7bcf4601db
Still to do. Tidy the Pt100 analysis Make a table of CC for all chapter 5 examples other stuiff
97 lines
3.4 KiB
TeX
97 lines
3.4 KiB
TeX
% %%%% FORMAL DEFINITIONS %%%% THESE MIGHT BE MOVED TO AN APPENDIX
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%
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%
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%
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% \chapter{Formal Definitions}
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% \label{sec:formalfmmd}
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% \section{An algebraic notation for identifying FMMD enitities}
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% Consider all `components' to exist as
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% members of a set $\mathcal{C}$.
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% %
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% Each component $c$ has an associated set of failure modes.
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% We can define a function $fm$ that returns a
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% set of failure modes $F$, for the component $c$.
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%
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% Let the set of all possible components be $\mathcal{C}$
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% and let the set of all possible failure modes be $\mathcal{F}$.
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%
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% We now define the function $fm$
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% as
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% \begin{equation}
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% \label{eqn:fm}
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% fm : \mathcal{C} \rightarrow \mathcal{P}\mathcal{F}.
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% \end{equation}
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% This is defined by, where $c$ is a component and $F$ is a set of failure modes,
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% $ fm ( c ) = F. $
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%
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% We can use the variable name $\FG$ to represent a {\fg}. A {\fg} is a collection
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% of components.
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% %We thus define $FG$ as a set of chosen components defining
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% %a {\fg}; all functional groups
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% We can state that
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% {\FG} is a member of the power set of all components, $ \FG \in \mathcal{P} \mathcal{C}. $
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%
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% We can overload the $fm$ function for a functional group {\FG}
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% where it will return all the failure modes of the components in {\FG}
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%
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%
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% given by
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%
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% $$ fm ({\FG}) = F. $$
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%
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% Generally, where $\mathcal{{\FG}}$ is the set of all functional groups,
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%
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% \begin{equation}
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% fm : \mathcal{{\FG}} \rightarrow \mathcal{P}\mathcal{F}.
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% \end{equation}
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% \section{Relationships between functional~groups and failure modes}
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%
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% Let the set of all possible components be $\mathcal{C}$
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% and let the set of all possible failure modes be $\mathcal{F}$, and $\mathcal{PF}$
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% is the power-set of $\mathcal{F}$.
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%
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% In order to analyse failure mode effects we need to be able to determine the
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% failure modes of a component. We define a function $fm$ to perform this (see equation~\ref{eqn:fmset}).
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% \label{fmdef}
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%
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% \begin{equation}
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% fm : \mathcal{C} \rightarrow \mathcal{P}\mathcal{F}
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% \label{eqn:fmset}
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% \end{equation}
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%
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% %%
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% % Above def gives below anyway
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% %
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% %The is defined by equation \ref{eqn:fminstance}, where C is a component and F is a set of failure modes.
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% %
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% %\begin{equation}
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% % fm ( C ) = F
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% % \label{eqn:fminstance}
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% %\end{equation}
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%
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% \paragraph{Finding all failure modes within the functional group.}
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%
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% For FMMD failure mode analysis %we need to consider the failure modes
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% from all the components in a functional~group.
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% In a functional group we have a collection of Components
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% which have associated failure mode sets.
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% we need to collect failure mode sets from the components and place them all
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% %modes
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% into a single set; this can be termed flattening the set of sets.
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% %%Consider the components in a functional group to be $C_1...C_N$.
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% The flat set of failure modes $FSF$ we are after can be found by applying function $fm$ to all the components
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% in the functional~group and taking the union of them thus:
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%
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% %%$$ FSF = \bigcup_{j=1}^{N} fm(C_j) $$
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% $$ FSF = \bigcup_{c \in FG} fm(c) \; .$$
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%
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% We can actually overload the notation for the function $fm$ % FM
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% and define it for the set components within a functional group $\mathcal{FG}$ (i.e. where $\mathcal{FG} \subset \mathcal{C} $)
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% in equation \ref{eqn:fmoverload}.
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%
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% \begin{equation}
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% fm : \mathcal{FG} \rightarrow \mathcal{F}
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% \label{eqn:fmoverload}
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% \end{equation}
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