components with independent failure modes

component_failure_modes_definition/component_failure_modes_definition.tex

@@ -734,6 +734,76 @@ component failure modes $\{ B_1 ... B_8, OK \}$ obeying unitary state conditions
  \label{fig:partitioncfm}
 \end{figure}
 
+\section{Components with Independent failure modes}
+
+Suppose that we have a component that can fail simultaneously
+with more than one failure mode.
+This would make it seemingly impossible to model as  `unitary state'.
+
+\paragraph{De-composition of complex component.}
+There are two ways in which we can deal with this.
+We could consider the component a composite
+of two simpler components, and model their interaction to
+create a derived component.
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=200pt,bb=0 0 353 247,keepaspectratio=true]{./component_failure_modes_definition/compco.jpg}
+ % compco.jpg: 353x247 pixel, 72dpi, 12.45x8.71 cm, bb=0 0 353 247
+ \caption{Component with three failure modes as partitioned sets}
+ \label{fig:combco}
+\end{figure}
+
+\paragraph{Combinations become new failure modes.}
+Alternatively, we could consider the combinations
+of the failure modes as new failure modes.
+We can model this using an Euler diagram representation of
+an example component with three failure modes $\{ B_1, B_2, B_3, OK \}$ see figure \ref{fig:combco}.
+
+For the purpose of example let us consider $\{ B_2, B_3 \}$
+to be intrinsically mutually exclusive, by $B_1$ to be independent.
+This means the we have the possibility of two new combinations
+$ B_1 \cap B_2$ and $ B_1 \cap B_3$.
+We can represent these 
+as shaded sections of figure \ref{fig:combco2}.
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=200pt,bb=0 0 353 247,keepaspectratio=true]{./component_failure_modes_definition/compco2.jpg}
+ % compco.jpg: 353x247 pixel, 72dpi, 12.45x8.71 cm, bb=0 0 353 247
+ \caption{Component with three failure modes where $B_1$ is independent}
+ \label{fig:combco2}
+\end{figure}
+
+
+
+We can calculate the probabilities for the shaded areas
+assuming the failure modes are statistically independent
+by multiplying the probabilities of the members of the intersection.
+We can use the function $P$ to return the probability of a 
+failure mode, or combination thereof.
+Thus for $P(B_1 \cap B_2) = P(B_1)P(B_2)$ and  $P(B_1 \cap B_3) = P(B_1)P(B_3)$.
+
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=200pt,bb=0 0 353 247,keepaspectratio=true]{./component_failure_modes_definition/compco3.jpg}
+ % compco.jpg: 353x247 pixel, 72dpi, 12.45x8.71 cm, bb=0 0 353 247
+ \caption{Component with two new failure modes}
+ \label{fig:combco3}
+\end{figure}
+
+
+We can now consider the shaded areas as new failure modes of the component.
+Because of the combinations, the probabilities for the failure modes
+$B_1, B_2$ and $B_3$ will now reduce.
+We can use the prime character ($/prime$), to represent the altered value for a failure mode, i.e.
+$B_1^\prime$ represents the altered value for $B_1$.
+Thus
+$$ P(B_1^\prime) = B_1 - P(B_1 \cap B_2) - P(B_1 \cap B_3)\; , $$
+$$ P(B_2^\prime) = B_2 - P(B_1 \cap B_2) \; and $$ 
+$$ P(B_3^\prime) = B_3 - P(B_1 \cap B_3) \; . $$
+
 
 
 
@@ -778,7 +848,7 @@ operational states.
 Some failure modes may only be active given specific environmental conditions
 or when other failures are already active.
 To model this, an `inhibit' class has been added.
-This is an optional atribute of 
+This is an optional attribute of 
 a failure mode. This inhibit class can be triggered
 on a combination of environmental or failure modes.