User:Egm6341.s10.team3.heejun/Meeting 32 transcript

 '''MTG 32: Tue. 23 Mar 10'''


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Note:  Typeset of Transparencies, not lecture transcript  Heejun Chung 20:58, 11 August 2010 (UTC)

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HW: (2) [[media: Egm6341.s10.mtg30.djvu | Page 30-3]] $$ \displaystyle \theta_{P} = 0 ^{\circ},\,\, \theta_{Q} = 60 ^{\circ} $$ Composite arclength(PQ) to $$ \displaystyle \theta(10^{-10}) $$ a) Use error estimate for composite Trapezoidal to find n b) Use successive numerical integration results as stopping criterion, e.g., $$ \displaystyle |I_{2n} - I_{n}| < 10^{-10} $$ c) Verify results of Romberg Integration, Clencurt Integration, Chebfun sum Integration with that of Composite Trapezoidal Rule
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Optimal Control continued, [[media: Egm6341.s10.mtg31.djvu | Page 31-3]]

$$ \displaystyle V= $$ velocity of $$ \displaystyle P $$ (airplane point) $$ \displaystyle T= $$ thrust || airplane body $$ \displaystyle \alpha= \angle (V,T)$$ = angle of attack $$ \displaystyle \gamma= \angle (x,V)$$ = angle between horizontal x-axis and velocity

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$$ \displaystyle D= $$ axial aerodynamic force || $$ \displaystyle T $$ (parallel body of airplane) $$ \displaystyle L= $$ transversal aerodynamic force $$ \displaystyle \perp T $$ $$ \displaystyle W= mg $$ = weight of aircraft

Equations of motion: Control input: $$ \displaystyle \underbrace{T,\, \alpha}_{\color{red}{time\,\, dependent:\,\, T(t),\, \alpha (t)}} $$ State variables: $$ \displaystyle x(t), y(t), V(t), \gamma(t) $$


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 * $$ \displaystyle D= \frac{1}{2} C_{d} \rho V^{2} S_{ref.} $$
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 * $$ \displaystyle L= \frac{1}{2} C_{l} \rho V^{2} S_{ref.} $$
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$$ \displaystyle C_{d}= $$ drag coefficient $$ \displaystyle C_{l}= $$ lift coefficient $$ \displaystyle \rho= $$ air density $$ \displaystyle S_{Ref.}= $$ reference area of aircraft

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 * $$ \displaystyle C_{d} = A_{1} \alpha^{2} + A_{2} \alpha + A_{3} $$
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$$ \displaystyle (A_{1}, A_{2}, A_{3}) = $$ curve fitting coefficients


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 * $$ \displaystyle C_{l} = B_{1} \alpha + B_{2} $$
 *   (2)
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$$ \displaystyle (B_{1}, B_{2}) = $$ curve fitting coefficients


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 * $$ \displaystyle \rho = C_{1} h^{2} + C_{2}h + C_{3} $$
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$$ \displaystyle (C_{1}, C_{2}, C_{3}) = $$ curve fitting coefficients

Ref: S + Z (2007)

Kinematics


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 * $$ \displaystyle \underbrace{\frac{dx}{dt}}_{\color{blue}{\dot{x}}} = V_{x} = V cos \gamma $$
 *   (4)
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 * $$ \displaystyle \dot{y} = V_{y} = V sin \gamma $$
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