# Design and Optimization of Modular Solid Rocket Grain Matching Multi-Thrust Performance Curve
## 2 Design Methods
### Burning Perimeter Approximation  
Figure 3 is the geometry of the star grain, where only half of the star angle is present. The relation between the burning perimeter (or burning surface area in 3D) of the star grain and the burned web thickness has the following analytical formula [18]:  
\[s_{\text{b,st}}(w)=\left\{ \begin{array}{l}2n_{\text{st}}\left(\begin{array}[ ]{c}\frac{l\sin(\varepsilon\theta_{\text{st}})}{\sin\left(\frac{\pi}{2}\right) }+l(1-\varepsilon)\beta_{\text{st}}+\\ \left(\frac{\pi}{2}+\beta_{\text{st}}-\frac{\theta}{2}-\cot\left(\frac{\theta }{2}\right)\right)(w+r)\end{array}\right),w<w*\\ 2n_{\text{st}}\left(\begin{array}{c}l(1-\varepsilon)\beta_{\text{st}}+\\ \left(\beta_{\text{st}}+\arcsin\left(\frac{l\sin(\varepsilon\theta_{\text{st} })}{w+r}\right)\right)(w+r)\end{array}\right),w\geq w*\end{array} \right. \tag{1}\]  
Figure 3: Geometry of the star grain (half star angle only). Figure 2: Principle of modular grain. where \(w*\) is web thickness defined by  
\[w*=l\frac{\sin(\varepsilon\beta_{\text{st}})}{\cos\left(\frac{\theta}{2}\right)}-r \tag{2}\]  
The maximum web thickness is  
\[w_{\text{max,st}}=R-l-r \tag{3}\]  
The formula is too complex to make further analysis. To simplify it, a quadric approximation is proposed in this work. During the burning, the star grain will always turn into a tube grain, so the asymptotic line (or linear approximation) of Equation (1) can be written as follows:  
\[s_{\text{b,st}}^{(1)}(w)=2\pi l\left(1+\frac{w+r}{l}\right) \tag{4}\]  
We assume the quadric approximation has the following form:  
\[s_{\text{b,st}}^{(2)}(w)=s_{\text{b,st}}^{(1)}(w)+A(w-(R-l-r))^{2} \tag{5}\]  
This quadric curve must cross the point \((0,s_{\text{b,star}}(0))\). Therefore, the coefficient to be determined (\(A\)) can be solved as follows:  
\[A=\frac{n_{\text{st}}\left(\pi r-r\theta-2r\cot\left(\frac{\theta}{2}\right)+2 l\csc\left(\frac{\theta}{2}\right)\sin(\varepsilon\beta_{\text{st}})\right)-2l \pi e}{\left(l+r-R\right)^{2}} \tag{6}\]  
In the end, the quadric approximation of star grain is written as  
\[s_{\text{b,st}}^{(2)}(w)=a_{\text{st}}+b_{\text{st}}w+c_{\text{st}}w^{2}, \text{where}\left\{ \begin{array}{l}a_{\text{st}}=2\pi(l+r)+A(l+r-R)^{2}\\ b_{\text{st}}=2(\pi+A(l+r-R))\\ c_{\text{st}}=A\end{array} \right. \tag{7}\]  
Figure 4 is the geometry of the slot grain, where only half of the slot is present. Similarly, the burning perimeter of the slot grain has the analytical formula  
\[s_{\text{b,sl}}(w)=2n_{\text{sl}}\left(\sqrt{R^{2}-(w+h)^{2}}-\frac{w+h}{\tan \beta_{\text{sl}}}\right) \tag{8}\]  
The maximum web thickness is  
\[w_{\text{max,sl}}=R\sin\beta_{\text{sl}}-h \tag{9}\]  
Figure 4: Geometry of the slot grain (half star angle only). After performing a first-order Taylor expansion at \(w=0\), the linear approximation of Equation (8) can be derived as follows:  
\[s_{\text{b,sl}}^{(1)}(w)=2n_{\text{sl}}\Big{(}\sqrt{R^{2}-h^{2}}-h\cot\beta_{ \text{sl}}\Big{)}-2n_{\text{sl}}\Big{(}\frac{h}{\sqrt{R^{2}-h^{2}}}+\cot\beta_{ \text{sl}}\Big{)}w \tag{10}\]  
We assume the quadric approximation has the following form:  
\[s_{\text{b,sl}}^{(2)}(w)=s_{\text{b,sl}}^{(1)}(w)+c_{\text{sl}}w^{2} \tag{11}\]  
This curve must go across the point \((R\sin\beta_{\text{sl}}-h,0)\), and one can solve the coefficient to be determined (\(c_{\text{slot}}\)) as  
\[c_{\text{sl}}=\frac{2n_{\text{sl}}R\Big{(}-R+\sqrt{R^{2}-h^{2}}\cos\beta_{ \text{sl}}+h\sin\beta_{\text{sl}}\Big{)}}{\sqrt{R^{2}-h^{2}}(h-R\sin\beta_{ \text{sl}})^{2}} \tag{12}\]  
In the end, the quadric approximation of slot grain is written as  
\[s_{\text{b,sl}}^{(2)}(w)=a_{\text{sl}}+b_{\text{sl}}w+c_{\text{sl}}w^{2},\ \text{where}\ \left\{ \begin{array}{l}a_{\text{sl}}=2n_{\text{sl}}\Big{(}\sqrt{R^{2}-h^{2}} -h\cot\beta_{\text{sl}}\Big{)}\\ b_{\text{sl}}=-2n_{\text{sl}}\Big{(}\frac{h}{\sqrt{R^{2}-h^{2}}}+\cot\beta_{ \text{sl}}\Big{)}\\ c_{\text{sl}}=\frac{2n_{\text{sl}}R\big{(}-R+\sqrt{R^{2}-h^{2}}\cos\beta_{ \text{sl}}+h\sin\beta_{\text{sl}}\big{)}}{\sqrt{R^{2}-h^{2}}(h-R\sin\beta_{ \text{sl}})^{2}}\end{array}