...

component_failure_modes_definition/component_failure_modes_definition.tex

@@ -701,11 +701,15 @@ associated with the test cases, complete coverage would be verified.
 A sample space is defined as the set of all possible outcomes.
 For a component in FMMD analysis, this set of all possible outcomes is its normal correct
 operating state and all its failure modes.
+We are thus considering the failure modes as events in the sample space.
+%
 When dealing with failure modes, we are not interested in 
 the state where the component is working perfectly or `OK' (i.e. operating with no error).
+%
 We are interested only in ways in which it can fail.
 By definition while all components in a system are `working perfectly' 
 that system will not exhibit faulty behaviour.
+We can say that the OK state corresponds to the empty set.
 Thus the statistical sample space $\Omega$ for a component or derived~component $C$ is
 %$$ \Omega = {OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N} $$
 $$ \Omega(C) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, \ldots ,failure\_mode_{N}\} . $$
@@ -718,13 +722,19 @@ $ \Omega(C) = fm(C) \cup \{OK\} $).
 The $OK$ statistical case is the largest in probability, and is therefore
 of interest when analysing systems from a statistical perspective.
 This is of interest for the application of conditional probability calculations
-such as Bayes theorem~\cite{probstat}.
+such as Bayes theorem~\cite{probstat}; 
+
+The current failure modelling methodologies (FMEA, FMECA, FTA, FMEDA) all use Bayesian
+statistics to justify their methodologies~\cite{nucfta}\cite{nasafta}.
+That is to say, a base component or a sub-system failure
+has a probability of causing given system level failures.
 
 Another way to view this is to consider the failure modes of
 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 $\{ B_1 ... B_8, OK \}$ obeying unitary state conditions.
+component failure modes $\{ B_1 ... B_8, OK \}$ : partitioned sets
+where the OK or empty set condition is included, obey unitary state conditions.
 Because the subsets of $\Omega$ are partitionned we can say these 
 failure modes are unitary state.
 

fmmd_concept/fmmd_concept.tex

@@ -272,12 +272,18 @@ not to rigorously detect all possible failures.
 Consequently it was not designed to guarantee to covering all component failure modes, 
 and has no rigorous in-built safeguards to ensure coverage of all possible
 system level outcomes.
+Also each system level error (or undesireable event) requires its own FTA tree.
+This increase the amount of work to do, and in the case of updates to
+particular sub-systems, introduces the requirement to update every FTA
+tree modelling that sub-system.
 
 \subsubsection{ FTA weaknesses }
 \begin{itemize}
 \item Complex component interaction effects are by definition modelled by FTA, but because of the top down approach, not all
 base component failure modes are guaranteed to be included in the model.
 \item Possibility to miss environmental affects.
+\item One FTA tree, per system failure mode. Thus there is not one model from which several FTA
+trees can be derived. Maintainability and consistency cannot therefore be automatically checked.
 \item No possibility to model base component level double failure modes.
 \end{itemize}
 

survey/survey.tex

@@ -64,10 +64,27 @@ Consequently it was not designed to guarantee to covering all component failure
 and has no rigorous in-built safeguards to ensure coverage of all possible
 system level outcomes.
 
+\paragraph{Outline of FTA Methodology}
+FTA works by taking an undesireable event 
+(or SYSTEM level failure mode or TOP level failure)
+and deciding top-down, what sub-systems it depends upon, and which
+failure events of those sub-systems could cause the top level failure.
+It then applies the same process to the sub-systems it identified
+from the top level, identifying level level sub-systems and events.
+It is not required to de-compose down to base component level.
+
+\paragraph{One FTA Tree per System Level Failure Mode.}
+This means that each system level error (or undesireable event) requires its own FTA tree.
+This increases the amount of work to do, and in the case of updates to
+particular sub-systems, introduces the requirement to update every FTA
+tree modelling that sub-system.
+
 \subsubsection{ FTA weaknesses }
 \begin{itemize}
 \item Possibility to miss component failure modes.
 \item Possibility to miss environmental affects.
+\item One FTA tree, per system failure mode. Thus there is not one model from which several FTA
+trees can be derived. Maintainability and consistency cannot therefore be automatically checked.
 \item No possibility to model base component level double failure modes.
 \end{itemize}
 
@@ -88,7 +105,7 @@ Consider an unused feature failing.}. Muliplying these
 together, 
 gives a risk probability number (RPN), given by $RPN = S \times O \times D$.
 This gives in effect
-a prioritised `todo list', with higher $RPN$ values being the most urgent.
+a prioritised `to~do~list', with higher $RPN$ values being the most urgent.
 
 
 \subsubsection{ FMEA weaknesses }
@@ -592,18 +609,26 @@ thus
  P(S|B) =  \frac{P(S) P(B|S)} {P(B)} .
 \end{equation}
 
-Equation \ref{eqn:bayes1} means, given the event $B$ what is the probability it was caused by $S$.
-Because we are interested in what base component failure modes could have caused $S$
-we need to re-arrange this 
 
-\begin{equation}
+This equation gives us the probability that if event B has occurred, of
-\label{eqn:bayes2}
+the event S occurring.
- P(B|S) =  \frac{P(B) P(S|B)} {P(S)} .
+In the context of failure mode analysis, the event B would 
-\end{equation}
+be the occurance of a component failure mode, and S would be a system level error.
+%
 
-Equation \ref{eqn:bayes2} can be read as given the system failure mode $S$
+%Equation \ref{eqn:bayes1} means, given the event $B$ what is the probability it was caused by $S$.
+%Because we are interested in what base component failure modes could have caused $S$
+%we need to re-arrange this 
 
-Typically a system level failure will have a number of possible causes, or base component failure
+%\begin{equation}
+%\label{eqn:bayes2}
+% P(B|S) =  \frac{P(B) P(S|B)} {P(S)} .
+%\end{equation}
+%
+%Equation \ref{eqn:bayes2} can be read as given the system failure mode $S$
+
+Typically a system level failure will have a number of possible causes, 
+or base component failure
 modes. Some base component failure modes may not be able to cause given system failures.
 We can represent the the base component failure modes as a partioned set~\cite{nucfta}[fig VI-7], and overlay 
 a given system failure mode on it.
@@ -654,8 +679,10 @@ will cause the system level error $S_k$
 %Prob $B_n$ caused $S_k$ is the prob $S_k$ caused by $B_n$ divided by prob of $B_n$
 
 $$
-% P(S_k|B_n) = \frac{P(S_k) \; P(B_n | S_k) }{P(B_n)} alternate form of no use to MEEEEEE
+ P(S_k|B_n) = \frac{P(S_k) \; P(B_n | S_k) }{P(B_n)} 
-P(B_n|S_k) = \frac{P(B_n) \; P(S_k | B_n) }{P(S_k)}
+
+%alternate form of no use to MEEEEEE
+%P(B_n|S_k) = \frac{P(B_n) \; P(S_k | B_n) }{P(S_k)}
 $$
 
 For example were we to have a component that has a failure mode $B_n$ with an MTTF of $10^{-7}$ hours