pagebreak[4]
\subsection{Complexity Comparison Examples} \subsection{Complexity Comparison Examples}
The potential divider discussed in section~\ref{potdivfmmd} has four failure modes and two components and therefore has $CC$ of 4. The potential divider discussed in section~\ref{subsec:potdiv} has four failure modes and two components and therefore has $CC$ of 4.
$$CC(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$ $$CC(potdiv) = \sum_{n=1}^{2} |2|.(|1|) = 4 $$
Even considering a $example$ system with just 81 components (with these components %Even considering a $example$
A system, $example$, with just 81 components (with these components
having 3 failure modes each) we would have an $CC$ of having 3 failure modes each) we would have an $CC$ of
$$CC(example) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$ $$CC(example) = \sum_{n=1}^{81} |3|.(|80|) = 19440 .$$
@ -228,16 +229,23 @@ rigorous checking feasible.
\pagebreak[4] \pagebreak[4]
%\subsection{Using the concept of Complexity Comparison to compare RFMEA with FMMD} %\subsection{Using the concept of Complexity Comparison to compare RFMEA with FMMD}
\begin{figure} % \begin{figure}
% \centering
% \includegraphics[width=400pt,keepaspectratio=true]{CH5_Examples/three_tree.png}
% % three_tree.png: 851x385 pixel, 72dpi, 30.02x13.58 cm, bb=0 0 851 385
% \caption{FMMD Hierarchy with number of components in {\fg} fixed to 3 $(|G| = 3)$ } % \wedge (|fm(c)| = 3)$}
% \label{fig:three_tree}
% \end{figure}
\begin{figure}[h]
\centering \centering
\includegraphics[width=400pt,keepaspectratio=true]{CH5_Examples/three_tree.png} \includegraphics[width=400pt]{./CH6_Evaluation/components_81_euler.png}
% three_tree.png: 851x385 pixel, 72dpi, 30.02x13.58 cm, bb=0 0 851 385 % components_81_euler.png: 3056x2532 pixel, 72dpi, 107.81x89.32 cm, bb=0 0 3056 2532
\caption{FMMD Hierarchy with number of components in {\fg} fixed to 3 $(|G| = 3)$ } % \wedge (|fm(c)| = 3)$} \caption{FMMD Hierarchy with number of compnents in each $FG$ fixed to three ($|FG|=3$)}
\label{fig:three_tree} \label{fig:three_tree}
\end{figure} \end{figure}
\subsection{Comparing FMMD and RFMEA comparison complexity} \subsection{Comparing FMMD and RFMEA comparison complexity}
Because components have variable numbers of failure modes, Because components have variable numbers of failure modes,