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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 burning
- is important to optimize rocket motor performance. The star
- configuration have been widely used to achieve this goal. In this
- report, I will present an analysis of the burning comportement of star
- shape as well as parameter recommandation to achieve better
- performance.
-
- <P>
-
- <H1><A NAME="SECTION00030000000000000000">
- Geometric definition</A>
- </H1>
-
- <P>
- The star could be characterize by seven independant variable as
- defined in figure <A HREF="star.html#star">2</A>. As every star points are identical,
- only one is necessary for the analysis.
-
- <P>
-
- <P></P>
- <DIV ALIGN="CENTER"><A NAME="variable"></A><A NAME="12"></A>
- <TABLE>
- <CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
- Geometric definition of star.</CAPTION>
- <TR><TD><DIV ALIGN="CENTER">
- <!-- MATH
- $\includegraphics[height=5in]{img/variable.ps}$
- -->
- <IMG
- WIDTH="498" HEIGHT="401" ALIGN="BOTTOM" BORDER="0"
- SRC="img3.gif"
- ALT="\includegraphics[height=5in]{img/variable.ps}">
-
- </DIV></TD></TR>
- </TABLE>
- </DIV><P></P>
-
- <P>
- <P></P>
- <DIV ALIGN="CENTER"><TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
- <TR VALIGN="MIDDLE">
- <TD NOWRAP ALIGN="RIGHT"><IMG
- WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img4.gif"
- ALT="$\displaystyle w$"></TD>
- <TD NOWRAP ALIGN="LEFT"><IMG
- WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img5.gif"
- ALT="$\displaystyle =$"> Web thickness</TD>
- <TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </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="$\displaystyle =$"> Radius</TD>
- <TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </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="$\displaystyle =$"> Tip radius</TD>
- <TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </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="$\displaystyle =$"> External radius</TD>
- <TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </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="$\displaystyle =$"> angle</TD>
- <TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </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="$\displaystyle =$"> Secant fillet angle</TD>
- <TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </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="$\displaystyle =$"> Number of star points</TD>
- <TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
- </TABLE></DIV>
- <BR CLEAR="ALL"><P></P>
-
- <P>
-
- <P></P>
- <DIV ALIGN="CENTER"><A NAME="star"></A><A NAME="28"></A>
- <TABLE>
- <CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
- Burning zone of the star configuration.</CAPTION>
- <TR><TD><DIV ALIGN="CENTER">
- <!-- MATH
- $\includegraphics[height=4in]{img/mainstar.ps}$
- -->
- <IMG
- WIDTH="516" HEIGHT="392" ALIGN="BOTTOM" BORDER="0"
- SRC="img12.gif"
- ALT="\includegraphics[height=4in]{img/mainstar.ps}">
-
- </DIV></TD></TR>
- </TABLE>
- </DIV><P></P>
-
- <P>
-
- <H1><A NAME="SECTION00040000000000000000">
- Analysis</A>
- </H1>
-
- <P>
- In this section, an expression for the perimeter of the star will be
- developp for each burning zone as a function of the web thickness
- burned (<IMG
- WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img13.gif"
- ALT="$ w_x$">).
-
- <P>
-
- <H2><A NAME="SECTION00041000000000000000">
- Zone 1</A>
- </H2>
-
- <P>
- The perimeter in the zone one could be divide in three
- sections. Starting by the right, we have the section before the
- radius <IMG
- WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img14.gif"
- ALT="$ r_1$">, which have a radius equal to <IMG
- WIDTH="94" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
- SRC="img15.gif"
- ALT="$ R-w+w_x$">. The length of
- this section is then: <!-- MATH
- $(R-w+w_x)(\pi/N - \varepsilon)$
- -->
- <IMG
- WIDTH="185" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
- SRC="img16.gif"
- ALT="$ (R-w+w_x)(\pi/N - \varepsilon)$">.
-
- <P>
- Then, we have the perimeter of the arc of initial radius <IMG
- WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img14.gif"
- ALT="$ r_1$">. The
- angle will remain constant to <IMG
- WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
- SRC="img17.gif"
- ALT="$ a$">. The length is then: <!-- MATH
- $(r_1+w_x)a$
- -->
- <IMG
- WIDTH="85" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
- SRC="img18.gif"
- ALT="$ (r_1+w_x)a$">.
-
- <P>
- The third section is more complicated. The lenght of the line
- starting at the end of the radius <IMG
- WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img14.gif"
- ALT="$ r_1$"> and crossing the vertical
- line will be evaluated first. Then, the perimeter of the radius
- <IMG
- WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$ r_2$"> will be add to the result, and the length of the line starting
- at the beginning of the radius will be substract.
-
- <P>
-
- <P></P>
- <DIV ALIGN="CENTER"><A NAME="len"></A><A NAME="37"></A>
- <TABLE>
- <CAPTION ALIGN="BOTTOM"><STRONG>Figure 3:</STRONG>
- Determination of the length <IMG
- WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
- SRC="img1.gif"
- ALT="$ L$">.</CAPTION>
- <TR><TD><DIV ALIGN="CENTER">
- <!-- MATH
- $\includegraphics[height=5in]{img/starL.ps}$
- -->
- <IMG
- WIDTH="255" HEIGHT="529" ALIGN="BOTTOM" BORDER="0"
- SRC="img19.gif"
- ALT="\includegraphics[height=5in]{img/starL.ps}">
-
- </DIV></TD></TR>
- </TABLE>
- </DIV><P></P>
-
- <P>
- In order to determine the length, refer to the figure <A HREF="star.html#len">3</A>. The
- lenght <IMG
- WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
- SRC="img1.gif"
- ALT="$ L$"> we are looking for will be equal <IMG
- WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img20.gif"
- ALT="$ x + y$">.
-
- <P>
- <P></P>
- <DIV ALIGN="CENTER"><TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
- <TR VALIGN="MIDDLE">
- <TD NOWRAP ALIGN="RIGHT"><IMG
- WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
- SRC="img21.gif"
- ALT="$\displaystyle b + z$"></TD>
- <TD NOWRAP ALIGN="LEFT"><IMG
- WIDTH="156" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
- SRC="img22.gif"
- ALT="$\displaystyle = (R-w-r_1)\sin{\varepsilon}$"></TD>
- <TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
- <TR VALIGN="MIDDLE">
- <TD NOWRAP ALIGN="RIGHT"><IMG
- WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img23.gif"
- ALT="$\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}$"></TD>
- <TD NOWRAP WIDTH="10" ALIGN="RIGHT">
- </TD></TR>
- <TR VALIGN="MIDDLE">
- <TD NOWRAP ALIGN="RIGHT"><IMG
- WIDTH="40" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img25.gif"
- ALT="$\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>
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