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Mum proof read for english

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Robin Clark 2010-08-17 21:04:00 +01:00
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38
component_failure_modes_definition/component_failure_modes_definition.tex
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@ -62,8 +62,8 @@ Let us first define a component. This is anything which we use to build a
product or system with. This could be something quite complicated product or system with. This could be something quite complicated
like an integrated microcontroller, or quite simple like the humble resistor. like an integrated microcontroller, or quite simple like the humble resistor.
We can define a We can define a
component by its name, a manufacturers part number and perhaps component by its name, a manufacturers' part number and perhaps
a vendors reference number. a vendors' reference number.
What these components all have in common is that they can fail, and fail in What these components all have in common is that they can fail, and fail in
a number of well defined ways. For common components a number of well defined ways. For common components
there is established literature for the failure modes for the system designer to consider (often with accompanying statistical there is established literature for the failure modes for the system designer to consider (often with accompanying statistical
@ -120,7 +120,7 @@ When building from the bottom up, it is more meaningful to call them `derived~co
%% Paragraph using failure modes to build from bottom up %% Paragraph using failure modes to build from bottom up
%% %%
\section{Fault Mode Analysis, top down or bottom up?} \section{Fault Mode Analysis, \\ top down or bottom up?}
Traditional static fault analysis methods work from the top down. Traditional static fault analysis methods work from the top down.
They identify faults that can occur in a system, and then work down They identify faults that can occur in a system, and then work down
@ -167,16 +167,16 @@ all the failure modes of all the components in the group,
and analysing it is called `symptom abstraction' and and analysing it is called `symptom abstraction' and
is dealt with in detail in chapter \ref{symptom_abstraction}. is dealt with in detail in chapter \ref{symptom_abstraction}.
In terms of our UML model the symptom abstraction process takes a functional~group, In terms of our UML model, the symptom abstraction process takes a {\fg}
and creates a new derived component from it. and creates a new {\dc} from it.
%To do this it first creates %To do this it first creates
%a new set of failure modes, representing the fault behaviour %a new set of failure modes, representing the fault behaviour
%of the functional group. This is a human process and to do this the analyst %of the functional group. This is a human process and to do this the analyst
%must consider all the failure modes of the components in the functional %must consider all the failure modes of the components in the functional
%group. %group.
The newly created derived~component requires a set of failure modes of its own. The newly created {\dc} requires a set of failure modes of its own.
These failure modes are the failure mode behaviour of the functional group that it was derived from. These failure modes are the failure mode behaviour of the {\fg} that it was derived from.
Because these new failure modes were determined from a derived component we can call Because these new failure modes were derived from a {\fg} we can call
these `derived~failure~modes'. these `derived~failure~modes'.
%It then creates a new derived~component object, and associates it to this new set of derived~failure~modes. %It then creates a new derived~component object, and associates it to this new set of derived~failure~modes.
We thus have a `new' component, or system building block, but with a known and traceable We thus have a `new' component, or system building block, but with a known and traceable
@ -185,7 +185,7 @@ fault behaviour.
The UML representation shows a `functional group' having a one to one relationship with a derived~component. The UML representation shows a `functional group' having a one to one relationship with a derived~component.
We can represent this using a UML diagram in figure \ref{fig:cfg}. We can represent this using a UML diagram in figure \ref{fig:cfg}.
Using the symbol $\bowtie$ to indicate the analysis process that takes a The symbol $\bowtie$ is used to indicate the analysis process that takes a
functional group and converts it into a new component. functional group and converts it into a new component.
\[ \bowtie ( FG ) \mapsto DerivedComponent \] \[ \bowtie ( FG ) \mapsto DerivedComponent \]
@ -258,7 +258,7 @@ $$ FM ( C ) = F $$
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 as a flat set.
Consider the components in a functional group to be $C$ indexed by j thus $C_j$. Consider the components in a functional group to be $C$ indexed by j thus $C_j$.
Thflat set of failure modes we are after can be found by applying function $FM$ to all the components The flat set of failure modes 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:
$$ FunctionalGroupAllFailureModes = \bigcup_{j \in \{1...n\}} FM(C_j) $$ $$ FunctionalGroupAllFailureModes = \bigcup_{j \in \{1...n\}} FM(C_j) $$
@ -271,12 +271,12 @@ FM : FG \mapsto \mathcal{F}
\end{equation} \end{equation}
\section{Unitary State Component Failure Mode sets} \section{Unitary State Component \\ Failure Mode sets}
\paragraph{Design Descision/Constraint} \paragraph{Design Descision/Constraint}
An important factor in defining a set of failure modes is that they An important factor in defining a set of failure modes is that they
should be as clearly defined as possible. should be as clearly defined as possible.
It should not be possible for instance for It should not be possible, for instance for
a component to have two or more failure modes active at once. a component to have two or more failure modes active at once.
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.
@ -327,9 +327,9 @@ Electrical resistors can fail by going OPEN or SHORTED.
For a given resistor R we can apply the For a given resistor R we can apply the
the function $FM$ to find its set of failure modes thus $ FM(R) = \{R_{SHORTED},R_{OPEN}\} $. the function $FM$ to find its set of failure modes thus $ FM(R) = \{R_{SHORTED},R_{OPEN}\} $.
A resistor cannot fail with both conditions open and short active at the same time ! The conditions A resistor cannot fail with both conditions open and short active at the same time! The conditions
OPEN and SHORT are thus mutually exclusive. OPEN and SHORT are thus mutually exclusive.
Because of this the failure mode set $F=FM(R)$ is `unitary~state'. Because of this, the failure mode set $F=FM(R)$ is `unitary~state'.
Thus because both fault modes cannot be active at the same time, the intersection of $ R_{SHORTED} $ and $ R_{OPEN} $ cannot exist. Thus because both fault modes cannot be active at the same time, the intersection of $ R_{SHORTED} $ and $ R_{OPEN} $ cannot exist.
@ -365,9 +365,9 @@ Note where there are more than two failure~modes,
by banning any pairs from being active at the same time, by banning any pairs from being active at the same time,
we have banned larger combinations as well. we have banned larger combinations as well.
\section{Handling Simultaneous Component Faults} \section{Handling Simultaneous \\ Component Faults}
For some integrity levels of static analysis there is a need to consider not only single For some integrity levels of static analysis, there is a need to consider not only single
failure modes in isolation, but cases where more then one failure mode may occur failure modes in isolation, but cases where more then one failure mode may occur
simultaneously. simultaneously.
It is an implied requirement of EN298\cite{en298} for instance to consider double simultaneous faults. It is an implied requirement of EN298\cite{en298} for instance to consider double simultaneous faults.
@ -377,7 +377,7 @@ 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 sub-set 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.
@ -388,7 +388,7 @@ is proposed and described in the next section.
A Cardinality Constrained powerset is one where sub-sets of a cardinality greater than a threshold A Cardinality Constrained powerset is one where sub-sets 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:
@ -537,7 +537,7 @@ associated with the test cases, complete coverage would be verified.
\pagebreak[1] \pagebreak[1]
\section{Component Failure Modes and Statistical Sample Space} \section{Component Failure Modes \\ and Statistical Sample Space}
%\paragraph{NOT WRITTEN YET PLEASE IGNORE} %\paragraph{NOT WRITTEN YET PLEASE IGNORE}
A sample space is defined as the set of all possible outcomes. A sample space is defined as the set of all possible outcomes.
For a component in FMMD analysis, this set of all possible outcomes is its normal correct For a component in FMMD analysis, this set of all possible outcomes is its normal correct