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Put nested bowtie equation in

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Your Name 2012-04-17 12:39:21 +01:00
parent 827470c88f
commit 313b7f89de
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papers/software_fmea/software_fmea.tex
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@ -65,6 +65,8 @@
\newcommand{\fm}{failure~mode} \newcommand{\fm}{failure~mode}
\newcommand{\fms}{failure~modes} \newcommand{\fms}{failure~modes}
\newcommand{\fg}{functional~group} \newcommand{\fg}{functional~group}
\newcommand{\FG}{\mathcal{G}}
\newcommand{\DC}{\mathcal{DC}}
\newcommand{\fgs}{functional~groups} \newcommand{\fgs}{functional~groups}
\newcommand{\dc}{derived~component} \newcommand{\dc}{derived~component}
\newcommand{\dcs}{derived~components} \newcommand{\dcs}{derived~components}
@ -345,7 +347,12 @@ of the {\fg} that it was derived from.
% a {\fg}. Our use of it as a building block corresponds to a {\dc}. % a {\fg}. Our use of it as a building block corresponds to a {\dc}.
We can use the symbol `$\bowtie$' to represent the creation of a derived component We can use the symbol `$\bowtie$' to represent the creation of a derived component
from a {\fg}. We show an FMMD hierarchy in figure~\ref{fig:fmmdh}. from a {\fg}. This symbol is convenient for drawn hierarchy diagrams. % (see figure~\ref{fmmdh}).
We define the $\bowtie$ function, where $\FG$ is the set of all {\fgs} and $\DC$ is the set of all {\dcs},
$$ \bowtie ( {\FG} ) \mapsto {\DC} .$$
We show an FMMD hierarchy in figure~\ref{fig:fmmdh}.
Using this diagram, we can follow the creation of the hierarchy in Using this diagram, we can follow the creation of the hierarchy in
a theoretical system. a theoretical system.
% %
@ -920,9 +927,14 @@ as a hierarchical diagram, see figure~\ref{fig:hd}.
We can represent the hierarchy in figure~\ref{fig:hd} algebraically, using the $\bowtie$ function
definition nested thus:
$$ \bowtie \Big( \bowtie \big( \bowtie(R,ADC), read\_4\_20\_input \big), read\_4\_20\_input \Big). $$
This nested structure means that we have multiple traceable
stages of failure mode reasoning in our analysis. Traditional FMEA would have only one stage
of reasoning for each component failure mode.
%\clearpage %\clearpage
\section{Conclusion} \section{Conclusion}