From aec4ad9a53aed977310a79c24705bdfbb2da6acb Mon Sep 17 00:00:00 2001 From: Robin Clark Date: Sun, 10 Feb 2013 16:12:28 +0000 Subject: [PATCH] prinout-red pen-edit --- submission_thesis/CH6_Evaluation/copy.tex | 59 +++++++++++++---------- 1 file changed, 33 insertions(+), 26 deletions(-) diff --git a/submission_thesis/CH6_Evaluation/copy.tex b/submission_thesis/CH6_Evaluation/copy.tex index 87fd40d..75f1223 100644 --- a/submission_thesis/CH6_Evaluation/copy.tex +++ b/submission_thesis/CH6_Evaluation/copy.tex @@ -14,12 +14,16 @@ and then formulae are presented for calculating the 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. - +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 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 -the cardinality constrained power-set. Using this in combination with unitary state failure modes -we can establish an expression for calculated the number of failure scenarios to +the cardinality constrained power-set. +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. % % 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 used with FMMD. % -This is followed by some critiques i.e. possible areas of difficulty when performing FMMD, and then -a general evaluation. % comparing it with traditional FMEA. +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. % % 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 all {\fgs} on each level. 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 -co-domain of a set of {\fgs} (specifically all the {\fgs} on the given level), -that returns -the sum of all complexity comparison -applied to {\fgs} at a particular hierarchy level in \hh, - +% 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), +% that returns +% the sum of all complexity comparison +% 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} g: \hh \times \mathbb{N} \rightarrow \mathbb{N} . \end{equation} @@ -384,7 +389,7 @@ $$ $$ %\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} 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 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 \textbf{Hierarchy} & \textbf{Derived} & \textbf{Complexity} & $|fm(c)|$: \textbf{number} \\ \textbf{Level} & \textbf{Component} & \textbf{Comparison} & \textbf{of derived} \\ - & & & \textbf{failure modes} OK \\ + & & & \textbf{failure modes} \\ %\hline \hline %\multicolumn{3}{ |c| }{Complexity Comparison against RFMEA for examples in Chapter~\ref{sec:chap5}} \\ %\hline \hline @@ -555,7 +562,7 @@ by more than a factor of ten. \hline \textbf{Hierarchy} & \textbf{Derived} & \textbf{Complexity} & $|fm(c)|$: \textbf{number} \\ \textbf{Level} & \textbf{Component} & \textbf{Comparison} & \textbf{of derived} \\ - & & & \textbf{failure modes} OK \\ + & & & \textbf{failure modes} \\ %\hline \hline %\multicolumn{3}{ |c| }{Complexity Comparison against RFMEA for examples in Chapter~\ref{sec:chap5}} \\ %\hline \hline @@ -730,7 +737,7 @@ but potentially $2^N$. % 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. %%- 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 component, with the $OK$ state, as a universal set $\Omega$, where all sets within $\Omega$ are partitioned. -Figure \ref{fig:partitioncfm} shows a partitioned set representing -component failure modes $\{ B_1 ... B_8, OK \}$: partitioned sets +Figure \ref{fig:combco} shows a partitioned set representing +component failure modes $\{ B_1 ... B_3, OK \}$: partitioned sets where the OK or empty set condition is included, obey unitary state conditions. Because the subsets of $\Omega$ are partitioned, we can say these failure modes are unitary state. - -\begin{figure}[h] - \centering - \includegraphics[width=350pt,keepaspectratio=true]{./CH4_FMMD/partitioncfm.png} - % 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} - \label{fig:partitioncfm} -\end{figure} +% +% \begin{figure}[h] +% \centering +% \includegraphics[width=350pt,keepaspectratio=true]{./CH4_FMMD/partitioncfm.png} +% % 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} +% \label{fig:partitioncfm} +% \end{figure} \section{Components with Independent failure modes}