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gyrostat_presentation.tex

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\documentclass{beamer}

\usepackage{graphics}
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\usetheme[numbers,totalnumber,compress,sidebarshades]{PaloAlto}
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\usecolortheme[named=kugreen]{structure}
\useinnertheme{circles}
\usefonttheme[onlymath]{serif}
\setbeamercovered{transparent}
\setbeamertemplate{blocks}[rounded][shadow=true]

\newcommand{\bs}[1]{ \boldsymbol{ #1 } }

\title{Gyrostats -- an introduction}
\author{Dale Lukas Peterson}
\institute{University of California Davis}
\date{\today}

\begin{document}

\frame{\titlepage}

\section[Outline]{}
\frame{\tableofcontents}

\section{Introduction}
\subsection{Definition}
\frame
{
  \begin{block}{Definition of a Gyrostat}
      A system of rigid bodies and/or particles whose
      \begin{itemize}
          \item<1-> System center of mass location is independent of
              configuration
          \item<2-> System central inertia tensor is independent of
              configuration
      \end{itemize}
  \end{block}

%  When rotation axis is parallel to a symmetry axis and passes through mass
%  center, you have a gyrostat.
}

\subsection{Examples}
\frame
{
\frametitle{Types of gyrostats}
  \begin{columns}
      \column{.5\textwidth}
      \begin{block}{Cylindrical gyrostat}
          \includegraphics[width=\textwidth]{cylindrical_gyrostat.png}
      \end{block}
      \column{.5\textwidth}
      \begin{block}{Spherical gyrostat}
          \includegraphics[width=\textwidth]{spherical_gyrostat.png}
      \end{block}
  \end{columns}
}

\frame
{
  \frametitle{Everyday examples}
  \begin{itemize}
      \item<1-> Toy Gyroscopes
      \item<2-> Unicycles (symmetric wheel)
      \item<3-> Bicycles (two unicycles connected by a hinge)
      \item<4-> Helicoptors (main rotor and tail rotor)
      \item<5-> Flywheels
      \item<6-> Spherical dampers (satellite)
  \end{itemize}
}

\section{Theory}
\subsection{Nomenclature}
\frame
{
\frametitle{Nomenclature}
\begin{itemize}
    \item<1-> ``Carrier'' -- $A$
        \begin{itemize}
            \item mass $m_A$
            \item Central inertia tensor $\bs{I}^{A/A_o}$
            \item Angular velocity ${}^N\bs{\omega}^A$
        \end{itemize}
            \item<2-> ``Rotor'' -- $B$
        \begin{itemize}
            \item mass $m_B$
            \item Symmetry axis $\bs{b}$
            \item Central inertia tensor
                \begin{itemize}
                    \item $\bs{I}^{B/B_o} = K\bs{U} + (J - K)\bs{b}\bs{b}$
                        (cylindrical rotor)
                    \item $\bs{I}^{B/B_o} = I \bs{U}$ (spherical rotor)
                \end{itemize}
            \item ${}^A\bs{\omega}^{B} = \Omega \bs{b}$
        \end{itemize}

    \item<3-> ``Gyrostat'' -- $G$, mass $m_G$, Central inertia tensor
        $\bs{I}^{G/G_o}$
    \item<4-> ``Rigid Gyrostat'' -- $RG$, mass $m_G$, Central inertia tensor
        $\bs{I}^{G/G_o}$, \textit{same angular velocity as $A$}
\end{itemize}
}

\subsection{Useful formulas}
\frame
{
\frametitle{Mass and Inertia Relationships}
\begin{itemize}
    \item<1-> $m_G = m_A + m_B$
    \item<2-> $\bs{I}^{G/G_o} = \bs{I}^{A/A_o} + \bs{I}^{A_o/G_o} + \bs{I}^{B/B_o} + \bs{I}^{B_o/G_o}$
    \item<3-> $\bs{I}^{CG/G_o} \triangleq \bs{I}^{G/G_o} - J \bs{b} \bs{b}$
    \item<4-> $\bs{I}^{SG/G_o} \triangleq \bs{I}^{G/G_o} - I \bs{b} \bs{b}$
\end{itemize}
}

\frame
{
\frametitle{Angular Momentum}

\begin{block}{Free rotors (traditional) -- $\omega_s \triangleq {}^A\bs{\omega}^B \cdot
    \bs{b}$}
        \begin{align*}
            {}^N\bs{H}^{RG/G_o} &\triangleq \bs{I}^{G/G_o} \cdot {}^N\bs{\omega}^A\\
            {}^N\bs{H}^{G/G_o} &= {}^N\bs{H}^{RG/G_o}  + J\omega_s\bs{b}
        \end{align*}
    \end{block}

\begin{block}{Free rotors (``efficient'') -- $\omega_s \triangleq {}^N\bs{\omega}^B \cdot
    \bs{b}$}
        \begin{align*}
            {}^N\bs{H}^{CG/G_o} &\triangleq \bs{I}^{CG/G_o} \cdot {}^N\bs{\omega}^A\\
            {}^N\bs{H}^{G/G_o} &= {}^N\bs{H}^{CG/G_o}  + J\omega_s\bs{b}
        \end{align*}
    \end{block}
}

\frame
{
\frametitle{Inertia Torque}
\begin{block}{Free rotors (traditional) -- $\omega_s \triangleq {}^A\bs{\omega}^B \cdot
    \bs{b}$}
        \begin{align*}
            {}^N\bs{T}^{RG} &\triangleq \bs{I}^{G/G_o} \cdot {}^N\bs{\alpha}^A
            + {}^N\bs{\omega}^A \times \bs{I}^{G/G_o} \cdot {}^N\bs{\omega}^A \\
            {}^N\bs{T}^{G} &= {}^N\bs{T}^{RG} + J(\dot{\omega}_s\bs{b}
            + {}^N\bs{\omega}^A \times \omega_s \bs{b})
        \end{align*}
    \end{block}

\begin{block}{Free rotors (``efficient'') -- $\omega_s \triangleq {}^N\bs{\omega}^B \cdot
    \bs{b}$}
        \begin{align*}
            {}^N\bs{T}^{CG} &\triangleq \bs{I}^{CG/G_o} \cdot {}^N\bs{\alpha}^A
            + {}^N\bs{\omega}^A \times \bs{I}^{CG/G_o} \cdot {}^N\bs{\omega}^A \\
            {}^N\bs{T}^{G} &= {}^N\bs{T}^{CG} + J(\dot{\omega}_s\bs{b}
            + {}^N\bs{\omega}^A \times \omega_s \bs{b})
        \end{align*}
    \end{block}

    \begin{block}{Inertia Torque of $G$ in $N$}
        \[\bs{T}^* \triangleq - {}^N\bs{T}^{G}\]
    \end{block}
}

\subsection{An example}
\frame
{
\frametitle{Example}
    \begin{block}{Traditional}
        \begin{description}
        \item[Carrier $A$]
            \begin{itemize}
                \item mass $m_A$
                \item $\bs{I}^{A/A_o} =
                    I_{xx}\bs{a}_x\bs{a}_x +
                    I_{yy}\bs{a}_y\bs{a}_y +
                    I_{zz}\bs{a}_z\bs{a}_z + I_{xz}\bs{a}_x\bs{a}_z +
                    I_{xz}\bs{a}_z\bs{a}_x$
            \end{itemize}
        \item[Rotor $B$]
            \begin{itemize}
                \item mass $m_B$
                \item $\bs{I}^{B/B_o} =
                    I\bs{a}_x\bs{a}_x +
                    J\bs{a}_y\bs{a}_y +
                    I\bs{a}_z\bs{a}_z$
            \end{itemize}
        \item[Geometry]
            $\bs{r}^{A_o / B_o} = x\bs{a}_x + z\bs{a}_z$
    \end{description}
    \end{block}
    \begin{block}{New}
        \begin{description}
        \item[Gyrostat $G$]
            \begin{itemize}
                \item mass $m_G$
                \item $\bs{I}^{G/G_o} =
                    I_{Gxx}\bs{a}_x\bs{a}_x +
                    I_{Gyy}\bs{a}_y\bs{a}_y +
                    I_{Gzz}\bs{a}_z\bs{a}_z + I_{Gxz}\bs{a}_x\bs{a}_z +
                    I_{Gxz}\bs{a}_z\bs{a}_x$
                \item Rotor spin moment of inertia $J$
            \end{itemize}
        \item[Geometry]
            $\bs{r}^{B_o / G_o} = x_g\bs{a}_x + z_g\bs{a}_z$
    \end{description}
    \end{block}
}

%\frame
%{
%\frametitle{Example (Angular momentum)}
%
%}

\section{Conclusions}
\frame
{
\frametitle{Conclusions}
To minimize complexity of motion equations:
\begin{itemize}
    \item<1-> For derivation of motion equations, define mass and inertia
        scalars in a gyrostat framework, rather than defining these scalars for
        each individual body
    \item<2-> Introduce geometric scalars which locate mass centers relative
        to the gyrostat mass center
    \item<3-> For free moving rotors, use generalized speeds which measure the
        component(s) of angular velocity about the axis of rotation,
        \textit{relative to the inertial frame}
    \item<4-> For rotors with a specified spin rate, use generalized speeds which measure the
        component(s) of angular velocity about the axis of rotation,
        \textit{relative to the carrier frame}
\end{itemize}
}
\end{document}
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