User:EGM6341.s11.team1.Chiu/Mtg34

Mtg 34: Wed, 23 Mar 11


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1) Supersonic intercaptor: Find angle of attack and thrust to reach a given altitude inmin time


 * Nm1.s11.Mtg34.pg1.fig1.svg

Reference: Bryson &amp; Denham 1962

2) Bunt maneuver of aircraft: min altitude to reach target with a bunt maneuver (inverted loop)


 * Nm1.s11.Mtg34.pg1.fig2.svg

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$${\color{blue}V} \ = \ velocity \ of \ P \ (airplane \ = \ pt)$$

$${\color{blue}T} \ = \ thrust \ // \ airplace \ axis$$

$$\alpha \ = \ \angle \ ({\color{blue}V,} \ {\color{red}T} ) \ = \ angle \ of \ attack$$

$$\gamma \ = \ \angle \ (x, \ {color{blue}V}) \ = \ angle \ between \ honz \ x \ axis \ and \ veloctity$$

$$D \ = \ axial \ aero \ force \ // \ {\color{red}T}$$

$$(parall. \ airplane \ axis)$$

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$$L \ = \ Tramsv. \ aero. \ force \ \bot \ {\color{red}T}$$

$$W \ = \ mg \ = \ airplane \ weight.$$

$$Equation \ of \ motion \ (EOM)$$

$${color{blue}Control \ input:} \ {\color{blue} \underset{time \ dependent}{ \underbrace{{\color{red}T}(t)}\, \ \alpha(t)} }$$

$${\color{blue}state \ var,:} \ x(t), \ y(t) \ v(t), \ r(t)$$

$${\color{blue} \begin{cases} & \ {\color{black}D=\frac{L}{2} \ C_{d} \ \rho \ V^{2} \ S_{reft}}  \\ & \ {\color{black}L=\frac{L}{2} \ C_{l} \ \rho \ V^{2} \ S_{reft}} \end{cases}}$$

$$C_{d}=drg \ coefficient \ {\color{green}|} \ \rho=air \ density$$

$$C_{l}=lift \ coefficient \ {\color{green}|} \ \S_{ref}=ref. \ area \ of \ airplane$$

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$$C_{d}=A_{1}\alpha^{2}+A_{2}\alpha+A_{3} \ {\color{red}(1)}$$

$$(A_{1}, \ A_{2}, \ A_{3})=curve \ fitting \ coefficient$$

$$C_{l}=B_{1}\alpha+B_{2} \ {\color{red}(2)}$$

$$(B_{1}, \ B_{2})=curve \ fitting \ coefficient$$

$$\rho=C_{1}h^{2}+C_{2}h+C_{3} \ {\color{red}(3)}$$

$$(C_{1}, \ C_{2}, \ C_{3})=curve \ fitting \ coefficient$$

Reference: S &amp; Z 2007

$${\underline { \color{blue} {Kinematics: }}} \ \frac{dx}{dt}=V_{x}=Vcosr \ {\color{red}(4)}$$

$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{dy}{dt}=V_{y}=Vsinr \ {\color{red}(5)}$$

$${\underline { \color{blue} {Kinematics: }}}{\color{blue}Euler \ equations:} \ \frac{d}{dt} {\underline {P}}= \sum_ {\underline {F}}_ \ {\color{red}(6)}$$

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Note: Parameteriztion of ellipse

$${\color{blue} \underset{cos^{2}t}{ \underbrace}}+{\color{blue} \underset{sin^{2}t}{ \underbrace}}=1 \ {\color{blue}ellipse}$$

<{\color{blue}dl={\color{red}[{\color{blue}dx^{2}+dy^{2}}]^{1/2}}}/math>

$$Eccentricity: \ e= \left(1-\frac{b^{2}}{a^{2}} \right)^{1/2}$$

$$C=\int dl=a \int_{t=0}^{2\pi}[1-e^{2}cos^{2}t]^{1/2}dt$$

$$=41\int_{\alpha=0}^{\frac{\pi}{2}}[1-e^{2}sin^{2}\alpha]^{\frac{1}{2}}d\alpha=4aE(e)$$