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R. J. Dev. Kwan eb05416991 Initial commit of "Rocket Workbench". 
Taken from sources in CVS at:
    https://sourceforge.net/projects/rocketworkbench/

Sources extracted in two steps:
1. Pull entire project tree into a subdir "rwb" via "rsync":
    rsync -a a.cvs.sourceforge.net::cvsroot/rocketworkbench/ rwb/.
2. Export sources:
    export CVSROOT=$(pwd)/rwb
    SUBDIRS="analyser cpropep cpropep-web CVSROOT data libcompat libcpropep libnum libsimulation libthermo prop rocketworkbench rockflight"
    mkdir rwbx; cd rwbx
    cvs export -D now ${SUBDIRS}

After this (and some backups for safety), the directory content was
added to a Git repo:
    git init .
    git add *
2021-01-20 15:50:36 -08:00

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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 99.2beta8 (1.42)
original version by: Nikos Drakos, CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Burning analysis of star configuration</TITLE>
<META NAME="description" CONTENT="Burning analysis of star configuration">
<META NAME="keywords" CONTENT="star">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<META NAME="Generator" CONTENT="LaTeX2HTML v99.2beta8">
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
<LINK REL="STYLESHEET" HREF="star.css">
</HEAD>
<BODY >
<P>
<H1 ALIGN=CENTER>Burning analysis of star configuration</H1>
<P ALIGN=CENTER><STRONG>Antoine Lefebvre</STRONG></P>
<P ALIGN=LEFT></P>
<P>
<BR>
<H2><A NAME="SECTION00010000000000000000">
Contents</A>
</H2>
<!--Table of Contents-->
<UL>
<LI><A NAME="tex2html19"
HREF="star.html">Contents</A>
<LI><A NAME="tex2html20"
HREF="star.html#SECTION00020000000000000000">Introduction</A>
<LI><A NAME="tex2html21"
HREF="star.html#SECTION00030000000000000000">Geometric definition</A>
<LI><A NAME="tex2html22"
HREF="star.html#SECTION00040000000000000000">Analysis</A>
<UL>
<LI><A NAME="tex2html23"
HREF="star.html#SECTION00041000000000000000">Zone 1</A>
<LI><A NAME="tex2html24"
HREF="star.html#SECTION00042000000000000000">Zone 2</A>
<LI><A NAME="tex2html25"
HREF="star.html#SECTION00043000000000000000">Zone 3</A>
<LI><A NAME="tex2html26"
HREF="star.html#SECTION00044000000000000000">Zone 4</A>
</UL>
<LI><A NAME="tex2html27"
HREF="star.html#SECTION00050000000000000000">Design example</A>
<LI><A NAME="tex2html28"
HREF="star.html#SECTION00060000000000000000">conclusion</A>
<LI><A NAME="tex2html29"
HREF="star.html#SECTION00070000000000000000">Bibliography</A>
<LI><A NAME="tex2html30"
HREF="star.html#SECTION00080000000000000000">About this document ...</A>
</UL>
<!--End of Table of Contents-->
<P>
<P>
<H1><A NAME="SECTION00020000000000000000">
Introduction</A>
</H1>
<P>
The design of solid propellant grain that provide neutral burning
is important to optimize rocket motor performance. The star
configuration have been widely used to achieve this goal. In this
report, I will present an analysis of the burning comportement of star
shape as well as parameter recommandation to achieve better
performance.
<P>
<H1><A NAME="SECTION00030000000000000000">
Geometric definition</A>
</H1>
<P>
The star could be characterize by seven independant variable as
defined in figure <A HREF="star.html#star">2</A>. As every star points are identical,
only one is necessary for the analysis.
<P>
<P></P>
<DIV ALIGN="CENTER"><A NAME="variable"></A><A NAME="12"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
Geometric definition of star.</CAPTION>
<TR><TD><DIV ALIGN="CENTER">
<!-- MATH
$\includegraphics[height=5in]{img/variable.ps}$
-->
<IMG
WIDTH="498" HEIGHT="401" ALIGN="BOTTOM" BORDER="0"
SRC="img3.gif"
ALT="\includegraphics[height=5in]{img/variable.ps}">
</DIV></TD></TR>
</TABLE>
</DIV><P></P>
<P>
<P></P>
<DIV ALIGN="CENTER"><TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img4.gif"
ALT="$\displaystyle w$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img5.gif"
ALT="$\displaystyle =$">&nbsp; &nbsp;Web thickness</TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img6.gif"
ALT="$\displaystyle r_1$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img5.gif"
ALT="$\displaystyle =$">&nbsp; &nbsp;Radius</TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img7.gif"
ALT="$\displaystyle r_2$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img5.gif"
ALT="$\displaystyle =$">&nbsp; &nbsp;Tip radius</TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img8.gif"
ALT="$\displaystyle R$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img5.gif"
ALT="$\displaystyle =$">&nbsp; &nbsp;External radius</TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img9.gif"
ALT="$\displaystyle \eta$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img5.gif"
ALT="$\displaystyle =$">&nbsp; &nbsp;angle</TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img10.gif"
ALT="$\displaystyle \varepsilon$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img5.gif"
ALT="$\displaystyle =$">&nbsp; &nbsp;Secant fillet angle</TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img11.gif"
ALT="$\displaystyle N$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img5.gif"
ALT="$\displaystyle =$">&nbsp; &nbsp;Number of star points</TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
<P></P>
<DIV ALIGN="CENTER"><A NAME="star"></A><A NAME="28"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
Burning zone of the star configuration.</CAPTION>
<TR><TD><DIV ALIGN="CENTER">
<!-- MATH
$\includegraphics[height=4in]{img/mainstar.ps}$
-->
<IMG
WIDTH="516" HEIGHT="392" ALIGN="BOTTOM" BORDER="0"
SRC="img12.gif"
ALT="\includegraphics[height=4in]{img/mainstar.ps}">
</DIV></TD></TR>
</TABLE>
</DIV><P></P>
<P>
<H1><A NAME="SECTION00040000000000000000">
Analysis</A>
</H1>
<P>
In this section, an expression for the perimeter of the star will be
developp for each burning zone as a function of the web thickness
burned (<IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img13.gif"
ALT="$ w_x$">).
<P>
<H2><A NAME="SECTION00041000000000000000">
Zone 1</A>
</H2>
<P>
The perimeter in the zone one could be divide in three
sections. Starting by the right, we have the section before the
radius <IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img14.gif"
ALT="$ r_1$">, which have a radius equal to <IMG
WIDTH="94" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img15.gif"
ALT="$ R-w+w_x$">. The length of
this section is then: <!-- MATH
$(R-w+w_x)(\pi/N - \varepsilon)$
-->
<IMG
WIDTH="185" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img16.gif"
ALT="$ (R-w+w_x)(\pi/N - \varepsilon)$">.
<P>
Then, we have the perimeter of the arc of initial radius <IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img14.gif"
ALT="$ r_1$">. The
angle will remain constant to <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img17.gif"
ALT="$ a$">. The length is then: <!-- MATH
$(r_1+w_x)a$
-->
<IMG
WIDTH="85" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img18.gif"
ALT="$ (r_1+w_x)a$">.
<P>
The third section is more complicated. The lenght of the line
starting at the end of the radius <IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img14.gif"
ALT="$ r_1$"> and crossing the vertical
line will be evaluated first. Then, the perimeter of the radius
<IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img2.gif"
ALT="$ r_2$"> will be add to the result, and the length of the line starting
at the beginning of the radius will be substract.
<P>
<P></P>
<DIV ALIGN="CENTER"><A NAME="len"></A><A NAME="37"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3:</STRONG>
Determination of the length <IMG
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img1.gif"
ALT="$ L$">.</CAPTION>
<TR><TD><DIV ALIGN="CENTER">
<!-- MATH
$\includegraphics[height=5in]{img/starL.ps}$
-->
<IMG
WIDTH="255" HEIGHT="529" ALIGN="BOTTOM" BORDER="0"
SRC="img19.gif"
ALT="\includegraphics[height=5in]{img/starL.ps}">
</DIV></TD></TR>
</TABLE>
</DIV><P></P>
<P>
In order to determine the length, refer to the figure <A HREF="star.html#len">3</A>. The
lenght <IMG
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img1.gif"
ALT="$ L$"> we are looking for will be equal <IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$ x + y$">.
<P>
<P></P>
<DIV ALIGN="CENTER"><TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img21.gif"
ALT="$\displaystyle b + z$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="156" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img22.gif"
ALT="$\displaystyle = (R-w-r_1)\sin{\varepsilon}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img23.gif"
ALT="$\displaystyle x$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="134" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img24.gif"
ALT="$\displaystyle = (r_1+w_x)\tan{\eta}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="40" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img25.gif"
ALT="$\displaystyle \cos{\eta}$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="84" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
SRC="img26.gif"
ALT="$\displaystyle = \frac{r_1+w_x}{z}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="38" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img27.gif"
ALT="$\displaystyle \sin{\eta}$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="36" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
SRC="img28.gif"
ALT="$\displaystyle = \frac{b}{y}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="38" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img27.gif"
ALT="$\displaystyle \sin{\eta}$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="225" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
SRC="img29.gif"
ALT="$\displaystyle = \frac{(R-w-r_1)\sin{\varepsilon} - \frac{r_1+w_x}{\cos{\eta}}}{y}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="80" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img30.gif"
ALT="$\displaystyle L = y + x$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="388" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
SRC="img31.gif"
ALT="$\displaystyle = (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} - \frac{r_1+w_x}{\cos{\eta}\sin{\eta}} + (r_1+w_x)\tan{\eta}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
We could now simplify this equation using two trigonometric
identity:
<P>
<P></P>
<DIV ALIGN="CENTER"><TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="76" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img32.gif"
ALT="$\displaystyle \sin^2{\eta} -1$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="82" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img33.gif"
ALT="$\displaystyle = -\cos^2{\eta}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="42" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img34.gif"
ALT="$\displaystyle \tan{\eta}$"></TD>
<TD NOWRAP ALIGN="LEFT"><IMG
WIDTH="111" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
SRC="img35.gif"
ALT="$\displaystyle = \frac{1}{\tan{(\frac{\pi}{2}-\eta)}}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\begin{split}
L &= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} +
(r_1+w_x)\left[
\frac{\sin^2{\eta}-1}{\cos{\eta}\sin{\eta}} \right]\\
&= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} +
(r_1+w_x)\left[ \frac{-\cos{\eta}}{\sin{\eta}} \right]\\
&= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} +
(r_1+w_x)\left[ \frac{-1}{\tan{\eta}} \right]\\
&= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} -
(r_1+w_x)\tan{(\frac{\pi}{2}-\eta)}\\
\end{split}
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="356" HEIGHT="191" BORDER="0"
SRC="img36.gif"
ALT="\begin{displaymath}\begin{split}L &amp;= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta...
...sin{\eta}} - (r_1+w_x)\tan{(\frac{\pi}{2}-\eta)}\\ \end{split}\end{displaymath}"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(1)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
We could now determine the length of the arc and how much we should
substract from the length L. Refer to figure <A HREF="star.html#arc">4</A> for the
variables.
<P>
<P></P>
<DIV ALIGN="CENTER"><A NAME="arc"></A><A NAME="96"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4:</STRONG>
Arc of radius <IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img2.gif"
ALT="$ r_2$">.</CAPTION>
<TR><TD><DIV ALIGN="CENTER">
<!-- MATH
$\includegraphics[height=3in]{img/arc.ps}$
-->
<IMG
WIDTH="201" HEIGHT="283" ALIGN="BOTTOM" BORDER="0"
SRC="img37.gif"
ALT="\includegraphics[height=3in]{img/arc.ps}">
</DIV></TD></TR>
</TABLE>
</DIV><P></P>
<P>
<!-- MATH
\begin{displaymath}
\text{Arc length} = (r_2-wx)(\frac{\pi}{2}-\eta)
\end{displaymath}
-->
<P></P><DIV ALIGN="CENTER">
Arc length<IMG
WIDTH="154" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
SRC="img38.gif"
ALT="$\displaystyle = (r_2-wx)(\frac{\pi}{2}-\eta)$">
</DIV><P></P>
<P>
<!-- MATH
\begin{displaymath}
x = \frac{r_2-wx}{\tan{\eta}}
\end{displaymath}
-->
<P></P><DIV ALIGN="CENTER">
<IMG
WIDTH="101" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
SRC="img39.gif"
ALT="$\displaystyle x = \frac{r_2-wx}{\tan{\eta}}$">
</DIV><P></P>
<P>
We have now the complete expression of the perimeter of the star as a
function of web burned (<IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img13.gif"
ALT="$ w_x$">) in the zone one. This expression is
valid for <!-- MATH
$0 < w_x < r_2$
-->
<IMG
WIDTH="96" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img40.gif"
ALT="$ 0 &lt; w_x &lt; r_2$">.
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\begin{split}
\frac{S}{2N} &= (R-w+w_x)(\frac{\pi}{N} - \varepsilon) +
(r_1+w_x)a+(R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}}\\
& \quad - (r_1+w_x)\tan{(\frac{\pi}{2}-\eta)}
+ (r_2-w_x)(\frac{\pi}{2}-\eta) \\
& \quad - (r_2-wx)\tan{(\frac{\pi}{2}-\eta)}\\
&= (R-w+w_x)(\frac{\pi}{N} - \varepsilon) +
(r_1+w_x)a+(R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}}\\
& \quad - (r_1+r_2)\tan{(\frac{\pi}{2}-\eta)}
+ (r_2-w_x)(\frac{\pi}{2}-\eta)\\
\end{split}
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="473" HEIGHT="204" BORDER="0"
SRC="img41.gif"
ALT="\begin{displaymath}\begin{split}\frac{S}{2N} &amp;= (R-w+w_x)(\frac{\pi}{N} - \varep...
...c{\pi}{2}-\eta)} + (r_2-w_x)(\frac{\pi}{2}-\eta)\\ \end{split}\end{displaymath}"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(2)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
We could now determined the first derivative of this expression to
evaluate if it is progressive, regressive or neutral.
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\frac{\delta S}{\delta w_x} = 2N\left[ \frac{\pi}{N} - \varepsilon + a -
\frac{\pi}{2} + \eta \right]
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="251" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
SRC="img42.gif"
ALT="$\displaystyle \frac{\delta S}{\delta w_x} = 2N\left[ \frac{\pi}{N} - \varepsilon + a - \frac{\pi}{2} + \eta \right]$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(3)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
We could verify that:
<P>
<!-- MATH
\begin{displaymath}
a = \frac{\pi}{2} - \eta + \varepsilon
\end{displaymath}
-->
<P></P><DIV ALIGN="CENTER">
<IMG
WIDTH="111" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
SRC="img43.gif"
ALT="$\displaystyle a = \frac{\pi}{2} - \eta + \varepsilon$">
</DIV><P></P>
<P>
Our expression become:
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\frac{\delta S}{\delta w_x} = 2\pi
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="80" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
SRC="img44.gif"
ALT="$\displaystyle \frac{\delta S}{\delta w_x} = 2\pi$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(4)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
The perimeter in zone 1 will always be progressive. So, it is
important to minimize the radius <IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img2.gif"
ALT="$ r_2$"> in order to switch as fast as
possible to the zone 2.
<P>
<H2><A NAME="SECTION00042000000000000000">
Zone 2</A>
</H2>
<P>
The expression for the perimeter in the second zone is almost the
same as in the zone one. The difference is that the radius <IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img2.gif"
ALT="$ r_2$"> had
vanish and the expression reduce to a simpler one:
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\begin{split}
\frac{S}{2N} &= (R-w+w_x)(\frac{\pi}{N} - \varepsilon) +
(r_1+w_x)a+(R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}}\\
& \quad - (r_1+w_x)\tan{(\frac{\pi}{2}-\eta)}\\
\end{split}
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="473" HEIGHT="83" BORDER="0"
SRC="img45.gif"
ALT="\begin{displaymath}\begin{split}\frac{S}{2N} &amp;= (R-w+w_x)(\frac{\pi}{N} - \varep...
...\ &amp; \quad - (r_1+w_x)\tan{(\frac{\pi}{2}-\eta)}\\ \end{split}\end{displaymath}"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(5)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
The derivative of this expression is:
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\begin{split}
\frac{\delta S}{\delta w_x} &= 2N\left[ \frac{\pi}{N} -\varepsilon +
a - \tan{(\frac{\pi}{2} - \eta)} \right]\\
&= 2N\left[ \frac{\pi}{2} - \eta +
\frac{\pi}{N} - \tan{(\frac{\pi}{2}
- \eta)} \right]\\
\end{split}
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="289" HEIGHT="83" BORDER="0"
SRC="img46.gif"
ALT="\begin{displaymath}\begin{split}\frac{\delta S}{\delta w_x} &amp;= 2N\left[ \frac{\p...
...c{\pi}{N} - \tan{(\frac{\pi}{2} - \eta)} \right]\\ \end{split}\end{displaymath}"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(6)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
As we could see in this expression, the progressivity in zone 2 is
determined by the angle <IMG
WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img47.gif"
ALT="$ \eta$"> and by the number of star point
<IMG
WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img48.gif"
ALT="$ N$">. It is independant of the angle <!-- MATH
$\varepsilon$
-->
<IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img49.gif"
ALT="$ \varepsilon$">.
<P>
The zone 2 will be predominant during the motor burn time and we
would like to provide neutrallity in this zone. Neutrality is obtain
when the derivative of the perimeter is equal to zero. This lead to
the following equation:
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\frac{\delta S}{\delta w_x} = 0 = \frac{\pi}{2} - \eta +
\frac{\pi}{N} - \tan{(\frac{\pi}{2}
- \eta)}
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="287" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
SRC="img50.gif"
ALT="$\displaystyle \frac{\delta S}{\delta w_x} = 0 = \frac{\pi}{2} - \eta + \frac{\pi}{N} - \tan{(\frac{\pi}{2} - \eta)}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(7)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
Which reduce to the following implicit equation of <IMG
WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img47.gif"
ALT="$ \eta$"> as a
function of <IMG
WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img48.gif"
ALT="$ N$">:
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\eta = \frac{\pi}{N} - \tan{(\frac{\pi}{2} - \eta)} + \frac{\pi}{2}
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="200" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
SRC="img51.gif"
ALT="$\displaystyle \eta = \frac{\pi}{N} - \tan{(\frac{\pi}{2} - \eta)} + \frac{\pi}{2}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(8)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
Solution of this equation give values of the angle <IMG
WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img47.gif"
ALT="$ \eta$"> to obtain
neutrality in zone 2 as a function of the number of star points.
<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER"><IMG
WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img48.gif"
ALT="$ N$"></TD>
<TD ALIGN="CENTER"><IMG
WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img47.gif"
ALT="$ \eta$"> (deg)</TD>
<TD ALIGN="CENTER"><IMG
WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img52.gif"
ALT="$ \pi/N$"> (deg)</TD>
</TR>
<TR><TD ALIGN="CENTER">3</TD>
<TD ALIGN="CENTER">24.55</TD>
<TD ALIGN="CENTER">60.00</TD>
</TR>
<TR><TD ALIGN="CENTER">4</TD>
<TD ALIGN="CENTER">28.22</TD>
<TD ALIGN="CENTER">45.00</TD>
</TR>
<TR><TD ALIGN="CENTER">5</TD>
<TD ALIGN="CENTER">31.13</TD>
<TD ALIGN="CENTER">36.00</TD>
</TR>
<TR><TD ALIGN="CENTER">6</TD>
<TD ALIGN="CENTER">33.53</TD>
<TD ALIGN="CENTER">30.00</TD>
</TR>
<TR><TD ALIGN="CENTER">7</TD>
<TD ALIGN="CENTER">35.56</TD>
<TD ALIGN="CENTER">25.71</TD>
</TR>
<TR><TD ALIGN="CENTER">8</TD>
<TD ALIGN="CENTER">37.31</TD>
<TD ALIGN="CENTER">22.50</TD>
</TR>
<TR><TD ALIGN="CENTER">9</TD>
<TD ALIGN="CENTER">38.84</TD>
<TD ALIGN="CENTER">20.00</TD>
</TR>
</TABLE>
</DIV>
<P>
It is important to note that when the angle <!-- MATH
$\eta < \pi/N$
-->
<IMG
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img53.gif"
ALT="$ \eta &lt; \pi/N$">, a secant
fillet <!-- MATH
$\varepsilon < \pi/N$
-->
<IMG
WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img54.gif"
ALT="$ \varepsilon &lt; \pi/N$"> will be necessary to prevent star point
from overlapping. In general, <!-- MATH
$\varepsilon$
-->
<IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img49.gif"
ALT="$ \varepsilon$"> should always be smaller
that <IMG
WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img52.gif"
ALT="$ \pi/N$">.
<P>
<H2><A NAME="SECTION00043000000000000000">
Zone 3</A>
</H2>
<P>
The perimeter in the zone 3 begin when <IMG
WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img55.gif"
ALT="$ w_x = Y*$">. The angle <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img17.gif"
ALT="$ a$">
become progressivly smaller when propellant burned. Perimeter could
be expressed like this:
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\frac{S}{2N} = (R-w+w_x)(\frac{\pi}{N}-\varepsilon) +
(r_1+w_x)\left[ \varepsilon +
\arcsin{(\frac{R-w-r_1}{r_1+w_x}\sin{\varepsilon}}) \right]
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="550" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
SRC="img56.gif"
ALT="$\displaystyle \frac{S}{2N} = (R-w+w_x)(\frac{\pi}{N}-\varepsilon) + (r_1+w_x)\left[ \varepsilon + \arcsin{(\frac{R-w-r_1}{r_1+w_x}\sin{\varepsilon}}) \right]$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(9)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
The derivative of this expression become:
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\begin{split}
\frac{\delta S}{\delta w_x} &= 2N \biggl[ \frac{\pi}{N} +
\arcsin{\biggl(\frac{R-w-r_1}{r_1+w_x}\sin{\varepsilon}\biggr)} - \\
&\quad \frac{w_x(R-w-r_1)\sin{\varepsilon}}{(r_1+w_x)^2\sqrt{1-\frac{(R-w-r_1)^2\sin^2{\varepsilon}}{(r_1+w_x)^2}}} \biggr]\\
\end{split}
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="339" HEIGHT="110" BORDER="0"
SRC="img57.gif"
ALT="\begin{displaymath}\begin{split}\frac{\delta S}{\delta w_x} &amp;= 2N \biggl[ \frac{...
..._1)^2\sin^2{\varepsilon}}{(r_1+w_x)^2}}} \biggr]\\ \end{split}\end{displaymath}"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(10)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
It could be demonstrate that the perimeter is progressive in this
section. It would be interesting to eliminate the zone 3 in order to
keep neutrality as long as possible.
<P>
The condition for the elimination of zone 3 is:
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
Y* = (R-w-r_1)\frac{\sin{\varepsilon}}{\cos{\eta}} - r_1 = w
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="260" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
SRC="img58.gif"
ALT="$\displaystyle Y* = (R-w-r_1)\frac{\sin{\varepsilon}}{\cos{\eta}} - r_1 = w$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(11)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
This equation reduce to:
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\sin{\varepsilon} = \frac{w+r_1}{R-w-r_1}\cos{\eta}
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="187" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
SRC="img59.gif"
ALT="$\displaystyle \sin{\varepsilon} = \frac{w+r_1}{R-w-r_1}\cos{\eta}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(12)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
Now, the angle <!-- MATH
$\varepsilon$
-->
<IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img49.gif"
ALT="$ \varepsilon$"> is determine by the web thickness <IMG
WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img60.gif"
ALT="$ w$">,
the radius <IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img14.gif"
ALT="$ r_1$"> and the angle <IMG
WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img47.gif"
ALT="$ \eta$">. As <IMG
WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img47.gif"
ALT="$ \eta$"> was determine by
the number of star points <IMG
WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img48.gif"
ALT="$ N$"> and the radius may be dictate by
technical decision, the web thickness <IMG
WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img60.gif"
ALT="$ w$"> will determine
<!-- MATH
$\varepsilon$
-->
<IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img49.gif"
ALT="$ \varepsilon$">.
<P>
<H2><A NAME="SECTION00044000000000000000">
Zone 4</A>
</H2>
<P>
The analytical solution of the perimeter in the zone 4 could be
found with the help of the cosinus law:
<P>
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation*}
c^2 = a^2 + b^2 - 2ab\cos{\theta}
\end{equation*}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="180" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img61.gif"
ALT="$\displaystyle c^2 = a^2 + b^2 - 2ab\cos{\theta}$"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
&nbsp;&nbsp;&nbsp;</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
The perimeter is then:
<P></P>
<DIV ALIGN="CENTER"><!-- MATH
\begin{equation}
\begin{split}
\frac{S}{2N} = (r_1+w_x) \biggl[ & \varepsilon +
\arcsin{\biggl(\frac{R-w-r_1}{r_1+w_x}\sin{\varepsilon}\biggr)}
- \pi\\
& \quad + \arccos{\biggl(\frac{(r_1+w_x)^2+(R-r_1-w)^2-R^2}{2(r_1+w_x)(R-r_1-w)}\biggr)}
\biggr]\\
\end{split}
\end{equation}
-->
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE">
<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="501" HEIGHT="98" BORDER="0"
SRC="img62.gif"
ALT="\begin{displaymath}\begin{split}\frac{S}{2N} = (r_1+w_x) \biggl[ &amp; \varepsilon +...
...1-w)^2-R^2}{2(r_1+w_x)(R-r_1-w)}\biggr)} \biggr]\\ \end{split}\end{displaymath}"></TD>
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
(13)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
<H1><A NAME="SECTION00050000000000000000">
Design example</A>
</H1>
<P>
In this section, a star configuration will be design with the
theory developp in the previous sections for a motor of <IMG
WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img63.gif"
ALT="$ 3 inch$">
internal diameter.
<P>
The goal is to have a perimeter that will remain as constant as
possible to mainatin neutrality. It will also be interesting to
minimize the number of star points in order to reduce the difficulty
to cast the propellant. We could also try to optimize the volumetric
loading.
<P>
First of all, we could determine the number of star points. In order
to maximize the quantity of matter, the angle <!-- MATH
$\varepsilon$
-->
<IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img49.gif"
ALT="$ \varepsilon$"> should
be equal to <IMG
WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img52.gif"
ALT="$ \pi/N$">. In order to obtain this condition, the angle
<IMG
WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img47.gif"
ALT="$ \eta$"> should be larger than <IMG
WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img52.gif"
ALT="$ \pi/N$">.
<P>
If we refer to the table of the angle <IMG
WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img47.gif"
ALT="$ \eta$"> in function of <IMG
WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img48.gif"
ALT="$ N$">, to
obtain neutrality in zone 2, we must choose <IMG
WIDTH="52" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img64.gif"
ALT="$ N=6$"> to have <!-- MATH
$\eta >
\pi/N$
-->
<IMG
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img65.gif"
ALT="$ \eta &gt;
\pi/N$">.
<P>
Three conditions are now determine:
<P>
<!-- MATH
\begin{displaymath}
N = 6
\end{displaymath}
-->
<P></P><DIV ALIGN="CENTER">
<IMG
WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img66.gif"
ALT="$\displaystyle N = 6$">
</DIV><P></P>
<!-- MATH
\begin{displaymath}
\eta = 33.53 deg
\end{displaymath}
-->
<P></P><DIV ALIGN="CENTER">
<IMG
WIDTH="103" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img67.gif"
ALT="$\displaystyle \eta = 33.53 deg$">
</DIV><P></P>
<!-- MATH
\begin{displaymath}
\varepsilon = 30 deg
\end{displaymath}
-->
<P></P><DIV ALIGN="CENTER">
<IMG
WIDTH="79" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img68.gif"
ALT="$\displaystyle \varepsilon = 30 deg$">
</DIV><P></P>
<P>
We must now found the web thickness <IMG
WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img60.gif"
ALT="$ w$"> and radius <IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img14.gif"
ALT="$ r_1$"> that fit
the conditions. A radius <!-- MATH
$r_1 = 1/16 in$
-->
<IMG
WIDTH="95" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img69.gif"
ALT="$ r_1 = 1/16 in$"> is reasonable technically.
<P>
The equation to be solve is the following:
<P>
<!-- MATH
\begin{displaymath}
sin{\varepsilon} = \frac{w+r_1}{R-w-r_1}\cos{\eta}
\end{displaymath}
-->
<P></P><DIV ALIGN="CENTER">
<IMG
WIDTH="187" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
SRC="img59.gif"
ALT="$\displaystyle \sin{\varepsilon} = \frac{w+r_1}{R-w-r_1}\cos{\eta}$">
</DIV><P></P>
<P>
The value of <IMG
WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img60.gif"
ALT="$ w$"> that solve this equation is:
<P>
<!-- MATH
\begin{displaymath}
w = 0.500
\end{displaymath}
-->
<P></P><DIV ALIGN="CENTER">
<IMG
WIDTH="80" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img70.gif"
ALT="$\displaystyle w = 0.500$">
</DIV><P></P>
<P>
The seven independant variable are now fixed. The resulting shape
could be seen in figure <A HREF="star.html#res">5</A>.
<P>
<P></P>
<DIV ALIGN="CENTER"><A NAME="res"></A><A NAME="253"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5:</STRONG>
Resulting star configuration for the 3 inch motor.</CAPTION>
<TR><TD><DIV ALIGN="CENTER">
<!-- MATH
$\includegraphics[]{img/res.ps}$
-->
<IMG
WIDTH="348" HEIGHT="348" ALIGN="BOTTOM" BORDER="0"
SRC="img71.gif"
ALT="\includegraphics[]{img/res.ps}">
</DIV></TD></TR>
</TABLE>
</DIV><P></P>
<P>
With the functions developp in the report, the evolution of the
perimeter as a function of the web burned could be plot.
<P>
<P></P>
<DIV ALIGN="CENTER"><A NAME="grah"></A><A NAME="260"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6:</STRONG>
Graphic of the perimeter as a function of web burned.</CAPTION>
<TR><TD><DIV ALIGN="CENTER">
<!-- MATH
$\includegraphics[height=4in]{img/perimeter.ps}$
-->
<IMG
WIDTH="628" HEIGHT="447" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="\includegraphics[height=4in]{img/perimeter.ps}">
</DIV></TD></TR>
</TABLE>
</DIV><P></P>
<P>
<H1><A NAME="SECTION00060000000000000000">
conclusion</A>
</H1>
<P>
The star configuration offer the possibility to design rocket motor
that works at almost constant pressure. It is then possible to
optimize on case thickness and throat diameter in order to obtain the best
performance.
<P>
<H2><A NAME="SECTION00070000000000000000">
Bibliography</A>
</H2><DL COMPACT><DD><P></P><DT><A NAME="nasa">1</A>
<DD> NASA SP-8076, <EM>Solid Propellant Grain Design And
Internal Ballistics</EM>, March 1972
</DL>
<P>
<H1><A NAME="SECTION00080000000000000000">
About this document ...</A>
</H1>
<STRONG>Burning analysis of star configuration</STRONG><P>
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<P>
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Computer Based Learning Unit, University of Leeds.
<BR>Copyright &#169; 1997, 1998, 1999,
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Mathematics Department, Macquarie University, Sydney.
<P>
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