From 22386efd158746a03ce46d555ae82e6be9268ef4 Mon Sep 17 00:00:00 2001
From: Robin <robin@NovaProspect.(none)>
Date: Sat, 24 Apr 2010 10:36:33 +0100
Subject: [PATCH]  after meeting with Andrew Fish in Hove Friday

---
 component_failure_modes_definition/cfg.dia    | Bin 0 -> 2091 bytes
 component_failure_modes_definition/cfg.jpg    | Bin 0 -> 15652 bytes
 .../component.dia                             | Bin 0 -> 1197 bytes
 .../component.jpg                             | Bin 0 -> 9251 bytes
 .../component_failure_modes_definition.tex    | 424 +++++++++++-------
 .../componentpl.dia                           | Bin 0 -> 1336 bytes
 .../componentpl.jpg                           | Bin 0 -> 7031 bytes
 7 files changed, 260 insertions(+), 164 deletions(-)
 create mode 100644 component_failure_modes_definition/cfg.dia
 create mode 100644 component_failure_modes_definition/cfg.jpg
 create mode 100644 component_failure_modes_definition/component.dia
 create mode 100644 component_failure_modes_definition/component.jpg
 create mode 100644 component_failure_modes_definition/componentpl.dia
 create mode 100644 component_failure_modes_definition/componentpl.jpg

diff --git a/component_failure_modes_definition/cfg.dia b/component_failure_modes_definition/cfg.dia
new file mode 100644
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diff --git a/component_failure_modes_definition/component_failure_modes_definition.tex b/component_failure_modes_definition/component_failure_modes_definition.tex
index e5c95b7..17e74d3 100644
--- a/component_failure_modes_definition/component_failure_modes_definition.tex
+++ b/component_failure_modes_definition/component_failure_modes_definition.tex
@@ -3,190 +3,281 @@
 components, component fault modes and `unitary~state' component fault modes.
 The application of Bayes theorem  in current methodologies, and
 the suitability of the `null hypothesis' or `P' value statistical approach
-ar discussed.
+are discussed.
 Mathematical constraints and definitions are made using set theory.
 }
 
  
 \section{Introduction}
-When building a system from components, 
-we should be able to find all known failure modes for each component.
-For most common electrical and mechanical components, the failure modes
-for a given type of part can be obtained from standard literature\cite{mil1991}
-\cite{mech}. %The failure modes for a given component $K$ form a set $F$.
+
+%When building a system from components, 
+%we should be able to find all known failure modes for each component.
+%For most common electrical and mechanical components, the failure modes
+%for a given type of part can be obtained from standard literature\cite{mil1991}
+%\cite{mech}. %The failure modes for a given component $K$ form a set $F$.
 
 
-Using these failure modes we can build a `failure model' from the bottom-up.
+%%
+%% Paragraph component and its relationship to its failure modes
+%%
+
+\subsection{ What is a Component ?}
+
+
+Let us first define a component. This is anything we use to build a 
+product or system with. This could be something quite complicated 
+like an integrated microcontroller, or quite simple like the humble resistor.
+We can define a 
+component by its name, a manufacturers part number and perhaps
+a vendors reference number.
+What these components all have in common is that they can fail, and fail in
+a number of well defined ways. For common components
+there is established literature for the failure modes for the system designer consider (with accompanying statistical
+failure rates)\cite{mil1991}. For instance, a simple resistor is generally considered 
+to fail in two ways, it can go open circuit or it can short.  But we can also 
+associate it with a set of known failure modes. The UML diagram in figure 
+\ref{fig:component} shows a component as a simple data
+structure with its failure modes.
+
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=400pt,bb=0 0 437 141,keepaspectratio=true]{./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}
+\end{figure}
+
+% \begin{figure}[h+]
+%  \centering
+%  \includegraphics[width=400pt,bb=0 0 433 68,keepaspectratio=true]{component_failure_modes_definition/component.jpg}
+%  % component.jpg: 433x68 pixel, 72dpi, 15.28x2.40 cm, bb=0 0 433 68
+%  \caption{A Component and its failure modes}
+%  \label{fig:component}
+% \end{figure}
+
+A product naturally consists of many components and these are traditionally
+kept in a `parts list'. For safety critical product this is a usually formal document
+and is used by quality inspectors to ensure the correct parts are being fitted.
+For our UML diagram the parts list is simply a collection of components 
+as shown in figure \ref{fig:componentpl}.
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=400pt,bb=0 0 712 68,keepaspectratio=true]{./componentpl.jpg}
+ % componentpl.jpg: 712x68 pixel, 72dpi, 25.12x2.40 cm, bb=0 0 712 68
+ \caption{Parts List of Components}
+ \label{fig:componentpl}
+\end{figure}
+
+
+ 
+
+%%
+%% Paragraph using failure modes to build from bottom up
+%%
+
+\subsection{Fault Mode Analysis, top down or bottom up?}
+
 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}).
-Another top down technique is ato apply cost benifit analysis 
+determine the likelihood of component failures 
+causing specific system level errors (see Bayes theorem \ref{bayes}).
+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 this study is to produce complete  failure
 models of safety critical systems  from the bottom-up,
 starting, where possible with known component failure modes.
 
-
-\subsection{Systems, functional groups, sub-systems and failure modes}
-
-It is helpful here to define some terms, `system', `functional~group', `component', `base~component' and `sub-system'.
-
-A System, is really any coherent entity that would be sold as a safety critical product.
-A sub-system is a  part of some larger system.
-For instance a stereo amplifier separate is a sub-system. The 
-whole Sound System, consists perhaps of the following `sub-systems': 
-CD-player, tuner, amplifier~separate, loudspeakers and ipod~interface.
- 
-%Thinking like this is a top~down analysis approach
-%and is the way in which FTA\cite{nucfta} analyses a System
-%and breaks it down.
-
-A sub-system will be composed of component parts, which
-may themselves be sub-systems.
-
-Eventually by a recursive downwards process we would be able to identify
-sub-systems built from base component parts.
-Each `component part'
-will have a known fault/failure behaviour.
-That is to say, each base component has a set of known
-ways in which it can fail.
-
-If we look at the sound system again as an
-example; the CD~player could fail in serveral distinct ways, no matter
-what has happened to it or has gone wrong inside it.
-
-A top down approach has an intrinsic problem in that we cannot guess
-every possible failure mode at the SYSTEM level.
-Using the reasoning that working from the bottom up forces the consideration of all possible
-component failures (which could be missed in a top~down approach)
-we are presented with a problem. Which initial collections of base components should we choose ?
-
-For instance in the CD~player example; to start at the bottom; we are presented with
-a massive list of base~components, resistors, motors, user~switches, laser~diodes all sorts !
-Clearly, working from the bottom~up we need to pick small
-collections of components that work together in some way.
-These are termed `functional~groups'. For instance the  circuitry that powers the laser diode
-to illuminate the CD might contain a handful of components, and as such would make a good candidate
-to be one of the base level functional~groups.
-
-
-In choosing the lowest level (base component) sub-systems we would look 
-for the smallest `functional~groups' of components within a system. A functional~group is a set of components that interact
-to perform a specific function.
-
-When we have analysed the fault behaviour of a functional group, we can treat it as a `black box'.
-We can now call our functional~group a sub-system. The goal here is to  know how will behave under fault conditions !
-%Imagine buying one such `sub~system'  from a very honest vendor.
-%One of those sir, yes but be warned it may fail in these distinct ways, here
-%in the honest data sheet the set of failure modes is listed!
-This type of thinking is starting to become more commonplace in product literature, with the emergence
-of reliability safety standards such as IOC1508\cite{sccs},EN61508\cite{en61508}.
-FIT (Failure in Time - expected number of failures per billion hours of operation) values
-are published for some micro-controllers. A micro~controller
-is a complex sub-system in its self and could be considered a `black~box' with a given reliability.
-\footnote{Microchip sources give an FIT of 4 for their PIC18 series micro~controllers\cite{microchip}, The DOD
-1991 reliability manual\cite{mil1991} applies a FIT of 100 for this generic type of component}
-
-As electrical components have detailed datasheets a useful extension of this would
-be failure modes of the component, with environmental factors and MTTF statistics.
-
-Currently this sort of information is generally only  available for generic component types\cite{mil1991}.
-
- 
-%At higher levels of analysis, functional~groups are pre-analysed sub-systems that interact to
-%erform a given function.
-
-%\vspace{0.3cm}
-\begin{table}[h]
-\begin{tabular}{||l|l||} \hline \hline
-  {\em Definition } & {\em Description}    \\ \hline
-System & A product designed to  \\
-       & work as a coherent entity  \\  \hline   
-Sub-system & A part of a system, \\
-           & sub-systems may contain sub-systems \\    \hline 
-Failure mode & A way in which a System, \\
-             & Sub-system or component can fail \\     \hline  
-Functional Group & A collection of sub-systems and/or \\
-                 & components that interact to \\
-                 & perform a specific function  \\    \hline  
-Failure Mode     & The collection of all failure \\
-Group            & modes from all the members of a \\
-                 & functional group \\ \hline
-Derived      & A failure mode determined from the analysis \\
-Failure mode & of a `Failure Mode Group' \\ \hline
-Base Component & Any bought in component, which \\
-               & hopefully has a known set of failure modes  \\    \hline  
- \hline
-
-\end{tabular}
-\label{tab:def}
-\caption{Table of FMMD definitions}
-\end{table}
-%\vspace{0.3cm}
-
-\section{A UML Model of terms introduced}
-
+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.
+We can term this a `Functional~Group'. When we have a 
+`Functional~Group' we can look at the failure modes of all the components
+in it and decide how these will affect the Group.
+Or in other words we can determine the failure modes of the functional
+group. These failure modes are derived from the functional group, as so we can call
+them `derived failure modes'.
+We now have something very useful, because
+we can now treat this functional group as a component with a known set of failure modes.
+This newly derived component can be used as a higher level
+building block for the system we are analysing. 
+Derived components, can be used
+to form higher level functional groups.
+This process can continue until have build a hierarcy that converges to a failure model of the entire system.
+To differentiate the components derived from functional groups, we can
+add a new attribute to the class `Component', that of analysis
+level.
+We can represet this in a UML diagram see figure \ref{fig:cfg}
 
 \begin{figure}[h]
  \centering
- \includegraphics[width=350pt,bb=0 0 680 500,keepaspectratio=true]{component_failure_modes_definition/fmmd_uml.jpg}
- % fmmd_uml.jpg: 680x500 pixel, 72dpi, 23.99x17.64 cm, bb=0 0 680 500
- \caption{UML respresentation of Failure Mode Data types}
- \label{fig:fmmd_uml}
+ \includegraphics[width=400pt,bb=0 0 712 235,keepaspectratio=true]{./cfg.jpg}
+ % cfg.jpg: 712x205 pixel, 72dpi, 25.12x7.23 cm, bb=0 0 712 205
+ \caption{Components Derived from Functional Groups}
+ \label{fig:cfg}
 \end{figure}
 
-The diagram in figure \ref{fig:fmmd_uml}
-shows the relationships between the terms defined in table \ref{tab:def} as classes in a UML model.
-We can start with the functional group. This is a minimal collection
-of components that perform a simple given function.
-For our audio separates rig, this could be 
-the compoents that supply power to the laser diode.
-From the `Functional~Group' we can now collect
-all the `failure modes of the `components', and 
-produce a `Failure~Mode~Group'. This
-has a reference to the `Functional~Group', and is a collection
-of `failure modes.
-By analysing the effects of the failure modes in the `Failure~Mode~Group'
-we can determine the failure mode behaviour of the functional group.
-This failure mode behaviour is a collection of derived failure modes.
-We can now consider the Functional group as a component now, because
-we have a set of failure modes for it.
 
-\subsection{Sub-System Class Definition}
-A sub-system can be defined by the classes used to create it, and 
-its set of derived failure modes.
-In this way sub-systems naturally form trees, with the lower most leaf nodes being
-base components.
-Note that the UML model is recursive. We can build functional groups using sub-systems
-as components. This UML model naturally therefore, forms a hierarchy
-of failure mode analysis, which has a one top level entry, that being the SYSTEM.
-The TOP level entry will determine the failure modes 
-for the product/system under analysis.
-
-\subsection{Refining the UML model to use inheritance}
-We can refine this model a little by noticing that a system is merely the 
-top level sub-system. We can thus have System inherit sub-system.
-A derived failure mode, is simply a failure mode at a higher level of analysis
-it can therefore inherit `failure\_mode'.
-
-The modified UML diagram using inheritance is figure \ref{fig:fmmd_uml2}.
-\begin{figure}[h]
- \centering
- \includegraphics[width=350pt,bb=0 0 877 675,keepaspectratio=true]{./fmmd_uml2.jpg}
- % fmmd_uml2.jpg: 877x675 pixel, 72dpi, 30.94x23.81 cm, bb=0 0 877 675
- \caption{UML Representation of Failure Mode Data Types}
- \label{fig:fmmd_uml2}
-\end{figure}
+% 
+% \subsection{Systems, functional groups, sub-systems and failure modes}
+% 
+% It is helpful here to define some terms, `system', `functional~group', `component', `base~component' and `sub-system'.
+% 
+% A System, is really any coherent entity that would be sold as a safety critical product.
+% A sub-system is a  part of some larger system.
+% For instance a stereo amplifier separate is a sub-system. The 
+% whole Sound System, consists perhaps of the following `sub-systems': 
+% CD-player, tuner, amplifier~separate, loudspeakers and ipod~interface.
+%  
+% %Thinking like this is a top~down analysis approach
+% %and is the way in which FTA\cite{nucfta} analyses a System
+% %and breaks it down.
+% 
+% A sub-system will be composed of component parts, which
+% may themselves be sub-systems.
+% 
+% Eventually by a recursive downwards process we would be able to identify
+% sub-systems built from base component parts.
+% Each `component part'
+% will have a known fault/failure behaviour.
+% That is to say, each base component has a set of known
+% ways in which it can fail.
+% 
+% If we look at the sound system again as an
+% example; the CD~player could fail in serveral distinct ways, no matter
+% what has happened to it or has gone wrong inside it.
+% 
+% A top down approach has an intrinsic problem in that we cannot guess
+% every possible failure mode at the SYSTEM level.
+% Using the reasoning that working from the bottom up forces the consideration of all possible
+% component failures (which could be missed in a top~down approach)
+% we are presented with a problem. Which initial collections of base components should we choose ?
+% 
+% For instance in the CD~player example; to start at the bottom; we are presented with
+% a massive list of base~components, resistors, motors, user~switches, laser~diodes all sorts !
+% Clearly, working from the bottom~up we need to pick small
+% collections of components that work together in some way.
+% These are termed `functional~groups'. For instance the  circuitry that powers the laser diode
+% to illuminate the CD might contain a handful of components, and as such would make a good candidate
+% to be one of the base level functional~groups.
+% 
+% 
+% In choosing the lowest level (base component) sub-systems we would look 
+% for the smallest `functional~groups' of components within a system. A functional~group is a set of components that interact
+% to perform a specific function.
+% 
+% When we have analysed the fault behaviour of a functional group, we can treat it as a `black box'.
+% We can now call our functional~group a sub-system. The goal here is to  know how will behave under fault conditions !
+% %Imagine buying one such `sub~system'  from a very honest vendor.
+% %One of those sir, yes but be warned it may fail in these distinct ways, here
+% %in the honest data sheet the set of failure modes is listed!
+% This type of thinking is starting to become more commonplace in product literature, with the emergence
+% of reliability safety standards such as IOC1508\cite{sccs},EN61508\cite{en61508}.
+% FIT (Failure in Time - expected number of failures per billion hours of operation) values
+% are published for some micro-controllers. A micro~controller
+% is a complex sub-system in its self and could be considered a `black~box' with a given reliability.
+% \footnote{Microchip sources give an FIT of 4 for their PIC18 series micro~controllers\cite{microchip}, The DOD
+% 1991 reliability manual\cite{mil1991} applies a FIT of 100 for this generic type of component}
+% 
+% As electrical components have detailed datasheets a useful extension of this would
+% be failure modes of the component, with environmental factors and MTTF statistics.
+% 
+% Currently this sort of information is generally only  available for generic component types\cite{mil1991}.
+% 
+%  
+% %At higher levels of analysis, functional~groups are pre-analysed sub-systems that interact to
+% %erform a given function.
+% 
+% %\vspace{0.3cm}
+% \begin{table}[h]
+% \begin{tabular}{||l|l||} \hline \hline
+%   {\em Definition } & {\em Description}    \\ \hline
+% System & A product designed to  \\
+%        & work as a coherent entity  \\  \hline   
+% Sub-system & A part of a system, \\
+%            & sub-systems may contain sub-systems \\    \hline 
+% Failure mode & A way in which a System, \\
+%              & Sub-system or component can fail \\     \hline  
+% Functional Group & A collection of sub-systems and/or \\
+%                  & components that interact to \\
+%                  & perform a specific function  \\    \hline  
+% Failure Mode     & The collection of all failure \\
+% Group            & modes from all the members of a \\
+%                  & functional group \\ \hline
+% Derived      & A failure mode determined from the analysis \\
+% Failure mode & of a `Failure Mode Group' \\ \hline
+% Base Component & Any bought in component, which \\
+%                & hopefully has a known set of failure modes  \\    \hline  
+%  \hline
+% component_failure_modes_definition/
+% \end{tabular}
+% \label{tab:def}
+% \caption{Table of FMMD definitions}
+% \end{table}
+% %\vspace{0.3cm}
+% 
+% \section{A UML Model of terms introduced}
+% 
 % 
 % \begin{figure}[h]
 %  \centering
-%  \includegraphics[width=350pt,bb=0 0 680 500,keepaspectratio=true]{component_failure_modes_definition/fmmd_uml2.jpg}
+%  \includegraphics[width=350pt,bb=0 0 680 500,keepaspectratio=true]{component_failure_modes_definition/fmmd_uml.jpg}
 %  % fmmd_uml.jpg: 680x500 pixel, 72dpi, 23.99x17.64 cm, bb=0 0 680 500
 %  \caption{UML respresentation of Failure Mode Data types}
+%  \label{fig:fmmd_uml}
+% \end{figure}
+% 
+% The diagram in figure \ref{fig:fmmd_uml}
+% shows the relationships between the terms defined in table \ref{tab:def} as classes in a UML model.
+% We can start with the functional group. This is a minimal collection
+% of components that perform a simple given function.
+% For our audio separates rig, this could be 
+% the compoents that supply power to the laser diode.
+% From the `Functional~Group' we can now collect
+% all the `failure modes of the `components', and 
+% produce a `Failure~Mode~Group'. This
+% has a reference to the `Functional~Group', and is a collection
+% of `failure modes.
+% By analysing the effects of the failure modes in the `Failure~Mode~Group'
+% we can determine the failure mode behaviour of the functional group.
+% This failure mode behaviour is a collection of derived failure modes.
+% We can now consider the Functional group as a component now, because
+% we have a set of failure modes for it.
+% 
+% \subsection{Sub-System Class Definition}
+% A sub-system can be defined by the classes used to create it, and 
+% its set of derived failure modes.
+% In this way sub-systems naturally form trees, with the lower most leaf nodes being
+% base components.
+% Note that the UML model is recursive. We can build functional groups using sub-systems
+% as components. This UML model naturally therefore, forms a hierarchy
+% of failure mode analysis, which has a one top level entry, that being the SYSTEM.
+% The TOP level entry will determine the failure modes 
+% for the product/system under analysis.
+% 
+% \subsection{Refining the UML model to use inheritance}
+% We can refine this model a little by noticing that a system is merely the 
+% top level sub-system. We can thus have System inherit sub-system.
+% A derived failure mode, is simply a failure mode at a higher level of analysis
+% it can therefore inherit `failure\_mode'.
+% 
+% The modified UML diagram using inheritance is figure \ref{fig:fmmd_uml2}.
+% \begin{figure}[h]
+%  \centering
+%  \includegraphics[width=350pt,bb=0 0 877 675,keepaspectratio=true]{./fmmd_uml2.jpg}
+%  % fmmd_uml2.jpg: 877x675 pixel, 72dpi, 30.94x23.81 cm, bb=0 0 877 675
+%  \caption{UML Representation of Failure Mode Data Types}
 %  \label{fig:fmmd_uml2}
 % \end{figure}
+% % 
+% % \begin{figure}[h]
+% %  \centering
+% %  \includegraphics[width=350pt,bb=0 0 680 500,keepaspectratio=true]{component_failure_modes_definition/fmmd_uml2.jpg}
+% %  % fmmd_uml.jpg: 680x500 pixel, 72dpi, 23.99x17.64 cm, bb=0 0 680 500
+% %  \caption{UML respresentation of Failure Mode Data types}
+% %  \label{fig:fmmd_uml2}
+% % \end{figure}
 
 
 \subsection{Unitary State Component Failure Mode sets}
@@ -233,7 +324,10 @@ Thus if the failure modes of $F$ are unitary~state, we can say $F \in U$.
 A component with simple ``unitary~state'' failure modes is the electrical resistor.
 
 Electrical resistors can fail by going OPEN or SHORTED.
-However they cannot fail with both conditions active. The conditions
+
+For a given resistor R we can assign it the failure mode by applying
+the function $FM$ thus  $ FM(R) =  \{R_{SHORTED},R_{OPEN}\} $.
+Nothing can fail with both conditions open and short active at the same time ! The conditions
 OPEN and SHORT are mutually exclusive.
 Because of this the failure mode set $F=FM(R)$ is `unitary~state'.
 
@@ -244,31 +338,33 @@ $$ R_{SHORTED} \cap R_{OPEN} = \emptyset $$
 
 
 We can make this a general case by taking a set $C$ (where $c1, c2 \in C$) representing a collection
-of component failure modes,
+of component failure modes.
 We can now state that
 
 
 $$ c1 \cap c2 \neq \emptyset  | c1 \neq c2 \wedge c1,c2 \in C \wedge C \not\in U  $$
 
-That is to say that if it is impossible that any pair of failure modes can be active at the same time
-the failure mode set is not unitary~state and does not exist in the family of sets $U$
+That is to say that it is impossible that any pair of failure modes can be active at the same time
+for the failure mode set $C$ to exists in the family of sets $U$
 
  Note where that are more than two failure~modes, by banning pairs from happening at the same time
- we have banned larger combinations as well
+ we have banned larger combinations as well.
 
 
 
 \subsection{Component Failure Modes and Statistical Sample Space}
 %\paragraph{NOT WRITTEN YET PLEASE IGNORE}
 A sample space is defined as the set of all possible outcomes.
+Here the outcomes we are interested in are the failure modes 
+of the component.
 When dealing with failure modes, we are not interested in 
-the state where the compoent is working perfectly or `OK' (i.e. operating with no error).
+the state where the component is working perfectly or `OK' (i.e. operating with no error).
 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 behavuiour.
+that system will not exhibit faulty behaviour.
 Thus the statistical sample space $\Omega$ for a component/sub-system K is
 %$$ \Omega = {OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N} $$
-$$ \Omega(K) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N}\} $$
+$$ \Omega(K) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3}, ... ,failure\_mode_{N}\} $$
 The failure mode set for a given component or sub-system $F$
 is therefore
 $$ F = \Omega(K) \backslash OK $$
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