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Robin Clark 2010-12-01 09:05:39 +00:00
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97
fmmd_data_model/fmmd_data_model.tex
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@ -111,11 +111,14 @@ 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
type `K' have a de-graded performance
\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
at high temperatures \cite{tlp181}.}.
failure mode between
${{80}\oc}$ and ${{125}\oc}$.
We can term this
degraded performance of component `K' as failure mode `d'.
@ -125,8 +128,8 @@ Again for the sake of example, let us say that each functional
group has one or two symptoms again subscripted by $a$ and $b$.
%Applying symptom abstraction to $FG^0_1$ i.e. $\bowtie fm ( FG^0_1 ) = \{ FG^0_{1 a}, FG^0_{1 b} \} $
%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} \} $.
%We can now create a new derived component, $C^1_1$, whose failure
%modes are the symptoms of $FG^0_1 $ thus $ fm ( {C}^1_1 ) = \{ FG^0_{1 a}, FG^0_{1 b} \} $.
\paragraph{Building the Object Model}
@ -142,6 +145,22 @@ We shall begin with the $FG^0$ level functional groups $ FG^0_1, FG^0_2 $ and $
\label{fig:cfg2fmmd_data}
\end{figure}
The UML model shows the relationships between data types (or classes) that
are used in the FMMD process.
The purpose of failure mode analysis, is to tie SYSTEM level failures
to their possible causes in the base components.
By doing this accurate statistics can be obtained for SYSTEM level
failures, and an insight into how we can make the system safer
can be determined.
In order to do this, we need to be able to trace the component
failure modes from the functional groups, to the symptoms
they cause, and to the failure modes in the {\dcs}.
We can use graph theory to represent this.
As it would make no sense for a derived component to
derive failure modes form itsself, we can apply an acyclic constraint
to the graph. This means the graph must be a Directed Acylic
Graph (DAG).
% %\begin{figure}[h]
% \centering
% \includegraphics[width=400pt,keepaspectratio=true]{./fmmd_data_model/cfg2.jpg}
@ -158,9 +177,9 @@ We must check this against all components used.
For our example, we component `K' which has an extra
failure mode for degraded performance `d'. Thus applying the function $fm$
to component type `K' under these temperature range conditions
gives the following failure modes, $fm{K} =\{ K_a, K_b, K_d \}$.
gives the following failure modes, $fm{K} =\{ K^0_a, K^0_b, K^0_d \}$.
Were our system specified for a ${{0}\oc}$ to ${{80}\oc}$ range
we could say $fm{K} =\{ K_a, K_b \}$.
we could say $fm{K} =\{ K^0_a, K^0_b \}$.
\pagebreak[3]
\paragraph{Get the failure modes from the functional groups.}
@ -171,9 +190,9 @@ constraint applied to component type `K', yields
%%$$ FG^0_2 = \{C_1, C_3, K_4\},$$
%%$$ FG^0_3 = \{C_5, C_6, K_7\}.$$
$$ fm(FG^0_1) = \{C_{1 a}, C_{1 b}, C_{2 a}, C_{2 b}\},$$
$$ fm(FG^0_2) = \{C_{1 a}, C_{1 b}, C_{3 a}, C_{3 b}, K_{4 a}, K_{4 b}, K_{4 d}\},$$
$$ fm(FG^0_3) = \{C_{5 a}, C_{5 b}, C_{6 a}, C_{6 b}, K_{7 a}, K_{7 b}, K_{7 d}\}.$$
$$ fm(FG^0_1) = \{C^0_{1 a}, C^0_{1 b}, C^0_{2 a}, C^0_{2 b}\},$$
$$ fm(FG^0_2) = \{C^0_{1 a}, C^0_{1 b}, C^0_{3 a}, C^0_{3 b}, K^0_{4 a}, K^0_{4 b}, K^0_{4 d}\},$$
$$ fm(FG^0_3) = \{C^0_{5 a}, C^0_{5 b}, C^0_{6 a}, C^0_{6 b}, K^0_{7 a}, K^0_{7 b}, K^0_{7 d}\}.$$
The next stage is to look at the failure modes from the perspective of
the functional groups, rather than the components.
@ -188,9 +207,9 @@ 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} \}$.
$ s1 = \{ C^0_{1 a}, C^0_{2 b} \}$ and $ s2 = \{ C^0_{1 b}, C^0_{2 a} \}$.
We can represent the relationships between the failure modes, and desired failure modes or symptoms
as a directed acyclic graph (see figure \ref{fig:dag0}).
as a DAG (see figure \ref{fig:dag0}).
\def\layersep{2.5cm}
@ -288,14 +307,14 @@ In this case all components were base components and therefore have an $\alpha$
Our derived component can thus take an $\alpha$ value of one.
Our newly derived component can be
$$ DC^1_1 = \bowtie fm(FG^0_1) .$$
$$ C^1_1 = \bowtie fm(FG^0_1) .$$
Applying $fm$ to our new derived component will give us our symptoms from functional group $ FG^0_1 $
thus
$$ fm(DC^1_1) = \{s1, s2 \}.$$
$$ fm(C^1_1) = \{s1, s2 \}.$$
We can represent $ DC^1_1 $ as an addition to the DAG (see figure \ref{fig:dag1}).
We can represent $ C^1_1 $ as an addition to the DAG (see figure \ref{fig:dag1}).
@ -380,25 +399,25 @@ We can represent $ DC^1_1 $ as an addition to the DAG (see figure \ref{fig:dag1}
\node[annot,right of=s](dcl) {Derived Component};
\end{tikzpicture}
% End of code
\caption{DAG representing failure modes and symptoms $FG^0_1 \rightarrow DC^1_1$}
\caption{DAG representing failure modes and symptoms $FG^0_1 \rightarrow C^1_1$}
\label{fig:dag1}
\end{figure}
\clearpage
\subsection{ Creating Derived components from $FG^0_2$ and $FG^0_3$ }
Applying the FMMD process for $FG^0_2$ and $FG^0_3$.
\paragraph{Applying FMMD $ \bowtie fm(FG^0_2) $:}
The failure modes $fm(FG^0_2) = \{C_{1 a}, C_{1 b}, C_{3 a}, C_{3 b}, K_{4 a}, K_{4 b}, K_{4 d}\}.$
Let us say new symptom s3 can be caused by failure modes $\{C_{1 a}, C_{3 b}, K_{4 b} \}$
, let us say new symptom s4 can be caused by failure modes $\{C_{1 b}, C_{3 a}, K_{4 d} \}$
and let us say new symptom s5 can be caused by failure mode $\{K_{4 a} \}$.
The failure modes $fm(FG^0_2) = \{C^0_{1 a}, C^0_{1 b}, C^0_{3 a}, C^0_{3 b}, K^0_{4 a}, K^0_{4 b}, K^0_{4 d}\}.$
Let us say new symptom s3 can be caused by failure modes $\{C^0_{1 a}, C^0_{3 b}, K^0_{4 b} \}$
, let us say new symptom s4 can be caused by failure modes $\{C^0_{1 b}, C^0_{3 a}, K^0_{4 d} \}$
and let us say new symptom s5 can be caused by failure mode $\{K^0_{4 a} \}$.
We can create a derived component $DC^1_2$ using
$\bowtie fm(FG^0_2) = DC^1_2$.
Applying $fm$ to our {\dcs} gives $fm(DC^1_2) = \{ s3,s4,s5 \}$.
We can create a derived component $C^1_2$ using
$\bowtie fm(FG^0_2) = C^1_2$.
Applying $fm$ to our {\dcs} gives $fm(C^1_2) = \{ s3,s4,s5 \}$.
We can respresent this in the DAG in figure \ref{fig:dag2}.
@ -510,7 +529,7 @@ We can respresent this in the DAG in figure \ref{fig:dag2}.
\node[annot,right of=s](dcl) {Derived Component};
\end{tikzpicture}
% End of code
\caption{DAG representing failure modes and symptoms $FG^0_1 \rightarrow DC^1_1$ and $FG^0_2 \rightarrow DC^1_2$}
\caption{DAG representing failure modes and symptoms $FG^0_1 \rightarrow C^1_1$ and $FG^0_2 \rightarrow C^1_2$}
\label{fig:dag2}
\end{figure}
@ -518,16 +537,16 @@ We can respresent this in the DAG in figure \ref{fig:dag2}.
%/\clearpage
\paragraph{Applying FMMD $\bowtie fm(FG^0_3) $ :}
Let us say new symptom s6 can be caused by failure modes $\{C_{5 a}, C_{6 b}, K_{4 b} \}$
, let us say new symptom s7 can be caused by failure modes $\{C_{5 b}, C_{6 a}, K_{7 d} \}$
and let us say new symptom s8 can be caused by failure mode $\{K_{7 a} \}$.
We can create a derived component $DC^1_3$ using
$\bowtie fm(FG^0_3) = DC^1_3$
where $fm(DC^1_3) = \{ s6,s7,s8 \}$.
We can create a derived component $C^1_3$ using
$\bowtie fm(FG^0_3) = C^1_3$
where $fm(C^1_3) = \{ s6,s7,s8 \}$.
We can now represent the first stage of FMMD, all base component
failure modes analysed and our first set of derived components determined.
@ -575,9 +594,9 @@ This is shown in the DAG in figure \ref{fig:dag3}.
\node[failure] (C-5b) at (\layersep,-11) {b};
\node[failure] (C-6a) at (\layersep,-12) {a};
\node[failure] (C-6b) at (\layersep,-13) {b};
\node[failure] (K-7a) at (\layersep,-15) {a};
\node[failure] (K-7b) at (\layersep,-16) {b};
\node[failure] (K-7d) at (\layersep,-17) {d};
\node[failure] (K-7a) at (\layersep,-14) {a};
\node[failure] (K-7b) at (\layersep,-15) {b};
\node[failure] (K-7d) at (\layersep,-16) {d};
% Draw the output layer node
@ -672,14 +691,14 @@ This is shown in the DAG in figure \ref{fig:dag3}.
\node[annot,right of=s](dcl) {Derived Component};
\end{tikzpicture}
% End of code
\caption{DAG representing failure modes and symptoms $FG^0_1 \rightarrow DC^1_1$ and $FG^0_2 \rightarrow DC^1_2$}
\caption{DAG representing failure modes and symptoms $FG^0_1 \rightarrow C^1_1$ and $FG^0_2 \rightarrow C^1_2$}
\label{fig:dag3}
\end{figure}
\clearpage
%\clearpage
%\pagebreak[4]
\subsection{Using Derived Components in Functional Groups}
@ -689,9 +708,9 @@ We can apply $fm$ to the derived components and
this returns the failure modes. We can notate
these with $a$ and $b$ etc as before, but can give them
a subscript representing the symptom they were sourced from thus:
$$ fm(DC^1_1) = \{ a_{s1}, b_{s2} \}, $$
$$ fm(DC^1_2) = \{ a_{s3}, b_{s4}, c_{s5} \}, $$
$$ fm(DC^1_3) = \{ a_{s6}, b_{s7}, c_{s8} \}. $$
$$ fm(C^1_1) = \{ a_{s1}, b_{s2} \}, $$
$$ fm(C^1_2) = \{ a_{s3}, b_{s4}, c_{s5} \}, $$
$$ fm(C^1_3) = \{ a_{s6}, b_{s7}, c_{s8} \}. $$
In order to determine SYSTEM level symptoms, we need to
use the derived components to form a higher level functional
@ -702,10 +721,10 @@ can use all three derived components to
create a top~level functional group.
Let
$ FG^1_1 = \{ DC^1_1, DC^1_1, DC^1_1 \} $.
$ FG^1_1 = \{ C^1_1, C^1_1, C^1_1 \} $.
Applying $fm(FG^1_1) = \{ a_{s1}, b_{s2}, a_{s3}, b_{s4}, c_{s5}, a_{s6}, b_{s7}, c_{s8} \}$.
To get our system level derived component we can apply $ \bowtie fm(FG^1_1) = DC^2_1 $.
To get our system level derived component we can apply $ \bowtie fm(FG^1_1) = C^2_1 $.
NOW THINK ABOUT THIS