From f0434f9463115e042eb7ee5c9798d430c37621ed Mon Sep 17 00:00:00 2001 From: Robin Clark Date: Mon, 10 Jan 2011 14:16:57 +0000 Subject: [PATCH] components with independent failure modes --- component_failure_modes_definition/compco.dia | Bin 0 -> 1121 bytes component_failure_modes_definition/compco.jpg | Bin 0 -> 8570 bytes .../compco2.dia | Bin 0 -> 1206 bytes .../compco2.jpg | Bin 0 -> 14396 bytes .../compco3.dia | Bin 0 -> 1268 bytes .../compco3.jpg | Bin 0 -> 10012 bytes .../component_failure_modes_definition.tex | 72 +++++++++++++++++- 7 files changed, 71 insertions(+), 1 deletion(-) create mode 100644 component_failure_modes_definition/compco.dia create mode 100644 component_failure_modes_definition/compco.jpg create mode 100644 component_failure_modes_definition/compco2.dia create mode 100644 component_failure_modes_definition/compco2.jpg create mode 100644 component_failure_modes_definition/compco3.dia create mode 100644 component_failure_modes_definition/compco3.jpg diff --git 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a/component_failure_modes_definition/component_failure_modes_definition.tex b/component_failure_modes_definition/component_failure_modes_definition.tex index cf66f66..f7c3013 100644 --- a/component_failure_modes_definition/component_failure_modes_definition.tex +++ b/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.