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