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Robin Clark 2013-02-10 16:12:28 +00:00
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submission_thesis/CH6_Evaluation/copy.tex
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@ -14,12 +14,16 @@ and then formulae are presented for calculating the
complexity of applying FMEA to a group of components. complexity of applying FMEA to a group of components.
% %
These formulae are then used for a hypothetical example, which is analysed by both FMEA and FMMD. These formulae are then used for a hypothetical example, which is analysed by both FMEA and FMMD.
After analysing hypothetical examples, the FMMD examples from chapter~\ref{sec:chap5} are
compared against RFMEA.
%
Following on from the formal definitions, `unitary state failure modes' are defined. In short these Following on from the formal definitions, `unitary state failure modes' are defined. In short these
ensure that component failure modes are mutually exclusive. % Using the unitary state failure mode definition ensure that component failure modes are mutually exclusive. % Using the unitary state failure mode definition
%
Standard formulae for combinations are then used to develop the concept of Standard formulae for combinations are then used to develop the concept of
the cardinality constrained power-set. Using this in combination with unitary state failure modes the cardinality constrained power-set.
we can establish an expression for calculated the number of failure scenarios to Using this in combination with unitary state failure modes
we can establish an expression for calculating the number of failure scenarios to
check for in double failure analysis. check for in double failure analysis.
% %
% MOVE TO CH5 FMMD makes the claim that it can perform double simultaneous failure mode analysis without an undue % MOVE TO CH5 FMMD makes the claim that it can perform double simultaneous failure mode analysis without an undue
@ -28,8 +32,8 @@ check for in double failure analysis.
% MOVE TO CH5 temperature measurement sensor circuit. This example is also used to show how component failure rate statistics can be % MOVE TO CH5 temperature measurement sensor circuit. This example is also used to show how component failure rate statistics can be
% MOVE TO CH5 used with FMMD. % MOVE TO CH5 used with FMMD.
% %
This is followed by some critiques i.e. possible areas of difficulty when performing FMMD, and then This is followed by some critiques of FMMD. % in use.%i.e. possible areas of difficulty when performing FMMD, and then
a general evaluation. % comparing it with traditional FMEA. %a general evaluation. % comparing it with traditional FMEA.
% %
% Moving Pt100 to metrics % Moving Pt100 to metrics
@ -200,12 +204,13 @@ An FMMD Hierarchy will have reducing numbers of {\fgs} as we progress up the hie
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}, $\hh$. We can define an FMMD hierarchy as a set of {\fgs}, $\hh$.
We define a helper function $g$ with a domain of the level $Level$ in an FMMD hierarchy $\hh$, and a % We define a helper function $g$ with a domain of the level $Level$ in an FMMD hierarchy $\hh$, and a
co-domain of a set of {\fgs} (specifically all the {\fgs} on the given level), % co-domain of a set of {\fgs} (specifically all the {\fgs} on the given level),
that returns % that returns
the sum of all complexity comparison % the sum of all complexity comparison
applied to {\fgs} at a particular hierarchy level in \hh, % applied to {\fgs} at a particular hierarchy level in \hh,
We define a helper function, g, that applies $CC$ to all {\fgs} at a particular level, $\xi$ in an FMMD hierarchy {\hh}
and returns the sum of the comparison complexities,
\begin{equation} \begin{equation}
g: \hh \times \mathbb{N} \rightarrow \mathbb{N} . g: \hh \times \mathbb{N} \rightarrow \mathbb{N} .
\end{equation} \end{equation}
@ -384,7 +389,7 @@ $$
$$ $$
%\clearpage %\clearpage
\subsection{Complexity Comparison applied to previous FMMD Examples} \subsection{Complexity Comparison applied to FMMD electroinc circuits analysed in chapter~\ref{sec:chap5}.}
All the FMMD examples in chapters \ref{sec:chap5} All the FMMD examples in chapters \ref{sec:chap5}
and \ref{sec:chap6} showed a marked reduction in comparison and \ref{sec:chap6} showed a marked reduction in comparison
@ -460,7 +465,9 @@ are presented in table~\ref{tbl:firstcc}.
\end{table} \end{table}
% end table % end table
The complexity comparison figures for the example circuits in chapter~\ref{sec:chap5} show The complexity comparison figures for the example circuits in chapter~\ref{sec:chap5} show
that for increasing complexity the performance benefits from FMMD are apparent. that for the non trival examples, as we
use more levels in the FMMD hierarchy, the performance
gains over RFMEA become apparent. %for increasing complexity the performance benefits from FMMD are apparent.
@ -481,7 +488,7 @@ We use these two analyses to compare the effect on comparison complexity (see ta
\hline \hline
\textbf{Hierarchy} & \textbf{Derived} & \textbf{Complexity} & $|fm(c)|$: \textbf{number} \\ \textbf{Hierarchy} & \textbf{Derived} & \textbf{Complexity} & $|fm(c)|$: \textbf{number} \\
\textbf{Level} & \textbf{Component} & \textbf{Comparison} & \textbf{of derived} \\ \textbf{Level} & \textbf{Component} & \textbf{Comparison} & \textbf{of derived} \\
& & & \textbf{failure modes} OK \\ & & & \textbf{failure modes} \\
%\hline \hline %\hline \hline
%\multicolumn{3}{ |c| }{Complexity Comparison against RFMEA for examples in Chapter~\ref{sec:chap5}} \\ %\multicolumn{3}{ |c| }{Complexity Comparison against RFMEA for examples in Chapter~\ref{sec:chap5}} \\
%\hline \hline %\hline \hline
@ -555,7 +562,7 @@ by more than a factor of ten.
\hline \hline
\textbf{Hierarchy} & \textbf{Derived} & \textbf{Complexity} & $|fm(c)|$: \textbf{number} \\ \textbf{Hierarchy} & \textbf{Derived} & \textbf{Complexity} & $|fm(c)|$: \textbf{number} \\
\textbf{Level} & \textbf{Component} & \textbf{Comparison} & \textbf{of derived} \\ \textbf{Level} & \textbf{Component} & \textbf{Comparison} & \textbf{of derived} \\
& & & \textbf{failure modes} OK \\ & & & \textbf{failure modes} \\
%\hline \hline %\hline \hline
%\multicolumn{3}{ |c| }{Complexity Comparison against RFMEA for examples in Chapter~\ref{sec:chap5}} \\ %\multicolumn{3}{ |c| }{Complexity Comparison against RFMEA for examples in Chapter~\ref{sec:chap5}} \\
%\hline \hline %\hline \hline
@ -730,7 +737,7 @@ but potentially
$2^N$. $2^N$.
% %
This would make the job of analysing the failure modes This would make the job of analysing the failure modes
in a {\fg} impractical due to the sheer size of the task. in a {\fg} impractical due to state explosion. %the sheer size of the task.
%Note that the `unitary state' conditions apply to failure modes within a component. %Note that the `unitary state' conditions apply to failure modes within a component.
%%- Need some refs here because that is the way gastec treat the ADC on microcontroller on the servos %%- Need some refs here because that is the way gastec treat the ADC on microcontroller on the servos
@ -1095,19 +1102,19 @@ to the probability that a given part failure mode will cause a given system leve
Another way to view this is to consider the failure modes of a Another way to view this is to consider the failure modes of a
component, with the $OK$ state, as a universal set $\Omega$, where component, with the $OK$ state, as a universal set $\Omega$, where
all sets within $\Omega$ are partitioned. all sets within $\Omega$ are partitioned.
Figure \ref{fig:partitioncfm} shows a partitioned set representing Figure \ref{fig:combco} shows a partitioned set representing
component failure modes $\{ B_1 ... B_8, OK \}$: partitioned sets component failure modes $\{ B_1 ... B_3, OK \}$: partitioned sets
where the OK or empty set condition is included, obey unitary state conditions. where the OK or empty set condition is included, obey unitary state conditions.
Because the subsets of $\Omega$ are partitioned, we can say these Because the subsets of $\Omega$ are partitioned, we can say these
failure modes are unitary state. failure modes are unitary state.
%
\begin{figure}[h] % \begin{figure}[h]
\centering % \centering
\includegraphics[width=350pt,keepaspectratio=true]{./CH4_FMMD/partitioncfm.png} % \includegraphics[width=350pt,keepaspectratio=true]{./CH4_FMMD/partitioncfm.png}
% partition.png: 510x264 pixel, 72dpi, 17.99x9.31 cm, bb=0 0 510 264 % % partition.png: 510x264 pixel, 72dpi, 17.99x9.31 cm, bb=0 0 510 264
\caption{Base Component Failure Modes with OK mode as partitioned set} % \caption{Base Component Failure Modes with OK mode as partitioned set}
\label{fig:partitioncfm} % \label{fig:partitioncfm}
\end{figure} % \end{figure}
\section{Components with Independent failure modes} \section{Components with Independent failure modes}