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updates after andrew fish pdf with notes on in gmail

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Robin Clark 2010-05-21 13:51:57 +01:00
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component_failure_modes_definition/Makefile
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@ -4,7 +4,7 @@
#
paper: paper.tex component_failure_modes_definition.tex
paper: paper.tex component_failure_modes_definition_paper.tex
#latex paper.tex
#dvipdf paper pdflatex cannot use eps ffs
pdflatex paper.tex

104
component_failure_modes_definition/component_failure_modes_definition.tex
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@ -11,7 +11,7 @@ Mathematical constraints and definitions are made using set theory.
\section{Introduction}
This chapter describes the data types and concepts for the Failure Mode Modular De-composition (FMMD) method.
When analysing a safety critical system using the
When analysing a safety critical system using
this technique, we need clearly defined failure modes for
all the components that are used to model the system.
These failure modes have a constraint such that
@ -31,12 +31,12 @@ build hierarchical bottom-up models of failure mode behaviour.
%% Paragraph component and its relationship to its failure modes
%%
\section{ What is a Component ?}
\section{ Defining the term `Component' }
\begin{figure}[h]
\centering
\includegraphics[width=400pt,bb=0 0 437 141,keepaspectratio=true]{component_failure_modes_definition/component.jpg}
\includegraphics[width=300pt,bb=0 0 437 141,keepaspectratio=true]{component_failure_modes_definition/component.jpg}
% component.jpg: 437x141 pixel, 72dpi, 15.42x4.97 cm, bb=0 0 437 141
\caption{A Component and its Failure Modes}
\label{fig:component}
@ -57,9 +57,16 @@ Thus we can associate a set of faults to this component $ResistorFaultModes=\{OP
The UML diagram in figure
\ref{fig:component} shows a component as a data
structure with its associated failure modes.
From this diagram we see that each component must have at least one failure mode.
Also to clearly show that the failure modes are unique events associated with one component,
each failure mode is referenced back to only one component.
each failure mode is referenced back to only one component.
This modelling constraint is due to the fact that even generic components with the same
failure mode types, will have different statistical MTTF properties within the same circuitry.
%% sharing failure modes arrrgghh so irrelevant
%% wrong as well perhaps, as each component will have environmental constraints
%% that determine its statistical behaviour. A 1 Meg ohm resistor
%% is less stressed than a 100 ohm in the same circuit etc
% Perhaps talk here about the failure modes being shared, but by being referenced
% by the component ?
@ -78,6 +85,10 @@ as shown in figure \ref{fig:componentpl}.
\label{fig:componentpl}
\end{figure}
Parts in the parts list (bought in parts) will be termed `base~comonents'.
Parts derived from base~components may not require parts numbers, and will
not require a vendor reference, but must be named.
@ -91,7 +102,8 @@ Traditional static fault analysis methods work from the top down.
They identify faults that can occur in a system, and then work down
to see how they could be caused. Some apply statistical tequniques to
determine the likelihood of component failures
causing specific system level errors (see Bayes theorem \ref{bayes}).
causing specific system level errors. For example, Bayes theorem \ref{bayes}, the relation between a conditional probability and its inverse,
can be applied to specific failure modes in components and the probability of them causing given system level errors.
Another top down technique is to apply cost benifit analysis
to determine which faults are the highest priority to fix\cite{FMEA}.
The aim of FMMD analysis is to produce complete failure
@ -106,12 +118,18 @@ In order to analyse from the bottom-up, we need to take
small groups of components from the parts~list that naturally
work together to perform a simple function.
The components to include in a functional group are chosen by a human, the analyst.
We can term this a `Functional~Group' and represent it as a class. When we have a
%We can represent the `Functional~Group' as a class.
When we have a
`Functional~Group' we can look at the failure modes of all the components
in it and determine a failure mode model for that group.
Or in other words we can determine the failure modes of the functional
in it.
% and determine a failure mode model for that group.
The `Functional~Group' is seen by the analyst as a collection of component failures modes.
Each of these failure modes, and optionally combinations of them, are
analsyed for their effect on the failure mode behaviour of the `Functional~Group'.
From this we can determine a new set of failure modes, the failure modes of the
Or in other words we can determine the failure modes of the `Functional~Group'.
group. We can now consider the functional group as a sort of super component
with a know set of failure modes.
with a known set of failure modes.
\subsection{From functional group to newly derived component}
@ -137,7 +155,7 @@ We thus have a `new' component, or system building block, but with a known and t
fault behaviour.
The UML representation shows a `functional group' having a one to one relationship with a derived~component.
We can represet this using an UML diagram in figure \ref{fig:cfg}
We can represent this using an UML diagram in figure \ref{fig:cfg}.
\begin{figure}[h]
\centering
@ -147,14 +165,25 @@ We can represet this using an UML diagram in figure \ref{fig:cfg}
\label{fig:cfg}
\end{figure}
\subsection{Keeping track of the dereived components position in the hierarchy}
Using the symbol $\bowtie$ to indicate an analysis process that takes a
functional group and converts it into a new component.
The UML meta model in figure \ref{fig:cfg}, will build a hierarchy of
$$ \bowtie ( FG ) \mapsto DerivedComponent $$
\subsection{Keeping track of the derived \\ components position in the hierarchy}
The UML meta model in figure \ref{fig:cfg}, shows the relationships
between the classes and sub-classes.
In use we will build a hierarchy of
objects, with derived~components forming functional~groups, and creating
derived components higher up in the structure.
The level variable in each Component,
indicates the position in the hierarchy. Base or parts~list components
have a `level' of 0. Derived~components take a level based on the highest level
have a `level' of 0.
% I do not know how to make this simpler
Derived~components take a level based on the highest level
component used to build the functional group it was derived from plus 1.
So a derived component built from base level or parts list components
would have a level of 1.
@ -162,21 +191,21 @@ would have a level of 1.
\section{Set Theory Description}
$$ System \stackrel{has}{\longrightarrow} PartsList $$
$$ PartsList \stackrel{has}{\longrightarrow} Components $$
$$ Component \stackrel{has}{\longrightarrow} FailureModes $$
$$ FunctionalGroup \stackrel{has}{\longrightarrow} Components $$
Using the symbol $\bowtie$ to indicate an analysis process that takes a
functional group and converts it into a new component.
$$ \bowtie ( FG ) \mapsto DerivedComponent $$
% \section{Set Theory Description}
%
% $$ System \stackrel{has}{\longrightarrow} PartsList $$
%
% $$ PartsList \stackrel{has}{\longrightarrow} Components $$
%
% $$ Component \stackrel{has}{\longrightarrow} FailureModes $$
%
% $$ FunctionalGroup \stackrel{has}{\longrightarrow} Components $$
%
% Using the symbol $\bowtie$ to indicate an analysis process that takes a
% functional group and converts it into a new component.
%
% $$ \bowtie ( FG ) \mapsto DerivedComponent $$
%
\section{Unitary State Component Failure Mode sets}
@ -216,9 +245,9 @@ the component failure modes in each of its members are unitary~state.
Thus if the failure modes of $F$ are unitary~state, we can say $F \in U$.
\section{Component failure modes : Unitary State example}
\section{Component failure modes:\\ Unitary State example}
A component with an obvious set of ``unitary~state'' failure modes is the electrical resistor.
An example of a component with an obvious set of ``unitary~state'' failure modes is the electrical resistor.
Electrical resistors can fail by going OPEN or SHORTED.
@ -236,9 +265,9 @@ therefore
$$ FM(R) \in U $$
We can make this a general case by taking a set $F$ (where $f1, f2 \in F$) representing a collection
We can make this a general case by taking a set $F$ (where $f_1, f_2 \in F$) representing a collection
of component failure modes.
We can define a boolean function {\ensuremath{\mathcal{ACTIVE()}}} that returns
We can define a boolean function {\ensuremath{\mathcal{ACTIVE}}} that returns
whether a fault mode is active (true) or dormant (false).
We can say that if any pair of fault modes is active at the same time, then the failure mode set is not
@ -272,16 +301,15 @@ 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 or derived~component $K$ is
Thus the statistical sample space $\Omega$ for a component or derived~component $C$ is
%$$ \Omega = {OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N} $$
$$ \Omega(K) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, \ldots ,failure\_mode_{N}\} $$
The failure mode set $F$ for a given component or derived~component $K$
$$ \Omega(C) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, \ldots ,failure\_mode_{N}\} $$
The failure mode set $F$ for a given component or derived~component $C$
is therefore
$$ F = \Omega(K) \backslash OK $$
$$ F = \Omega(C) \backslash OK $$
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.
of interest when analysing systems from a statistical perspective.
\vspace{40pt}