Wednesday night edit

component_failure_modes_definition/component_failure_modes_definition.tex

@@ -292,9 +292,10 @@ would have an $\alpha$ value of 1.
 
 Let the set of all possible components  be $\mathcal{C}$
 and let the set of all possible failure modes be $\mathcal{F}$ and $\mathcal{PF}$ is the powerset of
-all $\mathcal{F}$..
+all $\mathcal{F}$.
 
 We can define a function $fm$ as equation \ref{eqn:fmset}.
+\label{fmdef}
 
 \begin{equation}
 fm : \mathcal{C} \rightarrow \mathcal{P}\mathcal{F}
@@ -763,6 +764,7 @@ operational states.
 The additional objects System, Environment and Operational States
 are added to  UML diagram in figure \ref{fig:cfg} and represented in figure \ref{fig:cfg2}.
 
+\label{completeuml}
 
 \begin{figure}[h]
  \centering

fmmd_data_model/fmmd_data_model.tex

@@ -8,31 +8,53 @@
 
 %% What I have done
 %%
+This paper presents a simple two stage Failure Mode Modular De-Composition (FMMD)
+model of a theoretical System.
+The Analysis model is then represented as a Directed Acyclic Graph (DAG), of the {\fg}s
+components and failure modes  represented in it.
 
-%% What I have found
+% What I have found
 %%
+From traversing the DAG, minimal cut sets (component level combinations
+that cause system level failures) are revealed.
+Common mode failure modes and same component dependencies
+can also be automatically determined.
 
 %% Sell it
 %%
-}
-}
+By having a clear data model, we can not only produce results
+for the traditional methodologies, we can trace common mode and 
+component dependency failures as well.
+Also, with statistical data, we can use the minimal cut set results
+to determine the likelihood of particular system failures, even
+if they have multiple causes.
+} % abstract
+} % ifthenelse
 {
  %%% CHAPTER INTO NEARLT THE SAME AS ABSTRACT
 \section{Introduction}
 
 This chapter 
+presents a simple two stage FMMD % Failure Mode Modular De-Composition (FMMD)
+model of a theoretical System.
+The Analysis model is then represented as a Directed Acyclic Graph (DAG), of the {\fg}s
+components and failure modes  represented in it.
 
-%% What I have done
+% What I have found
 %%
-
-%% What I have found
-%%
-%and considering some constraints determined from
-%the evaluation of the four established methodologies,
+From traversing the DAG, minimal cut sets (component level combinations
+that cause system level failures) are revealed.
+Common mode failure modes and same component dependencies
+can also be automatically determined.
 
 %% Sell it
 %%
-
+By having a clear data model, we can not only produce results
+for the traditional methodologies, we can trace common mode and
+component dependency failures as well.
+Also, with statistical data, we can use the minimal cut set results
+to determine the likelihood of particular system failures, even
+if they have multiple causes.
 }
 
 %{ \huge This might become a chapter in its own right after fmmdset }
@@ -60,7 +82,7 @@ represents the FMMD hierarchy level, or $\alpha$ value, thus:
 }
 {
 We can organise these into functional groups (where the superscript 
-represents the $\alpha$ value, see section \ref{alpha}), thus:
+represents the $\alpha$ value, or FMMD hierarchy level, see section \ref{alpha}), thus:
 }
 
 $$ FG^0_1 = \{C_1, C_2\},$$
@@ -68,10 +90,28 @@ $$ FG^0_2 = \{C_1, C_3, K_4\},$$
 $$ FG^0_3 = \{C_5, C_6, K_7\}.$$
 
 Note that in this model the base~component $C_1$ has been used in
-two separate functional groups.
+two separate functional groups. This could be a component that they 
+both commonly use. A real world example of a component included in
+more than one {\fg} could
+be a powersupply or DCDC\footnote{A DCDC (direct current to direct current)
+converter, is a common feature in modern PCBs, used to provide isolation
+and/or voltage supplies at a different EMF from the source of power.} 
+converter shared  to power
+the functional groups $FG^0_1$ and $FG^1_1$.
+
 Also that the component type $K$ has been used by 
 two different functional groups.
 
+For the sake of example let our temperature environment 
+for the SYSTEM be ${{0}\oc}$ to ${{125}\oc}$, but let the component
+type `K' have a de-graded performance failure mode between
+${{80}\oc}$ and ${{125}\oc}$\footnote{ A real world example of
+degraded performace with temperature is the isolating opto coupler.
+These can typically only cope with lower baud rate ranges
+at high temperatures \cite{tlp181}.}. We can term this
+degraded performce of component `K' as failure mode `d'.
+
+
 \paragraph{Symptom Extraction.}
 A processes of symptom extraction is now applied to the functional groups.
 Again for the sake of example, let us say that each functional
@@ -81,6 +121,83 @@ Applying symptom abstraction to $FG^0_1$ i.e. $\bowtie fm ( FG^0_1 ) = \{ FG^0_{
 We can now create a new derived component, $DC^1_1$, whose failure
 modes are the symptoms of $FG^0_1 $ thus $ fm ( {DC}^1_1 )  = \{ FG^0_{1 a}, FG^0_{1 b}  \} $.
 
+\paragraph{Building the Object Model}
+
+Using the  UML model in figure \ref{fig:cfg2fmmd_data} we will apply FMMD analysis stages
+to build a hierarchy representing the whole system, begining with the $FG^0$ level functional groups.
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=400pt,bb=0 0 702 464,keepaspectratio=true]{./fmmd_data_model/cfg2.jpg}
+ % cfg2.jpg: 702x464 pixel, 72dpi, 24.76x16.37 cm, bb=0 0 702 464
+ \caption{UML Class model for FMMD}
+ \label{fig:cfg2fmmd_data}
+\end{figure}
+
+% %\begin{figure}[h]
+%  \centering
+%  \includegraphics[width=400pt,keepaspectratio=true]{./fmmd_data_model/cfg2.jpg}
+%  % cfg2.jpg: 702x464 pixel, 72dpi, 24.76x16.37 cm, bb=0 0 702 464
+%  \caption{Complete UML diagram}
+%  \label{fig:cfg2fmmd_data}
+% \end{figure}
+
+\paragraph{Find Failure Modes}
+
+Consider the SYSTEM environment with its temperature range of ${{0}\oc}$ to ${{125}\oc}$.
+We must check this against all components used.
+For our example, we component `K' which has an extra 
+failure mode for degraded performance `d'.
+
+
+
+\ifthenelse {\boolean{paper}}
+{
+We can definine a `failure modes' function $fm$ that has a functional group as its range
+and returns a set of failure modes as its domain.
+We now use this to determine the failure modes
+in our functional groups.
+}
+{
+Using the overloaded function $fm$ from chapter \ref{fmdef} we can determine the failure modes
+in our functional groups.
+}
+
+Applying the function $fm$ to our functional groups, with the SYSTEM environmental
+constraint applied to component type `K',  yields
+
+%%//$$ FG^0_1 = \{C_1, C_2\},$$
+%%$$ FG^0_2 = \{C_1, C_3, K_4\},$$
+%%$$ FG^0_3 = \{C_5, C_6, K_7\}.$$
+
+$$ fm(FG^0_1) = \{C_{1 a}, C_{1 b}, C_{2 a}, C_{2 b}\},$$
+$$ fm(FG^0_2) = \{C_{1 a}, C_{1 b}, C_{2 a}, C_{2 b}, K_{4 a}, K_{4 b}, K_{4 d}\},$$
+$$ fm(FG^0_3) = \{C_{5 a}, C_{5 b}, C_{6 a}, C_{6 b}, K_{7 a}, K_{7 b}, K_{7 d}\}.$$
+
+The next stage is to look at the failure modes from the perspective of
+the functional groups, rather than the components.
+We can call these failures modes `symptoms'.
+As this is a theoretical example, we shall have to skip this step.
+The next stage is to collect the common symptoms, or the symtoms that
+are the same {\em from the perspective of a user of the {\fg}}.
+We can define this stage as the function $\bowtie$ which has a set of failure modes as 
+its range and {\dc} as its domain.
+
+For the sake of example let us determine some arbitary collections
+into symptoms. Let us group the symptoms from $ FG^0_1 $ as the following
+$ s1 = \{ C_{1 a}, C_{2 b} \}$ and $ s2  = \{ C_{1 b}, C_{2 a} \}$.
+We can now create a new {\dc}. This will have an $\alpha$ value higher
+than the {\fg} it was derived from.
+
+thus
+$$ DC^1_1 = \bowtie fm(FG^0_1) .$$
+
+Applying $fm$ to our new derived component will give us our symptoms from functional group $ FG^0_1 $
+thus
+
+$$ fm(DC^1_1) = \{s1, s2 \}.$$
+
+
 UML OBJECT MODEL OF  DERIVED COMPONENT TOO
 
 

fmmd_data_model/paper.tex

@@ -32,7 +32,7 @@
 \author{R.P.Clark}
 \title{FMMD Data Model}
 \maketitle 
-\input{fmmd_data_model}
+\input{fmmd_data_model_paper}
 
 \bibliographystyle{plain}
 \bibliography{../vmgbibliography,../mybib}

style.tex

@@ -66,7 +66,7 @@
 %\newcommand{\wlc}{{Water~Level~Controller~Unit}}
 %\newcommand{\ft}{{\em 4 $\rightarrow$ 20mA } }
 %\newcommand{\tds}{TDS Daughterboard}
-\newcommand{\oc}{$^{o}{C}$}
+\newcommand{\oc}{\ensuremath{^{o}{C}}}
 \newcommand{\adctw}{{${\mathcal{ADC}}_{12}$}}
 \newcommand{\adcten}{{${\mathcal{ADC}}_{10}$}}
 \newcommand{\ohms}[1]{\ensuremath{#1\Omega}}