From 2394eb67c444f743743d5340bfe7837dcea07f9a Mon Sep 17 00:00:00 2001 From: Robin Date: Sat, 20 Mar 2010 19:48:51 +0000 Subject: [PATCH] trying to get this fir for Andrew to see --- component_failure_modes_definition/Makefile | 17 ++++ .../component_failure_modes_definition.tex | 89 +++++++++++++------ component_failure_modes_definition/paper.tex | 27 ++++++ 3 files changed, 104 insertions(+), 29 deletions(-) create mode 100644 component_failure_modes_definition/Makefile create mode 100644 component_failure_modes_definition/paper.tex diff --git a/component_failure_modes_definition/Makefile b/component_failure_modes_definition/Makefile new file mode 100644 index 0000000..ff49043 --- /dev/null +++ b/component_failure_modes_definition/Makefile @@ -0,0 +1,17 @@ + +# +# Make the propositional logic diagram a paper +# + + +paper: paper.tex component_failure_modes_definition_paper.tex + #latex paper.tex + #dvipdf paper pdflatex cannot use eps ffs + pdflatex paper.tex + okular paper.pdf + + +# Remove the need for referncing graphics in subdirectories +# +component_failure_modes_definition_paper.tex: component_failure_modes_definition.tex + cat component_failure_modes_definition.tex | sed 's/component_failure_modes_definition\///' > component_failure_modes_definition_paper.tex diff --git a/component_failure_modes_definition/component_failure_modes_definition.tex b/component_failure_modes_definition/component_failure_modes_definition.tex index 0a69fe2..9fc0d8f 100644 --- a/component_failure_modes_definition/component_failure_modes_definition.tex +++ b/component_failure_modes_definition/component_failure_modes_definition.tex @@ -1,67 +1,98 @@ \abstract{ This chapter defines what is meant by the terms -components, component fault modess and atomic component fault modes. +components, component fault modess and `unitary~state' component fault modes. Mathematical constraints and definitions are made using set theory. } \section{Introduction} -When building a system the components used, will have known failure modes. +When building a system from components, +we should be able to find all known failure modes for each component. For most common electrical and mechanical components, the failure modes for a given type of part can be obtained from standard literature\cite{mil1991} -\cite{mech}. +\cite{mech}. %The failure modes for a given component $K$ form a set $F$. -We can define a function $FM()$ to represent thiss, where K is the component -and F is the set of failure modes and A represents the set of atomic failure mode sets. +An important factor in defining a failure mode is that they +should be as clearly defined as possible. +% +It should not be possible for instance for +a component to have two or more failure modes active at once. +Should this be the case, the failure modes have not been clearly analysed. +The combination could be represented by a new failure mode, or +the component should be re-analysed. A set of failure modes where only one fault mode +can be active at a time is termed a `unitary~state' failure mode set. -$$ FM : K \mapsto F | F \exits A $$ +We can define a function $FM()$ to +take a given component $K$ and return its set of failure modes $F$. + +$$ FM : K \mapsto F $$ + +We can further define a set $U$ which is a set of sets of failure modes, where +the component failure modes in each of its members are unitary~state. +Thus if the failure modes of $F$ are unitary~state, we can say $F \in U$. -\subsection{Component failure modes : Atomic definition} +\subsection{Component failure modes : Unitary State example} -Electrical resistors can fail by going OPEN or SHORTED for instance. +A component with simple ``unitary~state'' failure modes is the electrical resistor. + +Electrical resistors can fail by going OPEN or SHORTED. However they cannot fail with both conditions active. The conditions OPEN and SHORT are mutually exlusive. -Because of this these failure modes can be considered `atomic'. -A more complex component, say a micro controller could have several -faults active. It could for instance have a broken I/O output -and an unstable ADC input. Here the faults cannot be considered atomic. +Because of this the failure mode set $F=FM(R)$ is `unitary~state'. +%A more complex component, say a micro controller could have several +%faults active. It could for instance have a broken I/O output +%and an unstable ADC input. Here the faults cannot be considered `unitary~state'. -A set of failure modes, where only one or no failure modes -are active is termed an atomic failure mode set. This -will be donoted as set $A$. +% A set of failure modes, where only one or no failure modes +% are active is termed an `unitary~state' failure mode set. This +% will be donoted as set $A$. +% +To define `unitary~state' using set theory we can define a function +`active'. +The function $active(f)$ deontes that the failure mode $f$ (where $f$ is an element of $F$) is currently active. -The function $active(f)$ deontes that the failure mode f is currently active. - -Thus for the set $F$ to exist in $A$ the following condition must be true. +Thus for the set $F$ to exist in $U$ the following condition must be true. \begin{equation} -\label{atomic_def} - active(f) | f \in F \wedge f1 \in F | f1 \neq f \wedge \neg active(f1) +\label{unitarystate_def} + F \in U | f \in F \wedge active(f) \wedge f1 \in \wedge \neq f \wedge \neg active(f1) \end{equation} As an example the resistor $R$ -has two failure modes $_{open}$ and $R_{shorted}$. +has two failure modes $R_{open}$ and $R_{shorted}$. -$$ F = FM(R) = \{ R_{open}, R_{shorted} \} $$ +$$ FM(R) = F = \{ R_{open}, R_{shorted} \} $$ -Applying equation \ref{atomic_definition} to a resistor +Applying equation \ref{`unitarystate'_definition} to a resistor for both fault modes - $$ active(R_{short}) | R_{short} \in F \wedge R_{open} \in F | R_{open} \neq R_{short} \wedge \neg active(R_{open}) $$ - $$ active(R_{open}) | R_{open} \in F \wedge R_{short} \in F | R_{short} \neq R_{open} \wedge \neg active(R_{short}) $$ + $$ active(R_{short}) | R_{short} \in F \wedge R_{open} \in F \wedge R_{open} \neq R_{short} \wedge \neg active(R_{open}) $$ + $$ active(R_{open}) | R_{open} \in F \wedge R_{short} \in F \wedge R_{short} \neq R_{open} \wedge \neg active(R_{short}) $$ For the case of the resistor with only two failure modes the results above, being true, -show that the failure modes for a resistor of $ F = \{ R_{open}, R_{shorted} \} $ are atomic +show that the failure modes for a resistor of $ F = \{ R_{open}, R_{shorted} \} $ are `unitary~state' component failure modes. -A general case can be stated by taking equation \ref{atomnic_def} and making it a function thus. +Thus + $$ FM(R) = \{ R_{open}, R_{shorted} \} \in U $$ + + +A general case can be stated by taking equation \ref{unitary_state_def} and making it a function thus. \begin{equation} -\label{atomic_def} - Atomic(F) = \forall f \in F | active(f) \wedge f1 \in F \wedge f1 \neq f \wedge \neg active(f1) +\label{`unitarystate'_def} + UnitaryState(F) = \forall f \in F | active(f) \wedge f1 \in F \wedge f1 \neq f \wedge \neg active(f1) \end{equation} +%Which can be written + +%$$ UnitaryState(FM(K)) $$ + + + +% should this be a paragraph in Symptom Abstraction ???? + diff --git a/component_failure_modes_definition/paper.tex b/component_failure_modes_definition/paper.tex new file mode 100644 index 0000000..6a7022b --- /dev/null +++ b/component_failure_modes_definition/paper.tex @@ -0,0 +1,27 @@ + +\documentclass[a4paper,10pt]{article} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{tikz} +\usepackage{amsfonts,amsmath,amsthm} +\input{style} + +%\newtheorem{definition}{Definition:} + +\begin{document} +\pagestyle{fancy} + +\outerhead{{\small\bf Unitary State Failure Mode Sets}} +%\innerfoot{{\small\bf R.P. Clark } } + % numbers at outer edges +\pagenumbering{arabic} % Arabic page numbers hereafter +\author{R.P.Clark} +\title{Unitary State Failure Mode Sets} +\maketitle +\input{component_failure_modes_definition_paper} + +\bibliographystyle{plain} +\bibliography{vmgbibliography,mybib} + +\today +\end{document}