From 63a37e5e9c77d9d33ee3e911ffd4e8998c2a2f5e Mon Sep 17 00:00:00 2001 From: Robin Clark Date: Mon, 22 Nov 2010 14:20:47 +0000 Subject: [PATCH] . --- .../component_failure_modes_definition.tex | 28 +++++++++---------- 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/component_failure_modes_definition/component_failure_modes_definition.tex b/component_failure_modes_definition/component_failure_modes_definition.tex index 7b69545..40004ba 100644 --- a/component_failure_modes_definition/component_failure_modes_definition.tex +++ b/component_failure_modes_definition/component_failure_modes_definition.tex @@ -495,13 +495,13 @@ Consider the set $S = \{a,b,c\}$. The powerset of S: -$$ \mathcal{P} S = \{ \emptyset, \{a,b,c\}, \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$. +$$ \mathcal{P} S = \{ \emptyset, \{a,b,c\}, \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} .$$ $\mathcal{P}_{\le 2} S $ means all non-empty subsets of S where the cardinality of the subsets is less than or equal to 2 or less. -$$ \mathcal{P}_{\le 2} S = \{ \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$. +$$ \mathcal{P}_{\le 2} S = \{ \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} . $$ Note that $\mathcal{P}_{1} S $ (non-empty subsets where cardinality $\leq 1$) for this example is: @@ -514,9 +514,9 @@ The number of $k$ combinations (each of size $k$) from a set $S$ with $n$ elements (size $n$) is the binomial coefficient~\cite{probstat} shown in equation \ref{bico}. \begin{equation} -C^n_k = {n \choose k} = \frac{n!}{k!(n-k)!} +C^n_k = {n \choose k} = \frac{n!}{k!(n-k)!} . \label{bico} -\end{equation} . +\end{equation} To find the number of elements in a cardinality constrained subset S with up to $cc$ elements in each combination sub-set, @@ -530,9 +530,9 @@ from $1$ to $cc$ thus % \begin{equation} - |{\mathcal{P}_{cc}S}| = \sum^{cc}_{k=1} \frac{|{S}|!}{ k! ( |{S}| - k)!} + |{\mathcal{P}_{cc}S}| = \sum^{cc}_{k=1} \frac{|{S}|!}{ k! ( |{S}| - k)!} . \label{eqn:ccps} -\end{equation} . +\end{equation} @@ -556,7 +556,7 @@ $|{n \choose 2}|$ and $|{n \choose 3}|$ for each component in the functional~gro \subsubsection{Example: Two Component functional group cardinality Constraint of 2} For example: suppose we have a simple functional group with two components R and T, of which -$$fm(R) = \{R_o, R_s\}$$ and $$fm(T) = \{T_o, T_s, T_h\}$$. +$$fm(R) = \{R_o, R_s\}$$ and $$fm(T) = \{T_o, T_s, T_h\}.$$ This means that the functional~group $FG=\{R,T\}$ will have a component failure mode set of $fm(FG) = \{R_o, R_s, T_o, T_s, T_h\}$ @@ -564,7 +564,7 @@ of $fm(FG) = \{R_o, R_s, T_o, T_s, T_h\}$ For a cardinality constrained powerset of 2, because there are 5 error modes ( $|fm(FG)|=5$), applying equation \ref{eqn:ccps} gives :- -$$ | P_2 (fm(FG)) | = \frac{5!}{1!(5-1)!} + \frac{5!}{2!(5-2)!} = 15$$. +$$ | P_2 (fm(FG)) | = \frac{5!}{1!(5-1)!} + \frac{5!}{2!(5-2)!} = 15.$$ This is composed of ${5 \choose 1}$ five single fault modes, and ${5 \choose 2}$ ten double fault modes. @@ -585,7 +585,7 @@ $$ \mathcal{P}_{2}(fm(FG)) = \{ \} $$ -And whose cardinality is 11. % by inspection +and whose cardinality is 11. % by inspection %$$ %| %\{ @@ -628,18 +628,18 @@ components $C_j$ are in \begin{equation} |{\mathcal{P}_{cc}SU}| = {\sum^{cc}_{k=1} \frac{|{SU}|!}{k!(|{SU}| - k)!}} - - {\sum_{j \in J} {|FM({C_{j})}| \choose 2}} + - {\sum_{j \in J} {|FM({C_{j})}| \choose 2}} . \label{eqn:correctedccps} -\end{equation} . +\end{equation} Expanding the combination in equation \ref{eqn:correctedccps} \begin{equation} |{\mathcal{P}_{cc}SU}| = {\sum^{cc}_{k=1} \frac{|{SU}|!}{k!(|{SU}| - k)!}} -- {{\sum_{j \in J} \frac{|FM({C_j})|!}{2!(|FM({C_j})| - 2)!}} } +- {{\sum_{j \in J} \frac{|FM({C_j})|!}{2!(|FM({C_j})| - 2)!}} } . \label{eqn:correctedccps2} -\end{equation} . +\end{equation} \paragraph{Use of Equation \ref{eqn:correctedccps2} } Equation \ref{eqn:correctedccps2} is useful for an automated tool that @@ -707,7 +707,7 @@ By definition while all components in a system are `working perfectly' that system will not exhibit faulty behaviour. Thus the statistical sample space $\Omega$ for a component or derived~component $C$ is %$$ \Omega = {OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N} $$ -$$ \Omega(C) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, \ldots ,failure\_mode_{N}\} $$ +$$ \Omega(C) = \{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 $C$ is therefore $ fm(C) = \Omega(C) \backslash \{OK\} $