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3. Wing lift effectiveness  ${{C}_{{{L}_{\alpha }}}}$  at Mach number  ${{M}_{\infty }}=0.2$

 

 

 

Wing lift curve slope ${{C}_{{{L}_{\alpha }}}}$ in subsonic flow

→ lifting line theory를 수정해서 만들어짐

 

\[{{C}_{{{L}_{\alpha }}}}=\frac{2\pi A}{2+\sqrt{\frac{{{A}^{2}}{{\beta }^{2}}}{{{\kappa }^{2}}}\left( 1+\frac{{{\tan }^{2}}{{\Lambda }_{c/2}}}{{{\beta }^{2}}} \right)+4}}\left( /rad \right)\]

\[\beta =\sqrt{1-M_{\infty }^{2}}:\] Prandl-Glauert subsonic compressibility factor

$\kappa =\frac{{{C}_{{{l}_{\alpha }}}}}{2\pi }:$ 

ratio of section lift effectiveness to the theoretical value of $2\pi $

 

이 식의 물리적 의미는 sweep angle이 있으면 양력효율이 떯어지고

Aspect ratio가 크면 양력효율이 증가한다는 것이다. (특히 aspect ratio가 작을때 (10이하))

 

\[{{C}_{{{L}_{\alpha }}}}=\frac{2\pi \times 5.33}{2+\sqrt{\frac{{{5.33}^{2}}\times {{0.98}^{2}}}{{{0.994}^{2}}}\left( 1+\frac{{{\tan }^{2}}20.56}{{{0.98}^{2}}} \right)+4}}=4.20(/rad)\]

     \[\kappa =\frac{0.109\left( \frac{1}{\deg } \right)}{2\pi \left( \frac{1}{rad} \right)}\times \frac{180(\deg )}{\pi (rad)}=0.994\]

$\beta =0.98$

 

 

 

 

4. Wing zero-lift angle of attack ${{\alpha }_{{{0}_{wing}}}}$

 

\[{{\alpha }_{{{0}_{wing}}}}=\frac{2}{S}\int_{0}^{b/2}{\left( {{\alpha }_{0}}\left( y \right)-\varepsilon \left( y \right) \right)c\left( y \right)dy}\]

※ ${{\alpha }_{0}}$ = 0 대칭형 익형은 0양력 받음각이 0도이다

$=\frac{2}{168.75}\int_{0}^{15}{\left( \left( \frac{y}{5} \right)\left( 7.5-\frac{y}{4} \right) \right)}dy$

$= 1.33deg$

${{\varepsilon }_{t}}=-3(\deg )$

$\varepsilon \left( y \right)=\frac{-3}{15}y(\deg )$

 

 

 

5. Length and position of the mean aerodynamic (geometric) chord  $\bar{c}$

 

 

평균 공력 시위를 구하는 방법은 2가지가 있다. (앞전과 뒷전이 직선이여야 한다)

 

5-1.  $\bar{c}=\frac{2}{3}{{C}_{r}}\frac{1+\lambda +{{\lambda }^{2}}}{1+\lambda }$

\[=\frac{2}{3}\times 7.5ft\times \frac{1+0.5+{{0.5}^{2}}}{1+0.5}=5.833ft\]

 

5-2.  $\bar{c}=\frac{2}{S}\int_{0}^{b/2}{{{c}^{2}}dy}$

$=\frac{2}{168.75}\int_{0}^{15}{{{\left( 7.5-\frac{3.75}{15}y \right)}^{2}}dy=5.833ft}$

 

6. Axial location of the wing aerodynamic center ${{X}_{A{{C}_{wing}}}}$

 

${{X}_{A{{C}_{w\operatorname{in}g}}}}$은 $\bar{c}$(mean aerodynamic chord)에 위치한다.

mean aerodynamic chord의 정확한 위치를 먼저 알아야한다.

 

${{X}_{L{{E}_{MAC}}}}=\frac{2}{S}\int_{0}^{b/2}{{{x}_{LE}}\left( y \right)c\left( y \right)dy}$

※ ${{x}_{LE}}\left( y \right)$ = chordwise location of leading edge at span station y

 

fig.1 ${{x}_{LE}}\left( y \right)$의 위치

※ Taper ratio = $\lambda =\frac{{{C}_{t}}}{{{C}_{r}}}$

 

fig. 2 x에 대한 정의

 

${{X}_{L{{E}_{MAC}}}}=\frac{2}{S}\int_{0}^{b/2}{{{x}_{LE}}\left( y \right)c\left( y \right)dy}$

$=\frac{2}{S}\int_{0}^{b/2}{\lambda y\left( {{C}_{r}}+\frac{\left( {{C}_{t}}-{{C}_{r}} \right)}{\left( b/2 \right)}y \right)dy}$

$=\frac{2}{168.75}\int_{0}^{15}{0.5y\left( 7.5+\frac{\left( 3.75-7.5 \right)}{15}y \right)dy}$

$=3.233ft$

→ Axial position at the wing's MAC (mean aerodynamic chord)

 

${{Y}_{MAC}}=\frac{2}{S}\int_{0}^{b/2}{c\left( y \right)dy}$

$= 6.67ft$

→ Spanwise location of MAC

(equivalent to spanwise location of centroid of area)

 

 

$\therefore \,\,\,{{X}_{A{{C}_{wing}}}}={{X}_{L{{E}_{MAC}}}}+0.25\bar{c}$

$=\,3.233ft\,+\,0.25\times 5.833ft$

※ 여기서 MAC에 0.25를 곱한 이유는 공력중심은 chord에 25%에 위치하기 때문..(대부분)