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Robin Clark 2010-11-21 12:20:21 +00:00
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component_failure_modes_definition/component_failure_modes_definition.tex
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@ -258,7 +258,8 @@ derived components higher up in the structure.
To keep track of the level in the hierarchy (i.e. how many stages of component To keep track of the level in the hierarchy (i.e. how many stages of component
derivation `$\bowtie$' have lead to the current derived component) derivation `$\bowtie$' have lead to the current derived component)
we can add an attribute to the component data type. we can add an attribute to the component data type.
This can be a natural number called the level variable $\alpha \in \mathbb{N}_0$. This can be a natural number called the level variable $\alpha \in \mathbb{N}$.
% J. Howse says zero is a given in comp sci. This can be a natural number called the level variable $\alpha \in \mathbb{N}_0$.
The $\alpha$ level variable in each component, The $\alpha$ level variable in each component,
indicates the position in the hierarchy. Base or parts~list components indicates the position in the hierarchy. Base or parts~list components
have a `level' of $\alpha=0$. have a `level' of $\alpha=0$.
@ -290,7 +291,8 @@ would have an $\alpha$ value of 1.
\subsection{Relationships between functional~groups and failure modes} \subsection{Relationships between functional~groups and failure modes}
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 $\mathcal{F}$ and $\mathcal{PF}$ is the powerset of
all $\mathcal{F}$..
We can define a function $fm$ as equation \ref{eqn:fmset}. We can define a function $fm$ as equation \ref{eqn:fmset}.
@ -299,22 +301,30 @@ fm : \mathcal{C} \rightarrow \mathcal{P}\mathcal{F}
\label{eqn:fmset} \label{eqn:fmset}
\end{equation} \end{equation}
The is defined by equation \ref{eqn:fminstance}, where C is a component and F is a set of failure modes. %%
% Above def gives below anyway
\begin{equation} %
fm ( C ) = F %The is defined by equation \ref{eqn:fminstance}, where C is a component and F is a set of failure modes.
\label{eqn:fminstance} %
\end{equation} %\begin{equation}
% fm ( C ) = F
% \label{eqn:fminstance}
%\end{equation}
\paragraph{Finding all failure modes within the functional group} \paragraph{Finding all failure modes within the functional group}
For FMMD failure mode analysis we need to consider the failure modes For FMMD failure mode analysis we need to consider the failure modes
from all the components in a functional~group as a flat set. from all the components in a functional~group.
Consider the components in a functional group to be $C_1...C_N$. In a functional group we have a collection of Components
that hold failure mode sets.
We need to collect these failure mode sets and place all the failure
modes into a single set; this can be termed flattening the set of sets.
%%Consider the components in a functional group to be $C_1...C_N$.
The flat set of failure modes $FSF$ we are after can be found by applying function $fm$ to all the components The flat set of failure modes $FSF$ we are after can be found by applying function $fm$ to all the components
in the functional~group and taking the union of them thus: in the functional~group and taking the union of them thus:
$$ FSF = \bigcup_{j=1}^{N} FM(C_j) $$ %%$$ FSF = \bigcup_{j=1}^{N} fm(C_j) $$
$$ FSF = \bigcup_{c \in FG} fm(c) $$
We can actually overload the notation for the function $fm$ % FM We can actually overload the notation for the function $fm$ % FM
and define it for the set components within a functional group $\mathcal{FG}$ (i.e. where $\mathcal{FG} \subset \mathcal{C} $) and define it for the set components within a functional group $\mathcal{FG}$ (i.e. where $\mathcal{FG} \subset \mathcal{C} $)
@ -337,7 +347,7 @@ Were this to be the case, we would have to consider additional combinations of
failure modes within the component. failure modes within the component.
Having a set of failure modes where $N$ modes could be active simultaneously Having a set of failure modes where $N$ modes could be active simultaneously
would mean having to consider an additional $2^N-1$ failure mode scenarios. would mean having to consider an additional $2^N-1$ failure mode scenarios.
Should a component be analysed and simultaneous failure mode cases exit, Should a component be analysed and simultaneous failure mode cases exist,
the combinations could be represented by new failure modes, or the combinations could be represented by new failure modes, or
the component should be considered from a fresh perspective, the component should be considered from a fresh perspective,
perhaps considering it as several smaller components perhaps considering it as several smaller components
@ -350,11 +360,18 @@ probability theory~\cite{probstat}.
\begin{definition} \begin{definition}
A set of failure modes where only one failure mode A set of failure modes where only one failure mode
can be active at one time is termed a `unitary~state' failure mode set. can be active at one time is termed a {\textbf{unitary~state}} failure mode set.
\end{definition} \end{definition}
Let the set of all possible components to 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
here. What is required is to define a property for
a set of failure modes where only one failure mode can be active at a time,
or borrowing from the terms of statistics, the failure mode is an event, and it is mutually exclusive
with the a specific set $F$.
We can define a set of failure mode sets called $\mathcal{U}$ to represent this
property.
\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
@ -394,7 +411,7 @@ we state this formally
\begin{equation} \begin{equation}
\forall f_1,f_2 \in F \dot ( f_1 \neq f_2 \wedge \mathcal{ACTIVE}({f_1}) \wedge \mathcal{ACTIVE}({f_2}) ) \implies F \not\in \mathcal{U} \exists f_1,f_2 \in F \dot ( f_1 \neq f_2 \wedge \mathcal{ACTIVE}({f_1}) \wedge \mathcal{ACTIVE}({f_2}) ) \implies F \not\in \mathcal{U}
\end{equation} \end{equation}
@ -411,18 +428,31 @@ we have banned larger combinations as well.
\subsection{Design Rule: Unitary State} \subsection{Design Rule: Unitary State}
All components must have unitary state failure modes to be used with the FMMD methodology.
Where a complex component is used, for instance a microcontroller
All components must have unitary state failure modes to be used with the FMMD methodology,
for base~components, this is usually the case. Most simple components fail in one
clearly defined way and generally stay in that state.
However, where a complex component is used, for instance a microcontroller
with several modules that could all fail simultaneously, a process with several modules that could all fail simultaneously, a process
of reduction into smaller theoretical components will have to be made of reduction into smaller theoretical components will have to be made.
\footnote{A modern microcontroller will typically have several modules, which are configured to operate on This is sometimes termed `heuristic~de-composition'.
A modern microcontroller will typically have several modules, which are configured to operate on
pre-assigned pins on the device. Typically voltage inputs (\adcten / \adctw), digital input and outputs, pre-assigned pins on the device. Typically voltage inputs (\adcten / \adctw), digital input and outputs,
PWM (pulse width modulation), UARTs and other modules will be found on simple cheap microcontrollers~\cite{pic18f2523}}. PWM (pulse width modulation), UARTs and other modules will be found on simple cheap microcontrollers~\cite{pic18f2523}.
For instance the voltage reading functions which consist For instance the voltage reading functions which consist
of an ADC multiplexer and ADC can be considered to be components of an ADC multiplexer and ADC can be considered to be components
inside the microcontroller package. inside the microcontroller package.
The microcontroller thuis becomes a collection of smaller components The microcontroller thus becomes a collection of smaller components
the can be analysed separately. that can be analysed separately~\footnote{It is common for the signal paths
in a safety critical product to be traced, and when entering a complex
component like a micro controller, the process of heuristic de-compostion
applied to it}.
\paragraph{Reason for Constraint} Were this constraint to not be applied \paragraph{Reason for Constraint} Were this constraint to not be applied
each component could not have $N$ failure modes to consider but potentially each component could not have $N$ failure modes to consider but potentially
$2^N$. This would make the job of analysing the failure modes $2^N$. This would make the job of analysing the failure modes