99 lines
3.7 KiB
TeX
99 lines
3.7 KiB
TeX
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\abstract{ This chapter defines what is meant by the terms
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components, component fault modess and `unitary~state' component fault modes.
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Mathematical constraints and definitions are made using set theory.
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}
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\section{Introduction}
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When building a system from components,
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we should be able to find all known failure modes for each component.
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For most common electrical and mechanical components, the failure modes
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for a given type of part can be obtained from standard literature\cite{mil1991}
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\cite{mech}. %The failure modes for a given component $K$ form a set $F$.
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An important factor in defining a failure mode is that they
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should be as clearly defined as possible.
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%
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It should not be possible for instance for
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a component to have two or more failure modes active at once.
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Should this be the case, the failure modes have not been clearly analysed.
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The combination could be represented by a new failure mode, or
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the component should be re-analysed. A set of failure modes where only one fault mode
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can be active at a time is termed a `unitary~state' failure mode set.
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We can define a function $FM()$ to
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take a given component $K$ and return its set of failure modes $F$.
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$$ FM : K \mapsto F $$
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We can further define a set $U$ which is a set of sets of failure modes, where
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the component failure modes in each of its members are unitary~state.
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Thus if the failure modes of $F$ are unitary~state, we can say $F \in U$.
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\subsection{Component failure modes : Unitary State example}
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A component with simple ``unitary~state'' failure modes is the electrical resistor.
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Electrical resistors can fail by going OPEN or SHORTED.
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However they cannot fail with both conditions active. The conditions
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OPEN and SHORT are mutually exlusive.
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Because of this the failure mode set $F=FM(R)$ is `unitary~state'.
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%A more complex component, say a micro controller could have several
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%faults active. It could for instance have a broken I/O output
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%and an unstable ADC input. Here the faults cannot be considered `unitary~state'.
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% A set of failure modes, where only one or no failure modes
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% are active is termed an `unitary~state' failure mode set. This
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% will be donoted as set $A$.
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%
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To define `unitary~state' using set theory we can define a function
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`active'.
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The function $active(f)$ deontes that the failure mode $f$ (where $f$ is an element of $F$) is currently active.
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Thus for the set $F$ to exist in $U$ the following condition must be true.
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\begin{equation}
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\label{unitarystate_def}
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F \in U | f \in F \wedge active(f) \wedge f1 \in F \wedge f1 \neq f \wedge \neg active(f1)
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\end{equation}
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As an example the resistor $R$
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has two failure modes $R_{open}$ and $R_{shorted}$.
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$$ FM(R) = F = \{ R_{open}, R_{shorted} \} $$
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Applying equation \ref{`unitarystate'_definition} to a resistor
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for both fault modes
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$$ active(R_{short}) | R_{short} \in F \wedge R_{open} \in F \wedge R_{open} \neq R_{short} \wedge \neg active(R_{open}) $$
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$$ active(R_{open}) | R_{open} \in F \wedge R_{short} \in F \wedge R_{short} \neq R_{open} \wedge \neg active(R_{short}) $$
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For the case of the resistor with only two failure modes the results above, being true,
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show that the failure modes for a resistor of $ F = \{ R_{open}, R_{shorted} \} $ are `unitary~state'
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component failure modes.
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Thus
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$$ FM(R) = \{ R_{open}, R_{shorted} \} \in U $$
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A general case can be stated by taking equation \ref{unitary_state_def} and making it a function thus.
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\begin{equation}
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\label{`unitarystate'_def}
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UnitaryState(F) = \forall f \in F | active(f) \wedge f1 \in F \wedge f1 \neq f \wedge \neg active(f1)
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\end{equation}
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%Which can be written
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%$$ UnitaryState(FM(K)) $$
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% should this be a paragraph in Symptom Abstraction ????
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