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Robin Clark 2010-05-04 12:01:39 +01:00
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38
component_failure_modes_definition/component_failure_modes_definition.tex
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@ -44,8 +44,9 @@ What these components all have in common is that they can fail, and fail in
a number of well defined ways. For common components a number of well defined ways. For common components
there is established literature for the failure modes for the system designer consider (with accompanying statistical there is established literature for the failure modes for the system designer consider (with accompanying statistical
failure rates)\cite{mil1991}. For instance, a simple resistor is generally considered failure rates)\cite{mil1991}. For instance, a simple resistor is generally considered
to fail in two ways, it can go open circuit or it can short. But we can also to fail in two ways, it can go open circuit or it can short.
associate it with a set of known failure modes. The UML diagram in figure Thus we can associate a set of faults to this component $ResistorFaultModes=\{OPEN, SHORT\}$.
The UML diagram in figure
\ref{fig:component} shows a component as a simple data \ref{fig:component} shows a component as a simple data
structure with its failure modes. structure with its failure modes.
@ -106,7 +107,7 @@ We can term this a `Functional~Group'. When we have a
`Functional~Group' we can look at the failure modes of all the components `Functional~Group' we can look at the failure modes of all the components
in it and decide how these will affect the Group. in it and decide how these will affect the Group.
Or in other words we can determine the failure modes of the functional Or in other words we can determine the failure modes of the functional
group. These failure modes are derived from the functional group, we can therefore call group. These new failure modes are derived from the functional group, we can therefore call
these `derived failure modes'. these `derived failure modes'.
We now have something very useful, because We now have something very useful, because
we can now treat this functional group as a component with a known set of failure modes. we can now treat this functional group as a component with a known set of failure modes.
@ -118,7 +119,7 @@ This process can continue until have build a hierarcy that converges to a failur
To differentiate the components derived from functional groups, we can To differentiate the components derived from functional groups, we can
add a new attribute to the class `Component', that of analysis add a new attribute to the class `Component', that of analysis
level. The UML representation shows a `functional group' having a one to one relationship with a derived component. level. The UML representation shows a `functional group' having a one to one relationship with a derived component.
We can represet this in a UML diagram see figure \ref{fig:cfg} We can represet this using an UML diagram in figure \ref{fig:cfg}
\begin{figure}[h] \begin{figure}[h]
\centering \centering
@ -141,7 +142,7 @@ $$ FunctionalGroup \stackrel{has}{\longrightarrow} Components $$
Using the symbol $\bowtie$ to indicate an analysis process that takes a Using the symbol $\bowtie$ to indicate an analysis process that takes a
functional group and converts it into a new component. functional group and converts it into a new component.
$$ \bowtie ( FG ) \mapsto Component $$ $$ \bowtie ( FG ) \mapsto DerivedComponent $$
% %
@ -309,13 +310,10 @@ $$ \bowtie ( FG ) \mapsto Component $$
An important factor in defining a set of failure modes is that they An important factor in defining a set of failure modes is that they
should be as clearly defined as possible. should be as clearly defined as possible.
%
It should not be possible for instance for It should not be possible for instance for
a component to have two or more failure modes active at once. a component to have two or more failure modes active at once.
Having a set of failure modes where $N$ modes could be active simultaneously Having a set of failure modes where $N$ modes could be active simultaneously
would mean having to consider $2^N-1$ failure mode scenarios. would mean having to consider an additional $2^N-1$ failure mode scenarios.
%
Should a component be analysed and simultaneous failure mode cases exit, Should a component be analysed and simultaneous failure mode cases exit,
the combinations could be represented by new failure modes, or the combinations could be represented by new failure modes, or
the component should be considered from a fresh perspective, the component should be considered from a fresh perspective,
@ -346,7 +344,7 @@ Thus if the failure modes of $F$ are unitary~state, we can say $F \in U$.
\section{Component failure modes : Unitary State example} \section{Component failure modes : Unitary State example}
A component with simple ``unitary~state'' failure modes is the electrical resistor. A component with an obvious set of ``unitary~state'' failure modes is the electrical resistor.
Electrical resistors can fail by going OPEN or SHORTED. Electrical resistors can fail by going OPEN or SHORTED.
@ -360,6 +358,8 @@ Because of this the failure mode set $F=FM(R)$ is `unitary~state'.
Thus Thus
$$ R_{SHORTED} \cap R_{OPEN} = \emptyset $$ $$ R_{SHORTED} \cap R_{OPEN} = \emptyset $$
therefore
$$ FM(R) \in U $$
We can make this a general case by taking a set $C$ (where $c1, c2 \in C$) representing a collection We can make this a general case by taking a set $C$ (where $c1, c2 \in C$) representing a collection
@ -367,12 +367,15 @@ of component failure modes.
We can now state that We can now state that
$$ c1 \cap c2 \neq \emptyset | c1 \neq c2 \wedge c1,c2 \in C \wedge C \not\in U $$ \begin{equation}
c1 \cap c2 \neq \emptyset | c1 \neq c2 \wedge c1,c2 \in C \wedge C \not\in U
\end{equation}
That is to say that it is impossible that any pair of failure modes can be active at the same time That is to say that it is impossible that any pair of failure modes can be active at the same time
for the failure mode set $C$ to exists in the family of sets $U$ for the failure mode set $C$ to exists in the family of sets $U$
Note where that are more than two failure~modes, by banning pairs from being active at the same time Note where that are more than two failure~modes,
by banning pairs from being active at the same time
we have banned larger combinations as well. we have banned larger combinations as well.
@ -387,11 +390,16 @@ the state where the component is working perfectly or `OK' (i.e. operating with
We are interested only in ways in which it can fail. We are interested only in ways in which it can fail.
By definition while all components in a system are `working perfectly' By definition while all components in a system are `working perfectly'
that system will not exhibit faulty behaviour. that system will not exhibit faulty behaviour.
Thus the statistical sample space $\Omega$ for a component/sub-system K is Thus the statistical sample space $\Omega$ for a component or derived~component $K$ is
%$$ \Omega = {OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N} $$ %$$ \Omega = {OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N} $$
$$ \Omega(K) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, ... ,failure\_mode_{N}\} $$ $$ \Omega(K) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, \ldots ,failure\_mode_{N}\} $$
The failure mode set for a given component or sub-system $F$ The failure mode set $F$ for a given component or derived~component $K$
is therefore is therefore
$$ F = \Omega(K) \backslash OK $$ $$ F = \Omega(K) \backslash OK $$
The $OK$ statistical case is the largest in probability, and is therefore
of interest when analysing systems that have failed using techniques
such as bayes theorem to determine the likelyhood of the failure source.
\vspace{40pt}