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% Verbal description for system PinnedBeam (PinnedBeam_desc.tex)
% Generated by MTT on Mon Apr 19 07:04:54 BST 1999.
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% %% Revision 1.1 1999/10/11 05:08:14 peterg
% %% Initial revision
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% %% Revision 1.1 1999/05/18 04:01:50 peterg
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The acausal bond graph of system \textbf{PinnedBeam} is displayed in
Figure \Ref{fig:PinnedBeam_abg.ps} and its label file is listed in
Section \Ref{sec:PinnedBeam_lbl}. The subsystems are listed in Section
\Ref{sec:PinnedBeam_sub}.
This example represents the dynamics of a uniform beam with two pinned
ends. The left-hand end is driven by a torque input and the
corresponding collocated angular velocity is measured. The beam is
approximated by 20 equal lumps using the Bernoulli-Euler. Because the
two end lumps have different causality to the rest of the beam lumps,
they are represented seperately. The system has 40 states (20 modes
of vibration), 1 input and 1 output.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|l|}
\hline
Name & Value\\
\hline
Beam Length, $L$ & 0.60 m\\
Beam Width $w$ & 0.05 m\\
Beam Thickness $t_b$ & 0.003\\
Young's Modulus $E$ & $68.94 \times 10^9$ \\
Density $\rho$ & 2712.8\\
\hline
Derived quantities & \\
\hline
$EI$ & 7.76\\
$\rho A$ & 0.40692 \\
\hline
\end{tabular}
\caption{Beam parameters}
\label{tab:beam}
\end{center}
\end{table}
The beam was made of aluminium with physical dimensions and constants
given in Table \ref{tab:beam}. The derived beam constants are given by the
formulae:
\begin{equation}
\label{eq:formulae}
\begin{align}
EI &= E \times w \frac{1}{12} t_b^3\\
\rho A &= \rho \times w t_b
\end{align}
\end{equation}
The system parameters are also given in Section
\Ref{sec:PinnedBeam_numpar.tex}.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{||l|l|l|l||}
\hline
\hline
Index & Theory & Model & Theory & Model \\
\hline
1 & 19.05 & 19.01 & 29.72 & 31.28\\
2 & 76.24 & 75.57 & 96.50 & 100.80\\
3 & 171.58 & 168.29 & 200.73 & 208.20\\
4 & 304.76 & 294.89 & 344.13 & 350.88\\
5 & 476.34 & 452.25 & 524.98 & 525.23\\
\hline
\hline
\end{tabular}
\caption{Mode frequencies (rad $s^{-1}$)}
\label{tab:freq}
\end{center}
\end{table}
Standard modal analysis give the theoretical system resonant
frequencies (based on the Bernoulli-Euler beam with the same values of
$EI$ and $\rho A$). The system anti-resonances correspond to those of
the \emph{inverse} system with reversed causality, that the driven
pinned end is replaced by a clamped end; again modal analysis of the
inverse system gives the system anti resonances. The model and
theoretical values are compared in Table \ref{tab:freq} for the first
5 modes. (This table was generated using the script MakeFreqTable.m)