proof read

component_failure_modes_definition/component_failure_modes_definition.tex

@@ -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
 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 
-to fail in two ways, it can go open circuit or it can short.  But we can also 
-associate it with a set of known failure modes. The UML diagram in figure 
+to fail in two ways, it can go open circuit or it can short.  
+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
 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
 in it and decide how these will affect the Group.
 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'.
 We now have something very useful, because
 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
 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.
-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]
  \centering
@@ -141,7 +142,7 @@ $$ FunctionalGroup  \stackrel{has}{\longrightarrow} Components $$
 Using the symbol $\bowtie$ to indicate an analysis process that takes a
 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
 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.
-
 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,
 the combinations could be represented by new failure modes, or
 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}
 
-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.
 
@@ -360,6 +358,8 @@ Because of this the failure mode set $F=FM(R)$ is `unitary~state'.
 Thus
 
 $$ R_{SHORTED} \cap R_{OPEN} = \emptyset $$
+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
@@ -367,13 +367,16 @@ of component failure modes.
 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
 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
- we have banned larger combinations as well.
+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.
 
 
 
@@ -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.
 By definition while all components in a system are `working perfectly' 
 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
 %$$ \Omega = {OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N} $$
-$$ \Omega(K) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, ... ,failure\_mode_{N}\} $$
-The failure mode set for a given component or sub-system $F$
+$$ \Omega(K) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, \ldots ,failure\_mode_{N}\} $$
+The failure mode set $F$ for a given component or derived~component $K$
 is therefore
 $$ F = \Omega(K) \backslash OK $$
 
+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}