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lN#1ySU^w3*0YA@4AXxweTHAa74J?+?;RPu1&fx)$@qaHg8x8;f diff --git a/fmmd_concept/System_safety_2011/state_exp.tex b/fmmd_concept/System_safety_2011/state_exp.tex new file mode 100644 index 0000000..a5b6200 --- /dev/null +++ b/fmmd_concept/System_safety_2011/state_exp.tex @@ -0,0 +1,137 @@ +\documentclass{beamer} + +\usepackage[utf8x]{inputenc} +\usepackage{default} + +\begin{document} + +Consider the FMEA type methodologies +where we look at all the failure modes in a system, and then +see how they can affect all other components within it, +to determine its system level symptom or failure mode. +We need to look at a large number of failure scenarios +to do this completely (all failure modes against all components). +This is represented in equation~\ref{eqn:fmea_state_exp}, +where $N$ is the total number of components in the system, and +$cfm$ is the number of failure modes per component. + +\begin{equation} + \label{eqn:fmea_state_exp} + N.(N-1).cfm % \\ + %(N^2 - N).cfm +\end{equation} + + +The FMMD methodology breaks the analysis down into small stages, +by making the analyst choose functional groups, and then when analysed the groups +are treated as components to be used for a higher stage. +This is designed to address the state explosion (where $O$ is order +of complexity) $O=N^2$ inherent in equation~\ref{eqn:fmea_state_exp}. + +\clearpage + +We can view the functional groups in FMMD as forming a hierarchy. +If for the sake of example we consider each functional group to +be three components, figure~\ref{fig:three_tree} shows +how the levels work and converge to a top or system level. + +\begin{figure} + \centering + \includegraphics[width=300pt]{./three_tree.png} + % three_tree.png: 780x226 pixel, 72dpi, 27.52x7.97 cm, bb=0 0 780 226 + \caption{Functional Group Tree example} + \label{fig:three_tree} +\end{figure} + +\clearpage +We can represent the number of failure scenarios to check in an FMMD hierarchy +with equation~\ref{eqn:anscen}. + +\begin{equation} + \label{eqn:anscen} + \sum_{n=0}^{L} {fgn}^{n}.fgn.cfm.(fgn-1) +\end{equation} + +Where $fgn$ is the number of components in each functional group, +and $cfm$ is the number of failure modes per component +and L is the number of levels, the number of +analysis scenarios to consider is show in equation~\ref{eqn:anscen}. + + +So for a very simple analysis with three components forming a functional group where +each component has three failure modes, we have only one level (zero'th). +So to check every failure modes against the other components in the functional group +requires 18 checks. + +\begin{equation} + \label{eqn:anscen2} + \sum_{n=0}^{0} {3}^{0}.3.3.(3-1) = 18 +\end{equation} +\clearpage + + + +In other words, we have three components in our functional group, +and nine failure modes to consider. +So taking each failure mode and looking at how that could affect the functional group, +we must compare each failure mode against the two other components (the `$fgn-1$' term). + +For the one `zero' level FMMD case we are doing the same thing as FMEA type analysis +(but on a very simple small sub-system). +We are looking at how each failure~mode can effect the system/top level. +We can use equation~\ref{eqn:fmea_state_exp} to represent +the number of checks to rigorously perform FMEA, where $N$ is the total +number of components in the system, and $cfm$ is the number of failures per component. + + + +Where $N=3$ and $cfm=3$ we can see that the number of checks for this simple functional +group is the same for equation~\ref{eqn:fmea_state_exp} +and equation~\ref{eqn:anscen}. +\clearpage + +\section{Example} + +To see the effects of reducing `state~explosion' we need to look at a larger system. +Let us take a system with 3 levels and apply these formulae. +Having three levels (in addition to the top zero'th level) +will require 81 base level components. + +$$ +%\begin{equation} + \label{eqn:fmea_state_exp} + 81.(81-1).3 = 19440 % \\ + %(N^2 - N).cfm +%\end{equation} +$$ + +$$ +%\begin{equation} + % \label{eqn:anscen} + \sum_{n=0}^{3} {3}^{n}.3.3.(2) = 720 +%\end{equation} +$$ + +Thud for FMMD we needed to examine 720 failure mode scenarios, and for traditional FMEA +type analysis methods 19440. +% In practical example followed through, no more than 9 components have ever been required for a functional +% group and the largest known number of failure modes has been 6. +% If we take these numbers and double them (18 components per functional group +% and 12 failure modes per component) and apply the formulas for a 4 level analysis +% (i.e. + +\clearpage +Note that for double simultaneous failures the equation~\ref{eqn:fmea_state_exp} becomes +equation~\ref{eqn:fmea_state_exp2} essentially making the order $N^3$. +The FMMD case is cubic within the functional groups only, not all the components in the system. + + +\begin{equation} + \label{eqn:fmea_state_exp2} + N.(N-1).(N-2).cfm % \\ + %(N^2 - N).cfm +\end{equation} + + + +\end{document} diff --git 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