1282 lines
35 KiB
HTML
1282 lines
35 KiB
HTML
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!--Converted with LaTeX2HTML 99.2beta8 (1.42)
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<HTML>
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<HEAD>
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<TITLE>Burning analysis of star configuration</TITLE>
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<META NAME="description" CONTENT="Burning analysis of star configuration">
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<META NAME="keywords" CONTENT="star">
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<META NAME="resource-type" CONTENT="document">
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<META NAME="distribution" CONTENT="global">
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<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
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<META NAME="Generator" CONTENT="LaTeX2HTML v99.2beta8">
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<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
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<LINK REL="STYLESHEET" HREF="star.css">
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</HEAD>
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<BODY >
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<P>
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<H1 ALIGN=CENTER>Burning analysis of star configuration</H1>
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<P ALIGN=CENTER><STRONG>Antoine Lefebvre</STRONG></P>
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<P ALIGN=LEFT></P>
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<P>
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<BR>
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<H2><A NAME="SECTION00010000000000000000">
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Contents</A>
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</H2>
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<!--Table of Contents-->
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<UL>
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<LI><A NAME="tex2html19"
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HREF="star.html">Contents</A>
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<LI><A NAME="tex2html20"
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HREF="star.html#SECTION00020000000000000000">Introduction</A>
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<LI><A NAME="tex2html21"
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HREF="star.html#SECTION00030000000000000000">Geometric definition</A>
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<LI><A NAME="tex2html22"
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HREF="star.html#SECTION00040000000000000000">Analysis</A>
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<UL>
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<LI><A NAME="tex2html23"
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HREF="star.html#SECTION00041000000000000000">Zone 1</A>
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<LI><A NAME="tex2html24"
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HREF="star.html#SECTION00042000000000000000">Zone 2</A>
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<LI><A NAME="tex2html25"
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HREF="star.html#SECTION00043000000000000000">Zone 3</A>
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<LI><A NAME="tex2html26"
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HREF="star.html#SECTION00044000000000000000">Zone 4</A>
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</UL>
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<LI><A NAME="tex2html27"
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HREF="star.html#SECTION00050000000000000000">Design example</A>
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<LI><A NAME="tex2html28"
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HREF="star.html#SECTION00060000000000000000">conclusion</A>
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<LI><A NAME="tex2html29"
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HREF="star.html#SECTION00070000000000000000">Bibliography</A>
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<LI><A NAME="tex2html30"
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HREF="star.html#SECTION00080000000000000000">About this document ...</A>
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</UL>
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<!--End of Table of Contents-->
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<P>
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<P>
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<H1><A NAME="SECTION00020000000000000000">
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Introduction</A>
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</H1>
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<P>
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The design of solid propellant grain that provide neutral burning
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is important to optimize rocket motor performance. The star
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configuration have been widely used to achieve this goal. In this
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report, I will present an analysis of the burning comportement of star
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shape as well as parameter recommandation to achieve better
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performance.
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<P>
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<H1><A NAME="SECTION00030000000000000000">
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Geometric definition</A>
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</H1>
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<P>
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The star could be characterize by seven independant variable as
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defined in figure <A HREF="star.html#star">2</A>. As every star points are identical,
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only one is necessary for the analysis.
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<P>
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<P></P>
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<DIV ALIGN="CENTER"><A NAME="variable"></A><A NAME="12"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
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Geometric definition of star.</CAPTION>
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<TR><TD><DIV ALIGN="CENTER">
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<!-- MATH
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$\includegraphics[height=5in]{img/variable.ps}$
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-->
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<IMG
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WIDTH="498" HEIGHT="401" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.gif"
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ALT="\includegraphics[height=5in]{img/variable.ps}">
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</DIV></TD></TR>
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</TABLE>
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</DIV><P></P>
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<P>
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<P></P>
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<DIV ALIGN="CENTER"><TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE">
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<TD NOWRAP ALIGN="RIGHT"><IMG
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WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img4.gif"
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ALT="$\displaystyle w$"></TD>
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<TD NOWRAP ALIGN="LEFT"><IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.gif"
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ALT="$\displaystyle =$"> Web thickness</TD>
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<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
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</TD></TR>
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<TR VALIGN="MIDDLE">
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<TD NOWRAP ALIGN="RIGHT"><IMG
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WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.gif"
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ALT="$\displaystyle r_1$"></TD>
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<TD NOWRAP ALIGN="LEFT"><IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.gif"
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ALT="$\displaystyle =$"> Radius</TD>
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<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
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</TD></TR>
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<TR VALIGN="MIDDLE">
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<TD NOWRAP ALIGN="RIGHT"><IMG
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WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img7.gif"
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ALT="$\displaystyle r_2$"></TD>
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<TD NOWRAP ALIGN="LEFT"><IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.gif"
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ALT="$\displaystyle =$"> Tip radius</TD>
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<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
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</TD></TR>
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<TR VALIGN="MIDDLE">
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<TD NOWRAP ALIGN="RIGHT"><IMG
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WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.gif"
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ALT="$\displaystyle R$"></TD>
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<TD NOWRAP ALIGN="LEFT"><IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.gif"
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ALT="$\displaystyle =$"> External radius</TD>
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<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
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</TD></TR>
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<TR VALIGN="MIDDLE">
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<TD NOWRAP ALIGN="RIGHT"><IMG
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WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img9.gif"
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ALT="$\displaystyle \eta$"></TD>
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<TD NOWRAP ALIGN="LEFT"><IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.gif"
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ALT="$\displaystyle =$"> angle</TD>
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<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
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</TD></TR>
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<TR VALIGN="MIDDLE">
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<TD NOWRAP ALIGN="RIGHT"><IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img10.gif"
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ALT="$\displaystyle \varepsilon$"></TD>
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<TD NOWRAP ALIGN="LEFT"><IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.gif"
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ALT="$\displaystyle =$"> Secant fillet angle</TD>
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<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
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</TD></TR>
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<TR VALIGN="MIDDLE">
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<TD NOWRAP ALIGN="RIGHT"><IMG
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WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img11.gif"
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ALT="$\displaystyle N$"></TD>
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<TD NOWRAP ALIGN="LEFT"><IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.gif"
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ALT="$\displaystyle =$"> Number of star points</TD>
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<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
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</TD></TR>
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</TABLE></DIV>
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<BR CLEAR="ALL"><P></P>
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<P>
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<P></P>
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<DIV ALIGN="CENTER"><A NAME="star"></A><A NAME="28"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
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Burning zone of the star configuration.</CAPTION>
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<TR><TD><DIV ALIGN="CENTER">
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<!-- MATH
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$\includegraphics[height=4in]{img/mainstar.ps}$
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-->
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<IMG
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WIDTH="516" HEIGHT="392" ALIGN="BOTTOM" BORDER="0"
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SRC="img12.gif"
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ALT="\includegraphics[height=4in]{img/mainstar.ps}">
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</DIV></TD></TR>
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</TABLE>
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</DIV><P></P>
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<P>
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<H1><A NAME="SECTION00040000000000000000">
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Analysis</A>
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</H1>
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<P>
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In this section, an expression for the perimeter of the star will be
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developp for each burning zone as a function of the web thickness
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burned (<IMG
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WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img13.gif"
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ALT="$ w_x$">).
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<P>
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<H2><A NAME="SECTION00041000000000000000">
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Zone 1</A>
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</H2>
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<P>
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The perimeter in the zone one could be divide in three
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sections. Starting by the right, we have the section before the
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radius <IMG
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WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img14.gif"
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ALT="$ r_1$">, which have a radius equal to <IMG
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WIDTH="94" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img15.gif"
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ALT="$ R-w+w_x$">. The length of
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this section is then: <!-- MATH
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$(R-w+w_x)(\pi/N - \varepsilon)$
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-->
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<IMG
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WIDTH="185" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img16.gif"
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ALT="$ (R-w+w_x)(\pi/N - \varepsilon)$">.
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<P>
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Then, we have the perimeter of the arc of initial radius <IMG
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WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img14.gif"
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ALT="$ r_1$">. The
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angle will remain constant to <IMG
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WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img17.gif"
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ALT="$ a$">. The length is then: <!-- MATH
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$(r_1+w_x)a$
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-->
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<IMG
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WIDTH="85" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img18.gif"
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ALT="$ (r_1+w_x)a$">.
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<P>
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The third section is more complicated. The lenght of the line
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starting at the end of the radius <IMG
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WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img14.gif"
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ALT="$ r_1$"> and crossing the vertical
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line will be evaluated first. Then, the perimeter of the radius
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<IMG
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WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.gif"
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ALT="$ r_2$"> will be add to the result, and the length of the line starting
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at the beginning of the radius will be substract.
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<P>
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<P></P>
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<DIV ALIGN="CENTER"><A NAME="len"></A><A NAME="37"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3:</STRONG>
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Determination of the length <IMG
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WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img1.gif"
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ALT="$ L$">.</CAPTION>
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<TR><TD><DIV ALIGN="CENTER">
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<!-- MATH
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$\includegraphics[height=5in]{img/starL.ps}$
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-->
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<IMG
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WIDTH="255" HEIGHT="529" ALIGN="BOTTOM" BORDER="0"
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SRC="img19.gif"
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ALT="\includegraphics[height=5in]{img/starL.ps}">
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</DIV></TD></TR>
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</TABLE>
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</DIV><P></P>
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<P>
|
||
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In order to determine the length, refer to the figure <A HREF="star.html#len">3</A>. The
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lenght <IMG
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WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img1.gif"
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ALT="$ L$"> we are looking for will be equal <IMG
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WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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||
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SRC="img20.gif"
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||
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ALT="$ x + y$">.
|
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<P>
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<P></P>
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<DIV ALIGN="CENTER"><TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE">
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<TD NOWRAP ALIGN="RIGHT"><IMG
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WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img21.gif"
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ALT="$\displaystyle b + z$"></TD>
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<TD NOWRAP ALIGN="LEFT"><IMG
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WIDTH="156" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img22.gif"
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ALT="$\displaystyle = (R-w-r_1)\sin{\varepsilon}$"></TD>
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<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
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</TD></TR>
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<TR VALIGN="MIDDLE">
|
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<TD NOWRAP ALIGN="RIGHT"><IMG
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WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img23.gif"
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ALT="$\displaystyle x$"></TD>
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<TD NOWRAP ALIGN="LEFT"><IMG
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WIDTH="134" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img24.gif"
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ALT="$\displaystyle = (r_1+w_x)\tan{\eta}$"></TD>
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|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
</TD></TR>
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="RIGHT"><IMG
|
||
|
|
WIDTH="40" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img25.gif"
|
||
|
|
ALT="$\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}$"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
</TD></TR>
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="RIGHT"><IMG
|
||
|
|
WIDTH="38" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img27.gif"
|
||
|
|
ALT="$\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}$"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
</TD></TR>
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="RIGHT"><IMG
|
||
|
|
WIDTH="38" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img27.gif"
|
||
|
|
ALT="$\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}$"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
</TD></TR>
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="RIGHT"><IMG
|
||
|
|
WIDTH="80" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img30.gif"
|
||
|
|
ALT="$\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}$"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
</TD></TR>
|
||
|
|
</TABLE></DIV>
|
||
|
|
<BR CLEAR="ALL"><P></P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
We could now simplify this equation using two trigonometric
|
||
|
|
identity:
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<P></P>
|
||
|
|
<DIV ALIGN="CENTER"><TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="RIGHT"><IMG
|
||
|
|
WIDTH="76" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img32.gif"
|
||
|
|
ALT="$\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}$"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
</TD></TR>
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="RIGHT"><IMG
|
||
|
|
WIDTH="42" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img34.gif"
|
||
|
|
ALT="$\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)}}$"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
</TD></TR>
|
||
|
|
</TABLE></DIV>
|
||
|
|
<BR CLEAR="ALL"><P></P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<P></P>
|
||
|
|
<DIV ALIGN="CENTER"><!-- MATH
|
||
|
|
\begin{equation}
|
||
|
|
\begin{split}
|
||
|
|
L &= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} +
|
||
|
|
(r_1+w_x)\left[
|
||
|
|
\frac{\sin^2{\eta}-1}{\cos{\eta}\sin{\eta}} \right]\\
|
||
|
|
&= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} +
|
||
|
|
(r_1+w_x)\left[ \frac{-\cos{\eta}}{\sin{\eta}} \right]\\
|
||
|
|
&= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} +
|
||
|
|
(r_1+w_x)\left[ \frac{-1}{\tan{\eta}} \right]\\
|
||
|
|
&= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} -
|
||
|
|
(r_1+w_x)\tan{(\frac{\pi}{2}-\eta)}\\
|
||
|
|
\end{split}
|
||
|
|
\end{equation}
|
||
|
|
-->
|
||
|
|
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="CENTER"><IMG
|
||
|
|
WIDTH="356" HEIGHT="191" BORDER="0"
|
||
|
|
SRC="img36.gif"
|
||
|
|
ALT="\begin{displaymath}\begin{split}L &= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta...
|
||
|
|
...sin{\eta}} - (r_1+w_x)\tan{(\frac{\pi}{2}-\eta)}\\ \end{split}\end{displaymath}"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
(1)</TD></TR>
|
||
|
|
</TABLE></DIV>
|
||
|
|
<BR CLEAR="ALL"><P></P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
We could now determine the length of the arc and how much we should
|
||
|
|
substract from the length L. Refer to figure <A HREF="star.html#arc">4</A> for the
|
||
|
|
variables.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
|
||
|
|
<P></P>
|
||
|
|
<DIV ALIGN="CENTER"><A NAME="arc"></A><A NAME="96"></A>
|
||
|
|
<TABLE>
|
||
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4:</STRONG>
|
||
|
|
Arc of radius <IMG
|
||
|
|
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img2.gif"
|
||
|
|
ALT="$ 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 < w_x < r_2$">.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<P></P>
|
||
|
|
<DIV ALIGN="CENTER"><!-- MATH
|
||
|
|
\begin{equation}
|
||
|
|
\begin{split}
|
||
|
|
\frac{S}{2N} &= (R-w+w_x)(\frac{\pi}{N} - \varepsilon) +
|
||
|
|
(r_1+w_x)a+(R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}}\\
|
||
|
|
& \quad - (r_1+w_x)\tan{(\frac{\pi}{2}-\eta)}
|
||
|
|
+ (r_2-w_x)(\frac{\pi}{2}-\eta) \\
|
||
|
|
& \quad - (r_2-wx)\tan{(\frac{\pi}{2}-\eta)}\\
|
||
|
|
&= (R-w+w_x)(\frac{\pi}{N} - \varepsilon) +
|
||
|
|
(r_1+w_x)a+(R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}}\\
|
||
|
|
& \quad - (r_1+r_2)\tan{(\frac{\pi}{2}-\eta)}
|
||
|
|
+ (r_2-w_x)(\frac{\pi}{2}-\eta)\\
|
||
|
|
\end{split}
|
||
|
|
\end{equation}
|
||
|
|
-->
|
||
|
|
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="CENTER"><IMG
|
||
|
|
WIDTH="473" HEIGHT="204" BORDER="0"
|
||
|
|
SRC="img41.gif"
|
||
|
|
ALT="\begin{displaymath}\begin{split}\frac{S}{2N} &= (R-w+w_x)(\frac{\pi}{N} - \varep...
|
||
|
|
...c{\pi}{2}-\eta)} + (r_2-w_x)(\frac{\pi}{2}-\eta)\\ \end{split}\end{displaymath}"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
(2)</TD></TR>
|
||
|
|
</TABLE></DIV>
|
||
|
|
<BR CLEAR="ALL"><P></P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
We could now determined the first derivative of this expression to
|
||
|
|
evaluate if it is progressive, regressive or neutral.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<P></P>
|
||
|
|
<DIV ALIGN="CENTER"><!-- MATH
|
||
|
|
\begin{equation}
|
||
|
|
\frac{\delta S}{\delta w_x} = 2N\left[ \frac{\pi}{N} - \varepsilon + a -
|
||
|
|
\frac{\pi}{2} + \eta \right]
|
||
|
|
\end{equation}
|
||
|
|
-->
|
||
|
|
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="CENTER"><IMG
|
||
|
|
WIDTH="251" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img42.gif"
|
||
|
|
ALT="$\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} &= (R-w+w_x)(\frac{\pi}{N} - \varep...
|
||
|
|
...\ & \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} &= 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 < \pi/N$">, a secant
|
||
|
|
fillet <!-- MATH
|
||
|
|
$\varepsilon < \pi/N$
|
||
|
|
-->
|
||
|
|
<IMG
|
||
|
|
WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img54.gif"
|
||
|
|
ALT="$ \varepsilon < \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]$"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
(9)</TD></TR>
|
||
|
|
</TABLE></DIV>
|
||
|
|
<BR CLEAR="ALL"><P></P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
The derivative of this expression become:
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<P></P>
|
||
|
|
<DIV ALIGN="CENTER"><!-- MATH
|
||
|
|
\begin{equation}
|
||
|
|
\begin{split}
|
||
|
|
\frac{\delta S}{\delta w_x} &= 2N \biggl[ \frac{\pi}{N} +
|
||
|
|
\arcsin{\biggl(\frac{R-w-r_1}{r_1+w_x}\sin{\varepsilon}\biggr)} - \\
|
||
|
|
&\quad \frac{w_x(R-w-r_1)\sin{\varepsilon}}{(r_1+w_x)^2\sqrt{1-\frac{(R-w-r_1)^2\sin^2{\varepsilon}}{(r_1+w_x)^2}}} \biggr]\\
|
||
|
|
\end{split}
|
||
|
|
\end{equation}
|
||
|
|
-->
|
||
|
|
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="CENTER"><IMG
|
||
|
|
WIDTH="339" HEIGHT="110" BORDER="0"
|
||
|
|
SRC="img57.gif"
|
||
|
|
ALT="\begin{displaymath}\begin{split}\frac{\delta S}{\delta w_x} &= 2N \biggl[ \frac{...
|
||
|
|
..._1)^2\sin^2{\varepsilon}}{(r_1+w_x)^2}}} \biggr]\\ \end{split}\end{displaymath}"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
(10)</TD></TR>
|
||
|
|
</TABLE></DIV>
|
||
|
|
<BR CLEAR="ALL"><P></P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
It could be demonstrate that the perimeter is progressive in this
|
||
|
|
section. It would be interesting to eliminate the zone 3 in order to
|
||
|
|
keep neutrality as long as possible.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
The condition for the elimination of zone 3 is:
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<P></P>
|
||
|
|
<DIV ALIGN="CENTER"><!-- MATH
|
||
|
|
\begin{equation}
|
||
|
|
Y* = (R-w-r_1)\frac{\sin{\varepsilon}}{\cos{\eta}} - r_1 = w
|
||
|
|
\end{equation}
|
||
|
|
-->
|
||
|
|
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="CENTER"><IMG
|
||
|
|
WIDTH="260" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img58.gif"
|
||
|
|
ALT="$\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}$"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
</TD></TR>
|
||
|
|
</TABLE></DIV>
|
||
|
|
<BR CLEAR="ALL"><P></P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
The perimeter is then:
|
||
|
|
<P></P>
|
||
|
|
<DIV ALIGN="CENTER"><!-- MATH
|
||
|
|
\begin{equation}
|
||
|
|
\begin{split}
|
||
|
|
\frac{S}{2N} = (r_1+w_x) \biggl[ & \varepsilon +
|
||
|
|
\arcsin{\biggl(\frac{R-w-r_1}{r_1+w_x}\sin{\varepsilon}\biggr)}
|
||
|
|
- \pi\\
|
||
|
|
& \quad + \arccos{\biggl(\frac{(r_1+w_x)^2+(R-r_1-w)^2-R^2}{2(r_1+w_x)(R-r_1-w)}\biggr)}
|
||
|
|
\biggr]\\
|
||
|
|
\end{split}
|
||
|
|
\end{equation}
|
||
|
|
-->
|
||
|
|
<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
|
||
|
|
<TR VALIGN="MIDDLE">
|
||
|
|
<TD NOWRAP ALIGN="CENTER"><IMG
|
||
|
|
WIDTH="501" HEIGHT="98" BORDER="0"
|
||
|
|
SRC="img62.gif"
|
||
|
|
ALT="\begin{displaymath}\begin{split}\frac{S}{2N} = (r_1+w_x) \biggl[ & \varepsilon +...
|
||
|
|
...1-w)^2-R^2}{2(r_1+w_x)(R-r_1-w)}\biggr)} \biggr]\\ \end{split}\end{displaymath}"></TD>
|
||
|
|
<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
|
||
|
|
(13)</TD></TR>
|
||
|
|
</TABLE></DIV>
|
||
|
|
<BR CLEAR="ALL"><P></P>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
|
||
|
|
<H1><A NAME="SECTION00050000000000000000">
|
||
|
|
Design example</A>
|
||
|
|
</H1>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
In this section, a star configuration will be design with the
|
||
|
|
theory developp in the previous sections for a motor of <IMG
|
||
|
|
WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img63.gif"
|
||
|
|
ALT="$ 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 >
|
||
|
|
\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>
|
||
|
|
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 © 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 © 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 Lefebvre
|
||
|
|
2001-07-12
|
||
|
|
</ADDRESS>
|
||
|
|
</BODY>
|
||
|
|
</HTML>
|