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408 lines
12 KiB
408 lines
12 KiB
\documentclass[11pt, titlepage]{article}
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\usepackage{amsmath, graphicx}
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\begin{document}
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\author{Antoine Lefebvre}
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\title{Burning analysis of star configuration}
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\maketitle
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\tableofcontents
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\newpage
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\section{Introduction}
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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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\section{Geometric definition}
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The star could be characterize by seven independant variable as
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defined in figure \ref{star}. As every star points are identical,
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only one is necessary for the analysis.
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\begin{figure}
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\begin{center}
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\includegraphics[height=5in]{img/variable.ps}
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\caption{Geometric definition of star.}\label{variable}
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\end{center}
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\end{figure}
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\begin{align*}
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w &= \text{Web thickness}\\
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r_1 &= \text{Radius}\\
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r_2 &= \text{Tip radius}\\
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R &= \text{External radius}\\
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\eta &= \text{angle}\\
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\varepsilon &= \text{Secant fillet angle}\\
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N &= \text{Number of star points}\\
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%S &= \text{burning perimeter as a function of } wx\\
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\end{align*}
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\begin{figure}
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\begin{center}
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\includegraphics[height=4in]{img/mainstar.ps}
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\caption{Burning zone of the star configuration.}\label{star}
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\end{center}
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\end{figure}
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\section{Analysis}
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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 ($w_x$).
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\subsection{Zone 1}
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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 $r_1$, which have a radius equal to $R-w+w_x$. The length of
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this section is then: $(R-w+w_x)(\pi/N - \varepsilon)$.
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Then, we have the perimeter of the arc of initial radius $r_1$. The
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angle will remain constant to $a$. The length is then: $(r_1+w_x)a$.
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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 $r_1$ and crossing the vertical
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line will be evaluated first. Then, the perimeter of the radius
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$r_2$ 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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\begin{figure}
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\begin{center}
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\includegraphics[height=5in]{img/starL.ps}
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\caption{Determination of the length $L$.}\label{len}
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\end{center}
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\end{figure}
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In order to determine the length, refer to the figure \ref{len}. The
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lenght $L$ we are looking for will be equal $x + y$.
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\begin{align*}
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b + z &= (R-w-r_1)\sin{\varepsilon}\\
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x &= (r_1+w_x)\tan{\eta}\\
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\cos{\eta} &= \frac{r_1+w_x}{z}\\
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\sin{\eta} &= \frac{b}{y}\\
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\sin{\eta} &= \frac{(R-w-r_1)\sin{\varepsilon} -
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\frac{r_1+w_x}{\cos{\eta}}}{y}\\
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L = y + x &= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} -
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\frac{r_1+w_x}{\cos{\eta}\sin{\eta}} +
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(r_1+w_x)\tan{\eta}
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\end{align*}
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We could now simplify this equation using two trigonometric
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identity:
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\begin{align*}
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\sin^2{\eta} -1 &= -\cos^2{\eta}\\
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\tan{\eta} &= \frac{1}{\tan{(\frac{\pi}{2}-\eta)}}
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\end{align*}
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\begin{equation}
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\begin{split}
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L &= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} +
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(r_1+w_x)\left[
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\frac{\sin^2{\eta}-1}{\cos{\eta}\sin{\eta}} \right]\\
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&= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} +
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(r_1+w_x)\left[ \frac{-\cos{\eta}}{\sin{\eta}} \right]\\
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&= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} +
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(r_1+w_x)\left[ \frac{-1}{\tan{\eta}} \right]\\
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&= (R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}} -
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(r_1+w_x)\tan{(\frac{\pi}{2}-\eta)}\\
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\end{split}
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\end{equation}
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We could now determine the length of the arc and how much we should
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substract from the length L. Refer to figure \ref{arc} for the
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variables.
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\begin{figure}
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\begin{center}
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\includegraphics[height=3in]{img/arc.ps}
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\caption{Arc of radius $r_2$.}\label{arc}
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\end{center}
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\end{figure}
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$$\text{Arc length} = (r_2-wx)(\frac{\pi}{2}-\eta)$$
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$$x = \frac{r_2-wx}{\tan{\eta}}$$
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We have now the complete expression of the perimeter of the star as a
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function of web burned ($w_x$) in the zone one. This expression is
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valid for $0 < w_x < r_2$.
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\begin{equation}
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\begin{split}
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\frac{S}{2N} &= (R-w+w_x)(\frac{\pi}{N} - \varepsilon) +
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(r_1+w_x)a+(R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}}\\
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& \quad - (r_1+w_x)\tan{(\frac{\pi}{2}-\eta)}
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+ (r_2-w_x)(\frac{\pi}{2}-\eta) \\
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& \quad - (r_2-wx)\tan{(\frac{\pi}{2}-\eta)}\\
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&= (R-w+w_x)(\frac{\pi}{N} - \varepsilon) +
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(r_1+w_x)a+(R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}}\\
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& \quad - (r_1+r_2)\tan{(\frac{\pi}{2}-\eta)}
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+ (r_2-w_x)(\frac{\pi}{2}-\eta)\\
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\end{split}
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\end{equation}
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We could now determined the first derivative of this expression to
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evaluate if it is progressive, regressive or neutral.
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\begin{equation}
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\frac{\delta S}{\delta w_x} = 2N\left[ \frac{\pi}{N} - \varepsilon + a -
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\frac{\pi}{2} + \eta \right]
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\end{equation}
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We could verify that:
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$$a = \frac{\pi}{2} - \eta + \varepsilon$$
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Our expression become:
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\begin{equation}
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\frac{\delta S}{\delta w_x} = 2\pi
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\end{equation}
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The perimeter in zone 1 will always be progressive. So, it is
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important to minimize the radius $r_2$ in order to switch as fast as
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possible to the zone 2.
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\subsection{Zone 2}
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The expression for the perimeter in the second zone is almost the
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same as in the zone one. The difference is that the radius $r_2$ had
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vanish and the expression reduce to a simpler one:
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\begin{equation}
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\begin{split}
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\frac{S}{2N} &= (R-w+w_x)(\frac{\pi}{N} - \varepsilon) +
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(r_1+w_x)a+(R-w-r_1)\frac{\sin{\varepsilon}}{\sin{\eta}}\\
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& \quad - (r_1+w_x)\tan{(\frac{\pi}{2}-\eta)}\\
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\end{split}
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\end{equation}
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The derivative of this expression is:
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\begin{equation}
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\begin{split}
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\frac{\delta S}{\delta w_x} &= 2N\left[ \frac{\pi}{N} -\varepsilon +
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a - \tan{(\frac{\pi}{2} - \eta)} \right]\\
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&= 2N\left[ \frac{\pi}{2} - \eta +
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\frac{\pi}{N} - \tan{(\frac{\pi}{2}
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- \eta)} \right]\\
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\end{split}
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\end{equation}
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As we could see in this expression, the progressivity in zone 2 is
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determined by the angle $\eta$ and by the number of star point
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$N$. It is independant of the angle $\varepsilon$.
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The zone 2 will be predominant during the motor burn time and we
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would like to provide neutrallity in this zone. Neutrality is obtain
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when the derivative of the perimeter is equal to zero. This lead to
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the following equation:
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\begin{equation}
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\frac{\delta S}{\delta w_x} = 0 = \frac{\pi}{2} - \eta +
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\frac{\pi}{N} - \tan{(\frac{\pi}{2}
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- \eta)}
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\end{equation}
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Which reduce to the following implicit equation of $\eta$ as a
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function of $N$:
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\begin{equation}
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\eta = \frac{\pi}{N} - \tan{(\frac{\pi}{2} - \eta)} + \frac{\pi}{2}
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\end{equation}
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Solution of this equation give values of the angle $\eta$ to obtain
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neutrality in zone 2 as a function of the number of star points.
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\begin{center}
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\begin{tabular}{||c|c|c||}
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\hline
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\hline
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$N$ & $\eta$ (deg) & $\pi/N$ (deg) \\
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\hline
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3 & 24.55 & 60.00\\
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4 & 28.22 & 45.00\\
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5 & 31.13 & 36.00\\
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6 & 33.53 & 30.00\\
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7 & 35.56 & 25.71\\
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8 & 37.31 & 22.50\\
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9 & 38.84 & 20.00\\
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\hline
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\end{tabular}
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\end{center}
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It is important to note that when the angle $\eta < \pi/N$, a secant
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fillet $\varepsilon < \pi/N$ will be necessary to prevent star point
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from overlapping. In general, $\varepsilon$ should always be smaller
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that $\pi/N$.
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\subsection{Zone 3}
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The perimeter in the zone 3 begin when $w_x = Y*$. The angle $a$
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become progressivly smaller when propellant burned. Perimeter could
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be expressed like this:
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\begin{equation}
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\frac{S}{2N} = (R-w+w_x)(\frac{\pi}{N}-\varepsilon) +
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(r_1+w_x)\left[ \varepsilon +
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\arcsin{(\frac{R-w-r_1}{r_1+w_x}\sin{\varepsilon}}) \right]
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\end{equation}
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The derivative of this expression become:
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\begin{equation}
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\begin{split}
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\frac{\delta S}{\delta w_x} &= 2N \biggl[ \frac{\pi}{N} +
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\arcsin{\biggl(\frac{R-w-r_1}{r_1+w_x}\sin{\varepsilon}\biggr)} - \\
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&\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]\\
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\end{split}
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\end{equation}
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It could be demonstrate that the perimeter is progressive in this
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section. It would be interesting to eliminate the zone 3 in order to
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keep neutrality as long as possible.
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The condition for the elimination of zone 3 is:
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\begin{equation}
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Y* = (R-w-r_1)\frac{\sin{\varepsilon}}{\cos{\eta}} - r_1 = w
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\end{equation}
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This equation reduce to:
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\begin{equation}
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\sin{\varepsilon} = \frac{w+r_1}{R-w-r_1}\cos{\eta}
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\end{equation}
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Now, the angle $\varepsilon$ is determine by the web thickness $w$,
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the radius $r_1$ and the angle $\eta$. As $\eta$ was determine by
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the number of star points $N$ and the radius may be dictate by
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technical decision, the web thickness $w$ will determine
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$\varepsilon$.
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\subsection{Zone 4}
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The analytical solution of the perimeter in the zone 4 could be
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found with the help of the cosinus law:
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\begin{equation*}
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c^2 = a^2 + b^2 - 2ab\cos{\theta}
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\end{equation*}
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The perimeter is then:
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\begin{equation}
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\begin{split}
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\frac{S}{2N} = (r_1+w_x) \biggl[ & \varepsilon +
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\arcsin{\biggl(\frac{R-w-r_1}{r_1+w_x}\sin{\varepsilon}\biggr)}
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- \pi\\
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& \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)}
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\biggr]\\
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\end{split}
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\end{equation}
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\section{Design example}
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In this section, a star configuration will be design with the
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theory developp in the previous sections for a motor of $3 inch$
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internal diameter.
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The goal is to have a perimeter that will remain as constant as
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possible to mainatin neutrality. It will also be interesting to
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minimize the number of star points in order to reduce the difficulty
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to cast the propellant. We could also try to optimize the volumetric
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loading.
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% \subsection{Solution}
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First of all, we could determine the number of star points. In order
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to maximize the quantity of matter, the angle $\varepsilon$ should
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be equal to $\pi/N$. In order to obtain this condition, the angle
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$\eta$ should be larger than $\pi/N$.
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If we refer to the table of the angle $\eta$ in function of $N$, to
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obtain neutrality in zone 2, we must choose $N=6$ to have $\eta >
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\pi/N$.
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Three conditions are now determine:
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$$N = 6$$
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$$\eta = 33.53 deg$$
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$$\varepsilon = 30 deg$$
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We must now found the web thickness $w$ and radius $r_1$ that fit
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the conditions. A radius $r_1 = 1/16 in$ is reasonable technically.
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The equation to be solve is the following:
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$$sin{\varepsilon} = \frac{w+r_1}{R-w-r_1}\cos{\eta}$$
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The value of $w$ that solve this equation is:
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$$w = 0.500$$
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The seven independant variable are now fixed. The resulting shape
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could be seen in figure \ref{res}.
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\begin{figure}
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\begin{center}
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\includegraphics[]{img/res.ps}
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\caption{Resulting star configuration for the 3 inch motor.}\label{res}
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\end{center}
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\end{figure}
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With the functions developp in the report, the evolution of the
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perimeter as a function of the web burned could be plot.
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\begin{figure}
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\begin{center}
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\includegraphics[height=4in]{img/perimeter.ps}
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\caption{Graphic of the perimeter as a function of web burned.}\label{grah}
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\end{center}
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\end{figure}
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\section{conclusion}
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The star configuration offer the possibility to design rocket motor
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that works at almost constant pressure. It is then possible to
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optimize on case thickness and throat diameter in order to obtain the best
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performance.
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\begin{thebibliography}{}
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\bibitem{nasa} NASA SP-8076, {\em Solid Propellant Grain Design And
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Internal Ballistics}, March 1972
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\end{thebibliography}
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\end{document}
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