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\begin{document}
\bibliographystyle{IEEEtran}
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\handout{}{Dan Ports}{2004/05/13}{drkp@mit.edu}{Final Project}
\renewcommand{\thepage}{\arabic{page}}
\tntitle{A Motorized Turntable System}

\tableofcontents
\listoffigures
\listoftables

\section{Overview}
\label{sec:overview}

This report presents a design for a generic turntable system capable
of use in a number of different theatrical situations. The turntable
is twenty feet in diameter, and designed to support a load of at least
ten thousand pounds. Some unusual features of the design include a purely
laminated-plywood platform design. By constructing the platform
from a number of identically-shaped sections, this speeds the process
of construction. In addition, since it uses modular sections, it can
be easily disassembled, stored, and reassembled for later uses. The
turntable is driven using a 2 HP motor and a roller chain drive
system. The use of a roller chain drive system instead of a friction
or cable drive system makes installing and adjusting the drive loop
simpler.

\section{Requirements Analysis}
\label{sec:requirements-analysis}

We begin by examining the requirements for this system. This turntable
not being designed for an actual production, we do not have specific
figures in mind. We make some general assumptions about the types of
loads the turntable will be subjected to, and its uses. This is a
reasonable design approach, since a turntable of the size we are
considering is a major scenic mechanism. One would like to be able to
reuse it for multiple productions --- the turntable itself can only be
reused if space permits it to be stored, but in any case we would like
the design to be reusable without too much modification. Again with
reusability in mind, we note that a mechanism of this scale may be
somewhat expensive, but this is to be expected; if the design is
reusable, then the expense is not such a great concern. Of course, we
would like to keep the budget as low as is reasonably possible, as
always.

The intended use of this turntable is to rotate a full set --- or at
least major portions of one. Hence, it will be a circular turntable
twenty feet in diameter (approximately 63 feet in circumference, and
approximately 315 square feet in area). It is to support a load of ten
thousand pounds. We assume the load will be evenly distributed over
the area of the turntable. This assumption is clearly not entirely
realistic, but it supports our goal of generality, and many designs
will approximate a uniform loading. Nevertheless, there are clearly
cases where this assumption becomes problematic: if all the load is at
the edges of the platform (a set of walls around the edge, perhaps),
the turntable's capacity will have to be de-rated.

We would also like to consider the speed and torque requirements.
Rather arbitrarily, we would like the turntable to be able to complete
one rotation per minute --- this corresponds to an average edge
velocity of about 63 feet per minute, or about 0.7 mph. We also
prepare for the worst cases in dynamic torque applied by the load: we
suppose the director, prompted by Murphy's law, will call for twenty
actors to walk briskly around the edge. To calculate this, we assume
that the contact force applied by the actor's feet will be
approximately at a 45$^\circ$ angle, i.e. the horizontal and vertical
components are equal\footnote{This assumption is really not
  justifiable for any precise applications; it serves only as a quick
  approximation.}, and that the actors average 150 pounds. Then the
torque applied is
\begin{align}
  \tau &= 20 r m g \\
  &= 20 \left(10'\right) \left(150 \mbox{ lbs}\right) \\
  %\left(32 \nicefrac{\mathrm{ft}}{\mathrm{sec}^2}\right)
   &= 30,000 \mbox{ ft lbs}\\
   &\approx 40,674 \;\mathrm{Joules} \label{eq:counter-torque-joules}
\end{align}


\section{Platform construction}
\label{sec:platform-construction}

The body of the turntable is essentially a platform that supports the
actors and is itself supported by a set of casters. A traditional
turntable design will use a typical platform design: a frame of sawn
lumber with plywood panels attached. Since the platform is so large,
it must be divided into sections --- typically either $4' \times 8'$
stock platforms plus special rounded edge platforms, or wedge-shaped
platforms. These platforms must be aligned and secured together. The
process of alignment is non-trivial, since various oddly-shaped
platforms will be required, and these will need to be carefully
constructed.

Instead, we use a variation on designs proposed by Sullivan
\cite{sullivan} and Holden \cite{holden}. This design calls for the
turntable to be constructed from laminated plywood sections. This
neatly avoids the problem of constructing frames by eliminating them
entirely. Moreover, it uses wedge-shaped sections that are identical,
which simplifies construction greatly. This is an advantage over
designs that use stock $4' \times 8'$ platforms, because those require
multiple different types of smaller rounded platforms at the edges
that must be carefully designed and constructed to fit together.

The design divides the 20' turntable into a 5' diameter disc-shaped
inner section and a ring-shaped\footnote{Or an annulus, if you prefer
  such terms} outer section that completes the 20' diameter of the
turntable. The outer section is divided into twenty wedge-shaped
segments. This is shown in Figure~\ref{fig:platform-overview}.

\begin{figure}[phtb]
  \centering
  \includegraphics{21m735-proj-platform-overview}
  \caption{Overview of platform design}
  \label{fig:platform-overview}
\end{figure}

The outer sections are constructed by cutting sixty identical
wedge-shaped sections of $\nicefrac{3}{4}''$ plywood. Initially, the
wedges should simply be cut with the straight edges shown as solid
lines in Figure~\ref{fig:wedges}; the dashed-line curved edges will be
cut later. The size of the sections was chosen so that they can be
easily cut from a $4' \times 8'$ sheet of plywood;
Figure~\ref{fig:wedges} shows how one wedge can be cut from the sheet,
and the remaining plywood on the sheet can be used to cut a second
identical wedge.

\begin{figure}[hptb]
  \centering
  \includegraphics[width=4in]{21m735-proj-wedges}
  \caption{Cutting a wedge from a $4' \times 8'$ sheet of plywood}
  \label{fig:wedges}
\end{figure}

Three wedges will be laminated together to form each section, as in
Figure~\ref{fig:lamination}. The three individual pieces are offset by
3 inches away from the inner edge and away from the factory edge of
the plywood. It is important to note that this is a translational, not
rotational offset, i.e. the next piece is shifted from the piece it
attaches to, with the edges still parallel. This ensures that there is
a constant 3-inch offset between each piece; if it were offset
rotationally, the offset would decrease toward the center, which would
make for a less secure joint between adjacent segments.

\begin{figure}[hptb]
  \centering
  \includegraphics[width=4in]{21m735-proj-lamination}
  \caption{Arrangement of wedge-pieces for lamination (not to scale)}
  \label{fig:lamination}
\end{figure}

The following method is probably the easiest way to construct the
outer section: first, lay out a full ring of 20 wedge pieces, using
only a single layer. This layer should then be trimmed into a ring
(probably using a router) with an outside diameter of $19'6''$ and an
inside diameter of $4'3''$. Next, the second layer can be added after
a thorough coating with wood glue. It should be trimmed to an outside
diameter of $19'9''$ and an inner diameter of $4'6''$. Following the
same procedure, the third layer can be added with outer diameter
$20'0''$ and inner diameter $4'9''$. This process will probably go
more smoothly if the wedge sections are actually cut
$\nicefrac{1}{8}''$ or so smaller than their nominal dimension, to
account for imprecisions and errors in alignment \cite{holden}.

The center section is a $5'$-diameter disk, also constructed from
laminated plywood. Since the disk cannot be cut from a single sheet of
plywood, it needs to be constructed using six semi-circles. The highest
disk should have a $5'$ diameter, and the two below it should have
diameters of $4'9''$ and $4'6''$ respectively. The semi-circles are to
be arranged such that the joins at each layer are perpendicular to the
one below. A plan is shown in Figure~\ref{fig:center}.

\begin{figure}[hptb]
  \centering
  \includegraphics[height=4.5in]{21m735-proj-center}
  \caption{Plan for the center section (bottom view)}
  \label{fig:center}
\end{figure}

Once constructed, the center disk section and the outer wedge sections
can then be joined together during put-in. Since the plywood-laminated
sections are constructed with offsets, it is simply necessary to
arrange the pieces with the overhang of the center disk section matching
the underhang of the outer wedge sections. Similarly, the layers of
the outer wedges should mesh together. This is shown in the section
view in Figure~\ref{fig:platform-section}. The joins can be made using
screws or bolts through each overlapped area.

\begin{figure}[hptb]
  \centering
  \includegraphics[width=5in]{21m735-proj-platform-section}
  \caption{Platform assembly (exploded section view, not to scale)}
  \label{fig:platform-section}
\end{figure}

The layered structure of the outer ring has the nice property that
sweeps can be added (as shown in Figure~\ref{fig:platform-section} to
the bottom layer, turning the outer edge of the middle layer into a
groove. This is ideal for the drive chain, since the upper layer and
bottom sweeps ensure that there is no danger of the tensioned chain
slipping out of its groove.

The total wood requirement is 30 $4' \times 8'$ sheets of
$\nicefrac{3}{4}''$ plywood, plus another 6 for the center section,
and smaller pieces to complete the sweeps. This is all that is
required to construct the platform; no sawn lumber is required.

Optionally, the platform may be surfaced with whatever decking is
appropriate for the specific use (perhaps a layer of masonite, etc.).

After the show, the sections may again be separated. The turntable
breaks down into sections of a reasonable size for storage (provided
space is available, of course). It can later be reassembled the same
way.

\section{Driving the turntable}
\label{sec:driving}

We now turn\footnote{Sorry. I couldn't resist.} to the problem of
making the turntable turn. As usual, we use casters to support the
turntable and allow it to rotate. The casters are mounted to the floor
rather than the turntable, so that they can be more easily adjusted to
account for a non-level floor\footnote{Not that we know of any
  theaters that have non-level floors...}. Since the bottom surface of
the turntable is flat --- it is, after all, made of plywood --- there
should be no problem with mounting the casters on the floor.

To ensure that the turntable moves smoothly, $4''$-diameter casters
are used. Forty of these casters are arranged in three circles
concentric with the center of the turntable: six at a radius of $6'$,
14 at a radius of $12'$, and 20 at a radius of $18'$. This ensures
that casters are spaced approximately three feet apart, both radially
and circumferentially.

The center pivot is simply constructed from a length of
$1~\nicefrac{1}{2}''$ Schedule~40 pipe and two flanges. These readily
available parts can be assembled into a pivot by attaching a flange to
the floor and another to the bottom of the turntable, then connecting
them with a heavily-greased $5~\nicefrac{1}{2}''$ section of
pipe. Holden \cite{holden} notes that power cables, if any are required
for practicals on the turntable, can be threaded through the pipe (if
so, a larger diameter pipe, e.g. $2''$ or $2~\nicefrac{1}{2}''$ might
be more appropriate, but the design is otherwise identical).

A chain drive system is used to operate the turntable. This is similar
to a cable-drive system, but uses roller chain instead of aircraft
cable. This has several important advantages. Foremost, the chain is
easier to work with in a few key ways: it does not need to be spliced,
but can simply be connected together using an extension link, which
makes it possible to construct the necessary loop without much effort,
or to later change the length of the loop. Also, it is no longer
necessary to deal with drums or sheaves; the only necessary hardware
for the chain are sprockets. Eighty feet of \#40 chain is specified. This is
enough length to make a full loop around the turntable plus an offset
for the motor, which can be located up to 8 feet away from the edge of
the turntable. If more distance is required, more chain can be
added.

\begin{figure}[hptb]
  \centering
  \includegraphics[width=4in]{21m735-proj-drive}
  \caption{Drive configuration (not to scale)}
  \label{fig:drive}
\end{figure}

The chain is wrapped around the turntable, then past two idler
sprockets to the motor, as in Figure~\ref{fig:drive}. The motor is
attached to a $3.196''$ sprocket through a $20:1$ speed reducer. This
provides a speed reduction factor of
\begin{equation}
  20 \frac{(20\pi)(12)}{3.196\pi} \approx 1502
\end{equation}
which can reduce a 1725 RPM motor output to 1.15 RPM, a quite
satisfactory rotation speed for our turntable. This gear arrangement
results in the following available torque with a 2 HP motor:
\footnote{In performing our calculations here, we implicitly convert
  from imperial to metric units. Though this adds the confusion of
  converting between units, it avoids the infinitely more confusing
  common usage of a \emph{pound} as both a unit of mass and a unit of
  force (``the gravitational force exerted on an object with a
  one-pound mass''). Since we deal with both mass and force here, it
  is far less confusing to use the metric system, where there is no
  confusion between the unit for mass (the kilogram) and the unit for
  force (the Newton).}:
\begin{align}
  \tau &= \frac{2 \mbox{ horsepower}}{1.15 \nicefrac{1}{\mathrm{min}}}
\\  &\approx 77,812 \; \mathrm{Joules}
\end{align}

We consider the moment of inertia of the turntable to be that of a
disk whose mass is evenly distributed over its area, and suppose that
it supports a mass of 15,000 pounds. This accounts for the specified
10,000 pound load, and provides a conservative overestimation for the
weight of the turntable\footnote{Again, this estimate is not
  particularly accurate, but we have chosen to err on the conservative
  side whenever making assumptions, to ensure safety.}. Using the
standard relation $I = \nicefrac{1}{2} m r^2$, this gives us the
following inertial moment:
\begin{align}
  I &= \frac{1}{2} 15,000\;\mathrm{lbs} (10\;\mathrm{feet})^2 = 750,000
  \;\mathrm{ft^2\;lbs} \\
  &= 31,605 \;\mathrm{m^2\:kg}
\end{align}
So with this moment of inertia, the turntable will be able to support
an angular acceleration
\begin{align}
  \alpha &= \frac{\tau}{I} =
  \frac{77812\;\mathrm{J}}{31605\mathrm{m^2\:kg}} \\
  &\approx 2.46 \frac{\mathrm{rotations}}{\mathrm{s}^2}
\end{align}
which is more than satisfactory for our requirements. Note also that
the available torque, 77,812 Joules, exceeds the torque applied by
actors moving around the edge of the stage, which we figured at 40,674
Joules in Equation~\ref{eq:counter-torque-joules}. This means that the
turntable's motor will not be overpowered by that adverse loading
condition. So our choice of motor is adequate --- two horsepower does
not sound like much, but between the gearbox and the
turntable-sprocket ratio, the speed reduction is massive.

Of course, this power will be useless if it cannot be transmitted to
the turntable. Roller chain has a high transmission efficiency in
general, but the turntable is not a sprocket. Hence, the chain must be
maintained under tension in order for it to grip the turntable and
turn it with the necessary torque. We accomplish this using a
tensioner design by Hendrickson \cite{hendrickson}. The mechanism is
shown in Figure~\ref{fig:tensioner}.

\begin{figure}[hptb]
  \centering
  \includegraphics[width=5in]{21m735-proj-tensioner}
  \caption{Tensioner mechanism assembly (section view, not to scale)}
  \label{fig:tensioner}
\end{figure}

This design mounts the motor, gearbox, and sprocket assembly to an
inner frame (not shown explicitly in the diagram), which is mounted
with slide rails to an outer frame. The inner and outer frames are
connected by two tensioners: a turnbuckle connected to a drawbar
spring. The turnbuckle can be tightened to stretch the spring. The
spring then pulls on the inner frame, pulling the
motor-gearbox-sprocket assembly away from the turntable. This applies
tension to the drive chain, allowing it to efficiently transfer torque
to the turntable. The rated maximum breaking strength for the chain is
4,290 pounds, which translates to a working load limit of 858 pounds
after applying a $5:1$ safety factor (appropriate since this chain is
not being used for overhead lifting or in a safety-critical
application). This is sufficiently greater than the tension that will
be applied to the system by the spring. A tension of approximately 100
pounds should be sufficient. Drawbar springs --- extension springs
that extend to a stop --- are used, to prevent the spring from
breaking if the turnbuckle is overtightened.

To prevent the turntable from moving when it is not supposed to move,
a brake is needed. We choose a 2 HP brakemotor, which both provides
enough torque to drive the turntable and the capability to brake it
when it is not moving. This brake is capable of holding the turntable
in position even when subjected to the worst-case torque loads such as
the twenty actors moving around the edge
(Equation~\ref{eq:counter-torque-joules}).

Since we do not have a specific application in mind for the turntable,
we cannot specify an application-specific control system for it. We
simply use a generic switched-motor configuration. The motor requires
three-phase AC power (while this is somewhat less convenient to obtain
a source of in the theater than single-phase power, it makes selecting
a motor much more straightforward), so the controller will need to be
able to switch all three phases. This calls for what is essentially a
heavy-duty three-pole switch. A motor starter could also be used,
which provides some degree of overcurrent protection, but this seems
like a bit of unnecessary complexity for this simple application. The
switch turns on the motor when desired, and activates the brake
otherwise. This process could also be automated somewhat by placing
limit microswitches at appropriate locations under the turntable and
triggering them with protrusions from the bottom of the turntable
(spaced between the rings of casters, of course). These microswitches
can be wired to a contactor in series with the motor to start and stop
the motor at the appropriate times. Since this depends so heavily on
the application, we cannot present a generic example here, as it would
not be very meaningful.

The mechanical components required to construct and drive this
turntable system are listed in Table~\ref{tab:drive-components}. The
only other requirement is the 36 sheets of $\nicefrac{3}{4}''$ plywood
mentioned above for constructing the platform. The design is very
simple in this respect.

\begin{table}[hptb]
  \centering
  \begin{tabular}{|c|c|c|c|c|}
    \hline
    \textbf{Item} & \textbf{Source} & \textbf{Price} &
    \textbf{Quantity} & \textbf{Total} \\
    \hline
    Roller chain, ANSI \#40, 10 ft & Grainger 2W093
    \cite{grainger} & \$23.65 & 8 & \$189.20 \\
    Connecting links, ANSI \#40, 5 ct. & Grainger 5X293
    \cite{grainger} & \$3.72 & 1 & \$3.72
    \\
    Sprocket, ANSI \#40, 3.196$''$ diam. & Grainger 1L142
    \cite{grainger} & \$17.25 & 1
    & \$17.25 \\
    Idler sprocket, ANSI \#40, 2.96$''$ OD & Grainger
    5A551 \cite{grainger} & \$31.64 & 1
    & \$31.64\\
    Brakemotor, 2 HP, 1725 RPM, 3-ph. & Grainger 4TD99 \cite{grainger}
    & \$587.50 & 1 & \$587.50 \\
    Inline speed reducer, $20:1$, 2 HP & Grainger 5YV07
    \cite{grainger} & \$412.25 & 1 & \$412.25 \\
    Motor switch, 3-ph, indoor & McM-C 7657K42 \cite{mcmaster}
    & \$46.98 & 1 & \$46.98 \\
    Drawbar spring, $8 \nicefrac{3}{4}''$ & McM-C 9630K2
    \cite{mcmaster} & \$15.26 & 2 & \$30.52 \\
    $3 \nicefrac{11}{16}''$ turnbuckle, 215 lb WLL & McM-C
    30125T603 \cite{mcmaster} & \$2.43 & 2 & \$4.86 \\
    $1 \nicefrac{1}{2}''$ Sch. 40 pipe flange, unthr. & McM-C
    68095K134 \cite{mcmaster} & \$8.84 & 2 & \$17.68 \\
    $5 \nicefrac{1}{2}'' \times 1 \nicefrac{1}{2}''$ Sch. 40 pipe
    nipple & McM-C 44615K163 \cite{mcmaster} & \$2.09 & 1 & \$2.09 \\
    Rigid caster, 4'' diam., 800 lb. & Grainger 2G131
    \cite{grainger} & \$18.32 & 40 & \$732.80 \\
    \hline
    \textbf{Total} & & & & \textbf{\$2076.49}\\
    \hline
  \end{tabular}
  \caption{Required mechanical components}
  \label{tab:drive-components}
\end{table}

\nocite{parker}
\nocite{arnold}
\nocite{gilette}
\nocite{holden}
\nocite{hendrickson}
\nocite{hendrickson2}
\nocite{sullivan}
\nocite{backstage-handbook}
\nocite{holden-and-sammler}
\bibliography{21m735-tns}
\end{document}