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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 approximation
with damping. 
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}
Because the two end lumps have different causality to the rest of the
beam lumps, they are represented seperately.


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 given in Section
\Ref{sec:PinnedBeam_numpar.tex}. 
The system parameters are also given in Section
\Ref{sec:PinnedBeam_numpar.tex}.

 The system has 20 states (10
modes of vibration), 1 inputs and 1 outputs.

The first 5 vibration frequencies are given in Table \ref{tab:freq}
togtherr with the theoretical (based on the Bernoulli-Euler beam with
the same values of $EI$ and $\rho A$. 
\begin{table}[htbp]
  \begin{center}
    \begin{tabular}{||l|l|l||}
    \begin{tabular}{||l|l|l|l||}
      \hline
      \hline
      Mode & Frequency & Theoretical frequency\\
      \hline
      1 & 119.44 & 119.69\\
      2 & 474.83 & 479.02\\
      3 &1057.41 &1078.09\\
      4 &1852.85 &1914.86\\
      5 &2841.54 &2992.95\\
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)