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> gyrostat_presentation.tex

# gyrostat_presentation.tex

gyrostat_presentation.tex — TeX document, 7Kb

## File contents

\documentclass{beamer}

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\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{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