staying late again

fmmdset/fmmdset.tex

@@ -232,9 +232,9 @@ iThe $\bowtie$ function takes a {\fg}
 as an argument and returns a newly created {\dc}.
 
 The $\bowtie$ analysis, a symptom extraction process, is described in chapter \ref{chap:sympex}.
-Using $\alpha$ to symbolise the fault abstraction level, we can now state:
+Using $\abslevel$ to symbolise the fault abstraction level, we can now state:
 
-$$ \bowtie(FG^{\alpha}) \rightarrow C^{{\alpha}+1}. $$ 
+$$ \bowtie(FG^{\abslevel}) \rightarrow C^{{\abslevel}+1}. $$ 
 
 \paragraph{The symptom abstraction process in outline.} The $\bowtie$ function processes each member (component) of the set $FG$ and 
 extracts all the component failure modes, which are used by the analyst to

style.tex

@@ -82,6 +82,7 @@
 %\newcommand{\pic}{\em pure~intersection~chain}
 \newcommand{\pic}{\em pair-wise~intersection~chain}
 \newcommand{\wrt}{\em with~respect~to}
+\newcommand{\abslevel}{\ensuremath{\Psi}}
 \newcommand{\fmmdgloss}{\glossary{name={FMMD},description={Failure Mode Modular De-Composition, a bottom-up methodolgy for incrementally building failure mode models, using a procedure taking functional groups of components and creating derived components representing them, and in turn using the derived components to create higher level functional groups, and so on, that are used to build a failure mode model of a SYSTEM}}}
 \newcommand{\fmodegloss}{\glossary{name={failure mode},description={The way in which a failure occurs. A component or sub-system may fail in a number of ways, and each of these is a 
 failure mode of the component or sub-system}}}

survey/survey.tex

@@ -250,20 +250,39 @@ FMEA described in this section (\ref{pfmea}) is sometimes called `production FME
 
 \subsection{FMECA}
 
-Failure mode, effects, and criticality analysis (FMECA) extends FMEA and adds a failure outcome criticallity factor.
-This is a bottom up methodology, which takes component failure modes
-and traces them to the SYSTEM level failures. 
+Failure mode, effects, and criticality analysis (FMECA)~\cite{FMD-91} extends FMEA
+by associaing failure probabilities with component failure modes.
+Essentially this adds a failure outcome criticallity factor to FMEA.
+This is a bottom up methodology, which builds on an existing FMEA
+analysis, which has already taken individual component failure modes
+and traced them to the SYSTEM level failures. 
 %
 Reliability data for components is used to predict the 
 failure statistics in the design stage.
 An openly published source for the reliability of generic
 electronic components was published by the DOD
-in 1991 (MIL HDK 1991 \cite{mil1991}) and is a typical 
+in 1991 (MIL~HDK~1991~\cite{mil1991}) and is a typical 
 source for MTFF data.
 %
-FMECA has a probability factor for a component error becoming % causing 
-a SYSTEM level error.
-This is termed the $\beta$ factor.
+FMECA has three probability factors for component failures.
+\paragraph{FMECA ${\lambda}_{p}$ value.}
+This is the overall failure rate of a base component.
+This will typically be the failure rate per million ($10^6$) or
+billion ($10^9$) hours of operation.
+
+\paragraph{FMECA $\alpha$ value.}
+The failure mode probability, usually dentoted by $\alpha$ is the  probability of 
+is the probability of a particular failure
+mode occuring within a component, should it fail.
+A component with N failure modes will thus have
+have an $\alpha$ value associated with each of those modes.
+As the $\alpha$ modes are probabilities, the sum of all $\alpha$ modes for a component must equal one.
+
+\paragraph{FMECA $\beta$ value.}
+The second probability factor $\beta$, is the probability that the failure mode
+will cause a given SYSTEM failure.
+This corresponds to Baysian probability, given a particular
+component failure mode, the probability of a system level failure.
 %\footnote{for a given component failure mode there will be a $\beta$ value, the
 %probability that the component failure mode will cause a given SYSTEM failure}.
 %
@@ -273,11 +292,19 @@ assigned a probability $\beta$ factor by the design engineer. The use of  a $\be
 is often justified using Bayes theorem \cite{probstat}.
 %Also, it can miss combinations of failure modes that will cause SYSTEM level errors.
 %
+\paragraph{Results of FMECA}
 The results of FMECA are similar to FMEA, in that component errors are
 listed according to importance, based on 
 probability of occurrence and criticallity.
 % to prevent the SYSTEM fault of given criticallity.
-Again this essentially produces a prioritised `to~do~list'.
+Again this essentially produces a prioritised `to~do~list'
+sorted by severity and liklihood.
+Each component failure mode has a criticallity number $C_m$, (where t is the operating time or product life time in hours), which can be calculated thus:
+
+
+\begin{equation}
+ C_m = \beta \alpha {\lambda}_p t  
+\end{equation}
 
 %%-WIKI- Failure mode, effects, and criticality analysis (FMECA) is an extension of failure mode and effects analysis (FMEA).
 %%-WIKI- FMEA is a a bottom-up, inductive analytical method which may be performed at either the functional or 
@@ -287,11 +314,23 @@ Again this essentially produces a prioritised `to~do~list'.
 %%-WIKI- FMECA tends to be preferred over FMEA in space and North Atlantic Treaty Organization (NATO) military applications, 
 %%-WIKI- while various forms of FMEA predominate in other industries.
 
+A second result, representing the overall reliability and safety of the product $P$, 
+, termed a criticallity number $C_r$ 
+(where we can consider $P$ to be a flat set of component failure modes
+which we can use the variable  $c_f$  to represent
+% where $f \in F$)
+can calculated thus
+\begin{equation}
+ C_r = \sum_{c_f \in P} {\beta \alpha {\lambda}_p t} c_f
+\end{equation}
+
 
 \subsubsection{ FMECA weaknesses }
 \begin{itemize}
 \item Possibility to miss the effects of failure modes at SYSTEM level.
 \item The $\beta$ factor is based on heuristics and does not reflect any rigourous calculations.
+\item The $\alpha$ factor is based on heuristics or general data, and may not to specific to the environmental or operational conditions
+under which the equipment is operating.
 \item Possibility to miss environmental affects.
 \item No possibility to model base component level double failure modes.
 \end{itemize}

symptom_ex_process/symptom_ex_process.tex

@@ -732,15 +732,15 @@ as a component with a known set of failure modes.
 
 
 \paragraph{Enumerating abstraction levels}
-We can assign an attribute  of abstraction level $\alpha$  to
-components, where $\alpha$ is a natural number, ($\alpha \in \mathbb{N}_0$).
+We can assign an attribute  of abstraction level $\abslevel$  to
+components, where $\abslevel$ is a natural number, ($\abslevel \in \mathbb{N}_0$).
 For a base component, let the abstraction level be zero.
 If we apply the symptom abstraction process $\bowtie$,
-the resulting derived~component will have an $\alpha$ value
-one higher that the highest $\alpha$ value of any of the components
+the resulting derived~component will have an $\abslevel$ value
+one higher that the highest $\abslevel$ value of any of the components
 in the functional group used to derive it.
 Thus a derived component sourced from base components
-will have an $\alpha$ value of 1. 
+will have an $\abslevel$ value of 1. 
 %
 %If $DC$ were to be included in a functional~group,
 %that functional~group must be considered to be at a higher level of
@@ -751,7 +751,7 @@ will have an $\alpha$ value of 1.
 %of the highest assigned to any of its components.
 %
 %With a derived component $DC$  having an abstraction level 
-The attribute $\alpha$ can be used to track the 
+The attribute $\abslevel$ can be used to track the 
 level of fault abstraction  of components in an FMMD hierarchy. Because base and derived components
 are collected to form functional groups, a hierarchy is
 naturally formed with the abstraction levels increasing with each tier.
@@ -759,8 +759,8 @@ naturally formed with the abstraction levels increasing with each tier.
 
 
 %\FORALL { $c \in FG $ } \COMMENT{Find the highest abstraction level of any component in the functional group}
-% \IF{$c.\alpha > \alpha_{max}$} 
-%   $\alpha_{max} = c.\alpha$
+% \IF{$c.\abslevel > \abslevel_{max}$} 
+%   $\abslevel_{max} = c.\abslevel$
 % \ENDIF 
 %\STATE { $ FM(c) \in FG_{cfm} $ } \COMMENT {Collect all failure modes from each component into the set $FM_{cfm}$}
 %\ENDFOR
@@ -1145,7 +1145,7 @@ $$ fcs(R) = SP $$
 %\end{algorithm}
 
 %Algorithm \ref{alg44}  
-This raises the failure~mode abstraction level, $\alpha$.
+This raises the failure~mode abstraction level, $\abslevel$.
 The failures have now been considered not from the component level, but from the sub-system or
 functional~group level.
 We now have a set $SP$ of the symptoms of failure.