\documentclass[11pt]{article}

\usepackage[english]{babel}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{psfrag}
\usepackage{pstricks}
\usepackage{amsmath}
\input{pst-plot}
\input{pst-node}
\input{pst-coil}

\input{pogbase_new_par}
\input{BG_base.tex}

\input{mathbase}
\input{VPPC09_POG_Automotive_Modelling_Base}

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\textwidth 7in
 \textheight 10in
 \topmargin -1in
 \oddsidemargin -0.3in

\begin{document}





\section{POG modeling of a planetary gear: only elastic part} \label{Planetary_Gears}


 \begin{figure}[h]
  \centering
  \schemaPOG{(-1.5,-11)(66,20)}{2.9mm}{
 \thicklines
 \bloin{\tau_{s}}{\omega_{s}}
 \scrivigiu{8}{3}{\footnotesize Sun}
 \bloivsuaor{b_{s}}{}{}
 \blokxyaor{2}{2}{r_{s}}{r_{s}}{}{}
 \scrivigiu{8}{3}{\footnotesize Elastic element}
 \put(1,8.3){\makebox(0,0){$F_{sc}$}}
 \blosuattdue{}{K_{sc}}{d_{sc}}{}{}{\dst}
%
 \bloivparingiu{
 \blokxyor{2}{2}{-r_{p}}{-r_{p}}{}{}
 \put(2,12.5){\vector(0,-1){2}}
 \put(2,12){$\tau_{p}$}
 \bloivgiumxm{\frac{1}{J_{p}}}{\omega_{p}}{\dst
 }{\frac{1}{s}}{}
 \bloivsuaor{b_{p}}{}{}
 \blokxyaor{2}{2}{r_{p}}{r_{p}}{}{}
}
{\blokxyor{2}{2}{r_{s} }{ r_{s}}{}{}
\addsu{\omega_{c}}{\snt}{\tau_{c}}{b_{c}}
\blokxyaor{2}{2}{r_{r}}{r_{r}}{}{}
}
{}{}{}{}{}
\put(15,0){\bloivparoutgiu{}{}{
 \scrivigiu{8}{3}{\footnotesize Elastic element}
 \put(1,8.3){\makebox(0,0){$F_{cr}$}}
 \blosuattdue{}{K_{cr}}{d_{cr}}{}{}{\dst}
 \scrivigiu{10}{3}{\footnotesize Ring}
 \blokxyor{2}{2}{r_{r}}{r_{r}}{}{}
 \bloivsuor{b_r}{}{\snt}
 \bloout{\tau_{r}}{\omega_{r}}}
{}{}}
 \scrivigiu{15}{11}{\footnotesize Planets: two-dimensional dynamics}
  \scrivigiu{10}{3}{}
 }
  \caption{The POG model of the considered elastic planetary gear: the sun, the planets and the ring interact through elastic elements.}
  \label{POG_epicicloidale}
\end{figure}
 %
 %

The POG state space dynamic model of the considered planetary gear:
only the inputs $\omega_{s}$,  $\omega_{c}$ and $\omega_{r}$ have been
considered.
\[
 %\footnotesize
 \underbrace{\mat{@{}c@{}c@{}c@{}}{
  \frac{1}{K_{sc}} &  0 &  0 \\
  0 &  J_{p} &  0 \\
  0 &  0 &  \frac{1}{K_{cr}}}}_{\ds \ooL}\!
 \underbrace{\mat{@{}c@{}}{\dot F_{sc} \\ \dot \omega_{p}\\ \dot F_{cr} }}_{\ds \dot\oox}\!
 =
 - \underbrace{\mat{@{}c@{\;}c@{\;}c@{}}{
  0 &  -r_{p} &  0 \\
  r_{p} &  r_{p}^{2} d_{sc} +b_{p}+r_{p}^{2} d_{cr} &  r_{p} \\
  0 &  -r_{p} &  0}}_{\ds\ooA}\!
 \underbrace{\mat{@{}c@{}}{F_{sc} \\ \omega_{p}\\  F_{cr} }}_{\ds\oox}\!
 +
 \underbrace{\mat{@{}ccc@{}}{
  r_{s} &  -r_{s} &  0 \\
  -r_{s}d_{sc}r_{p} &  r_{s}d_{sc}r_{p}-r_{r}d_{cr}r_{p}  &  r_{r}d_{cr}r_{p} \\
  0 &  r_{r} &  -r_{r}}}_{\ds\ooB}\!
  \underbrace{\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}\\ \omega_{r}}}_{\ds\u}
\]
\[
 \footnotesize
 \underbrace{\mat{@{}c@{}}{\tau_{s} \\ \tau_{c}\\ \tau_{r} }}_{\ds \ooy}\!
 =
 \underbrace{\mat{@{}c@{\;}c@{\;}c@{}}{
  r_{s} &  r_{s}d_{sc}r_{p} &  0 \\
  -r_{s} &  0 &  r_{r} \\
  0 &  -r_{r}d_{cr}r_{p} &  -r_{r}}}_{\ds\ooC}\!
 \underbrace{\mat{@{}c@{}}{F_{sc} \\ \omega_{p}\\  F_{cr} }}_{\ds\oox}\!
 +
 \underbrace{\mat{@{}ccc@{}}{
  b_{s}+r_{s}^2 d_{sc} &  -r_{s}^2 d_{sc} &  0 \\
  -r_{s}^2 d_{sc} &  b_{c}+r_{s}^2 d_{sc}+r_{r}^2 d_{cr}  &  -r_{r}^2 d_{cr} \\
  0 &  -r_{r}^2 d_{cr} &  b_{r}+-r_{r}^2 d_{cr}}}_{\ds\ooD}\!
  \underbrace{\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}\\ \omega_{r}}}_{\ds\u}
\]

 %
\end{document}
