125 lines
5.0 KiB
TeX
125 lines
5.0 KiB
TeX
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\abstract{ This chapter defines what is meant by the terms
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components, component fault modes and `unitary~state' component fault modes.
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The application of Bayes theorem in current methodologies, and
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the unsuitability of the `null hypothesis' or p value statistical approach.
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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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\subsection{Unitary State Component Failure Mode sets}
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An important factor in defining a set of failure modes 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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Having a set of failure modes where $N$ modes could be active simultaneously
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would mean having to consider $2^N$ failure mode scenarios.
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%
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Should a component be analysed and simultaneous failure mode cases exit,
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the combinations could be represented by a new failure modes, or
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the component should be considered from a fresh perspective,
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perhaps considering it as several smaller components
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within one package.
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\begin{definition}
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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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This is termed the $U$ set thoughout this study.
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This corresponds to the `mutually exclusive' definition in
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probability theory\cite{probandstat}.
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\end{definition}
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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 exclusive.
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Because of this the failure mode set $F=FM(R)$ is `unitary~state'.
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Thus
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$$ R_{SHORTED} \cap R_{OPEN} = \emptyset $$
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We can make this a general case by taking a set $C$ representing a collection
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of component failure modes,
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We can now state that
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$$ c1 \cap c2 \neq \emptyset | c1 \neq c2 \wedge c1,c2 \in C \wedge C \not\in U $$
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That is to say that if it is impossible that any pair of failure modes can be active at the same time
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the failure mode set is not unitary~state and does not exist in the family of sets $U$
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Note where that are more than two failure~modes, by banning pairs from happening at the same time
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we have banned larger combinations as well
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\subsection{Component Failure Modes and Statistical Sample Space}
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A sample space is defined as the set of all possible outcomes.
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When dealing with failure modes, we are not interested in
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the state where the compoent is working perfectly or `OK' (i.e. operating with no error).
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We are interested only in ways in which it can fail.
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By definition while all components in a system are `working perfectly'
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that system will not exhibit faulty behavuiour.
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Thus the statistical sample space $\Omega$ for a component/sub-system K is
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%$$ \Omega = {OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N} $$
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$$ \Omega(K) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N}\} $$
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The failure mode set for a given component or sub-system $F$
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is therefore
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$$ F = \Omega(K) \backslash OK $$
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\subsection{Bayes Theorem}
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Describe application - likely hood of faults being the cause of symptoms -
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probablistic approach - no direct causation paths to the higher~abstraction fault mode.
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Often for instance a component in a module within a module within a module etc
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that has a probability of causing a SYSTEM level fault.
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Used in FTA\cite{NASA}\cite{NUK}. Problems, difficult to get reliable stats
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for probability to cause because of small sample numbers...
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FMMD approach can by traversing down the tree use known component failure figures
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to
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%$$ c1 \cap c2 \eq \emptyset | c1 \neq c2 \wedge c1,c2 \in C \wedge C \in U $$
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%Thus if the failure~modes are pairwaise mutually exclusive they qualify for inclusion into the
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%unitary~state set family.
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\subsection{Tests of Hypotheses and Significance}
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In high reliability systems the fauls are often logged - strange occurances -
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processors resetting - what are the common factors - P values -
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for instance very high voltage spikes can reset micro controllers -
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but how do you corrollate that with unshielded suppressed contactors...
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Maybe looking at the equipment and seeing if there is a 5\%
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level of the error being caused ?
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i.e. using it to search for these conditions ?
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