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<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 burningis important to optimize rocket motor performance. The starconfiguration have been widely used to achieve this goal. In thisreport, I will present an analysis of the burning comportement of starshape as well as parameter recommandation to achieve betterperformance.
<P>
<H1><A NAME="SECTION00030000000000000000">Geometric definition</A></H1>
<P>The star could be characterize by seven independant variable asdefined 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 bedevelopp for each burning zone as a function of the web thicknessburned (<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 motorthat works at almost constant pressure. It is then possible tooptimize on case thickness and throat diameter in order to obtain the bestperformance.
<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 AndInternal Ballistics</EM>, March 1972</DL>
<P>
<H1><A NAME="SECTION00080000000000000000">About this document ...</A></H1> <STRONG>Burning analysis of star configuration</STRONG><P>This document was generated using the<A HREF="http://www-texdev.mpce.mq.edu.au/l2h/docs/manual/"><STRONG>LaTeX</STRONG>2<tt>HTML</tt></A> translator Version 99.2beta8 (1.42)<P>Copyright &#169; 1993, 1994, 1995, 1996,<A HREF="http://cbl.leeds.ac.uk/nikos/personal.html">Nikos Drakos</A>, Computer Based Learning Unit, University of Leeds.<BR>Copyright &#169; 1997, 1998, 1999,<A HREF="http://www.maths.mq.edu.au/~ross/">Ross Moore</A>, Mathematics Department, Macquarie University, Sydney.<P>The command line arguments were: <BR> <STRONG>latex2html</STRONG> <TT>-white -image_type gif -no_navigation -split 0 -dir html -mkdir star.tex</TT><P>The translation was initiated by Antoine Lefebvre on 2001-07-12<BR><HR><ADDRESS>Antoine Lefebvre2001-07-12</ADDRESS></BODY></HTML>