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Just need full UML digram now.

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Robin Clark 2010-11-21 19:42:34 +00:00
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component_failure_modes_definition/component_failure_modes_definition.tex
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@ -366,12 +366,13 @@ can be active at one time is termed a {\textbf{unitary~state}} failure mode set.
Let the set of all possible components be $ \mathcal{C}$ Let the set of all possible components be $ \mathcal{C}$
and let the set of all possible failure modes be $ \mathcal{F}$. and let the set of all possible failure modes be $ \mathcal{F}$.
The set of failure modes of a particular component are of interest The set of failure modes of a particular component are of interest
here. What is required is to define a property for here.
a set of failure modes where only one failure mode can be active at a time, What is required is to define a property for
or borrowing from the terms of statistics, the failure mode is an event, and it is mutually exclusive a set of failure modes where only one failure mode can be active at a time;
with the a specific set $F$. or borrowing from the terms of statistics, the failure mode being an event that is mutually exclusive
with a set $F$.
We can define a set of failure mode sets called $\mathcal{U}$ to represent this We can define a set of failure mode sets called $\mathcal{U}$ to represent this
property. property for a set of failure modes..
\begin{definition} \begin{definition}
We can define a set $\mathcal{U}$ which is a set of sets of failure modes, where We can define a set $\mathcal{U}$ which is a set of sets of failure modes, where
@ -475,36 +476,36 @@ to dealing with double simultaneous failure modes.}.
To generalise, we may need to consider $N$ simultaneous To generalise, we may need to consider $N$ simultaneous
failure modes when analysing a functional group. This involves finding failure modes when analysing a functional group. This involves finding
all combinations of failures modes of size $N$ and less. all combinations of failures modes of size $N$ and less.
The Powerset concept from Set theory is useful to model this. %The Powerset concept from Set theory is useful to model this.
The powerset, when applied to a set S is the set of all subsets of S, including the empty set The powerset, when applied to a set S is the set of all subsets of S, including the empty set
\footnote{The empty set ( $\emptyset$ ) is a special case for FMMD analysis, it simply means there \footnote{The empty set ( $\emptyset$ ) is a special case for FMMD analysis, it simply means there
is no fault active in the functional~group under analysis.} is no fault active in the functional~group under analysis.}
and S itself. and S itself.
In order to consider combinations for the set S where the number of elements in each sub-set of S is $N$ or less, a concept of the `cardinality constrained powerset' In order to consider combinations for the set S where the number of elements in each subset of S is $N$ or less, a concept of the `cardinality constrained powerset'
is proposed and described in the next section. is proposed and described in the next section.
%\pagebreak[1] %\pagebreak[1]
\subsection{Cardinality Constrained Powerset } \subsection{Cardinality Constrained Powerset }
\label{ccp} \label{ccp}
A Cardinality Constrained powerset is one where sub-sets of a cardinality greater than a threshold A Cardinality Constrained powerset is one where subsets of a cardinality greater than a threshold
are not included. This threshold is called the cardinality constraint. are not included. This threshold is called the cardinality constraint.
To indicate this, the cardinality constraint $cc$ is subscripted to the powerset symbol thus $\mathcal{P}_{cc}$. To indicate this, the cardinality constraint $cc$ is subscripted to the powerset symbol thus $\mathcal{P}_{cc}$.
Consider the set $S = \{a,b,c\}$. Consider the set $S = \{a,b,c\}$.
The powerset of S: 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}_{2} S $ means all non-empty subsets of S where the cardinality of the subsets is $\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. less than or equal to 2 or less.
$$ \mathcal{P}_{2} S = \{ \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$ $$ \mathcal{P}_{\le 2} S = \{ \{a,b\},\{b,c\},\{c,a\},\{a\},\{b\},\{c\} \} $$.
Note that $\mathcal{P}_{1} S $ (non-empty subsets where cardinality $\leq 1$) for this example is: Note that $\mathcal{P}_{1} S $ (non-empty subsets where cardinality $\leq 1$) for this example is:
$$ \mathcal{P}_{1} S = \{ \{a\},\{b\},\{c\} \} $$ $$ \mathcal{P}_{1} S = \{ \{a\},\{b\},\{c\} \} $$.
\paragraph{Calculating the number of elements in a cardinality constrained powerset} \paragraph{Calculating the number of elements in a cardinality constrained powerset}
@ -515,7 +516,7 @@ with $n$ elements (size $n$) is the binomial coefficient~\cite{probstat} shown i
\begin{equation} \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} \label{bico}
\end{equation} \end{equation} .
To find the number of elements in a cardinality constrained subset S with up to $cc$ elements To find the number of elements in a cardinality constrained subset S with up to $cc$ elements
in each combination sub-set, in each combination sub-set,
@ -531,7 +532,7 @@ from $1$ to $cc$ thus
\begin{equation} \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} \label{eqn:ccps}
\end{equation} \end{equation} .
@ -584,14 +585,14 @@ $$ \mathcal{P}_{2}(fm(FG)) = \{
\} \}
$$ $$
And % by inspection And whose cardinality is 11. % by inspection
$$ %$$
| %|
\{ %\{
\{R_o T_o\}, \{R_o T_s\}, \{R_o T_h\}, \{R_s T_o\}, \{R_s T_s\}, \{R_s T_h\}, \{R_o \}, \{R_s \}, \{T_o \}, \{T_s \}, \{T_h \} % \{R_o T_o\}, \{R_o T_s\}, \{R_o T_h\}, \{R_s T_o\}, \{R_s T_s\}, \{R_s T_h\}, \{R_o \}, \{R_s \}, \{T_o \}, \{T_s \}, \{T_h \}
\} %\}
| = 11 %| = 11
$$ %$$
\pagebreak[1] \pagebreak[1]
@ -600,29 +601,36 @@ cardinality calculation}
The cardinality constrained powerset in equation \ref{eqn:ccps}, can be modified for % corrected for The cardinality constrained powerset in equation \ref{eqn:ccps}, can be modified for % corrected for
unitary state failure modes. unitary state failure modes.
This is written as a general formula in equation \ref{eqn:correctedccps}. %This is written as a general formula in equation \ref{eqn:correctedccps}.
%\indent{ %\indent{
To define terms : %To define terms :
\begin{itemize} %\begin{itemize}
\item Let $C$ be a set of components (indexed by $j \in J$) %\item
Let $C$ be a set of components (indexed by $j \in J$)
that are members of the functional group $FG$ that are members of the functional group $FG$
i.e. $ \forall j \in J | C_j \in FG $ i.e. $ \forall j \in J | C_j \in FG $.
\item Let $|fm({C}_{j})|$
%\item
Let $|fm({C}_{j})|$
indicate the number of mutually exclusive fault modes of component $C_j$. indicate the number of mutually exclusive fault modes of component $C_j$.
\item Let $fm(FG)$ be the collection of all failure modes %\item
Let $fm(FG)$ be the collection of all failure modes
from all the components in the functional group. from all the components in the functional group.
\item Let $SU$ be the set of failure modes from the {\fg} where all $FG$ is such that %\item
Let $SU$ be the set of failure modes from the {\fg} where all $FG$ is such that
components $C_j$ are in components $C_j$ are in
`unitary state' i.e. $(SU = fm(FG)) \wedge (\forall j \in J | fm(C_j) \in \mathcal{U}) $ `unitary state' i.e. $(SU = fm(FG)) \wedge (\forall j \in J | fm(C_j) \in \mathcal{U}) $, then
\end{itemize} %\end{itemize}
%} %}
\begin{equation} \begin{equation}
|{\mathcal{P}_{cc}SU}| = {\sum^{cc}_{k=1} \frac{|{SU}|!}{k!(|{SU}| - k)!}} |{\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} \label{eqn:correctedccps}
\end{equation} \end{equation} .
Expanding the combination in equation \ref{eqn:correctedccps} Expanding the combination in equation \ref{eqn:correctedccps}
@ -631,7 +639,7 @@ Expanding the combination in equation \ref{eqn:correctedccps}
|{\mathcal{P}_{cc}SU}| = {\sum^{cc}_{k=1} \frac{|{SU}|!}{k!(|{SU}| - k)!}} |{\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} \label{eqn:correctedccps2}
\end{equation} \end{equation} .
\paragraph{Use of Equation \ref{eqn:correctedccps2} } \paragraph{Use of Equation \ref{eqn:correctedccps2} }
Equation \ref{eqn:correctedccps2} is useful for an automated tool that Equation \ref{eqn:correctedccps2} is useful for an automated tool that
@ -639,11 +647,12 @@ would verify that a single or double simultaneous failures model has complete fa
By knowing how many test cases should be covered, and checking the cardinality By knowing how many test cases should be covered, and checking the cardinality
associated with the test cases, complete coverage would be verified. associated with the test cases, complete coverage would be verified.
\paragraph{N Venn disallowed combinations} %\paragraph{Multiple simultaneous failure modes disallowed combinations}
The general case of equation \ref{eqn:correctedccps2}, involves not just dis-allowing pairs %The general case of equation \ref{eqn:correctedccps2}, involves not just dis-allowing pairs
of failure modes within components, but also ensuring that combinations across components %of failure modes within components, but also ensuring that combinations across components
do not involve any pairs of failure modes within the same component. %do not involve any pairs of failure modes within the same component.
A recursive algorithm and proof is described in appendix \ref{chap:vennccps}. %%%%- NOT SURE ABOUT THAT !!!!!
%%%- A recursive algorithm and proof is described in appendix \ref{chap:vennccps}.
%%\paragraph{Practicality} %%\paragraph{Practicality}
%%Functional Group may consist, typically of four or five components, which typically %%Functional Group may consist, typically of four or five components, which typically
@ -701,7 +710,9 @@ Thus the statistical sample space $\Omega$ for a component or derived~component
$$ \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}\} $$
The failure mode set $F$ for a given component or derived~component $C$ The failure mode set $F$ for a given component or derived~component $C$
is therefore is therefore
$$ F = \Omega(C) \backslash \{OK\} $$ $ fm(C) = \Omega(C) \backslash \{OK\} $
(or expressed as
$ \Omega(C) = fm(C) \cup \{OK\} $).
The $OK$ statistical case is the largest in probability, and is therefore The $OK$ statistical case is the largest in probability, and is therefore
of interest when analysing systems from a statistical perspective. of interest when analysing systems from a statistical perspective.