added atomic definition of failure mode sets

component_failure_modes_definition/component_failure_modes_definition.tex

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+
+\abstract{ This chapter defines what is meant by the terms
+components, component fault modess and atomic 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.
+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}.
+
+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.
+
+$$ FM : K \mapsto F | F \exits A $$
+
+
+\subsection{Component failure modes : Atomic definition}
+
+Electrical resistors can fail by going OPEN or SHORTED for instance.
+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.
+
+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 $A$.
+
+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.
+
+\begin{equation}
+\label{atomic_def}
+ active(f) | f \in F \wedge f1 \in F | f1 \neq f  \wedge \neg active(f1) 
+\end{equation}
+
+As an example the resistor $R$ 
+has two failure modes $_{open}$ and $R_{shorted}$. 
+
+$$ F = FM(R) = \{ R_{open}, R_{shorted} \} $$
+ 
+Applying equation \ref{atomic_definition} to a resistor
+for both fault modes
+
+ $$ active(R_{short}) | R_{short} \in F \wedge R_{open} \in F | R_{open} \neq R_{short}  \wedge \neg active(R_{open})  $$
+ $$ active(R_{open}) | R_{open} \in F \wedge R_{short} \in F | R_{short} \neq R_{open}  \wedge \neg active(R_{short})  $$
+
+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
+component failure modes.
+
+A general case can be stated by taking equation \ref{atomnic_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) 
+\end{equation}
+
+
+