Aerodynamic Stability and Performance of Next-Generation ...
Aerodynamic Stability and Performance of Next-Generation ...
Aerodynamic Stability and Performance of Next-Generation ...
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The derivatives <strong>of</strong> the tangential canopy velocity in the pitch plane ( γ θ<br />
) <strong>and</strong> the actual wind velocity in the pitch<br />
plane can be found by differentiating Eqs. (17), (18) <strong>and</strong> (19.1) <strong>and</strong> are calculated via Eqs. (27.1.1), (27.1.2), <strong>and</strong><br />
(27.1.3) respectively. The derivatives <strong>of</strong> the tangential canopy velocity in the yaw plane ( γ ψ ) <strong>and</strong> the actual wind<br />
velocity in the yaw plane can be found in the same way <strong>and</strong> are calculated via Eqs. (27.2.1), (27.2.2), <strong>and</strong> (27.2.3)<br />
respectively.<br />
V tθ<br />
= R <br />
cp<br />
θ (27.1.1)<br />
γ θ<br />
= θ <br />
(27.1.2)<br />
Downloaded by GEORGIA INST OF TECHNOLOGY on April 1, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-1356<br />
V wθ<br />
=<br />
V wψ<br />
=<br />
V tθ ( V tθ<br />
−V c<br />
cosγ θ ) + γ θ<br />
V c<br />
V tθ<br />
sinγ θ<br />
(27.1.3)<br />
V wθ<br />
V tψ<br />
V tψ<br />
= R <br />
cp<br />
θ (27.2.1)<br />
γ ψ<br />
= ψ (27.2.2)<br />
( V tψ<br />
−V c<br />
cosγ<br />
ψ ) + γ<br />
ψ<br />
V c<br />
V tψ<br />
sinγ<br />
ψ<br />
(27.2.3)<br />
V wψ<br />
The second derivative <strong>of</strong> the total angle <strong>of</strong> attack can be calculated by twice differentiating Eq. (23.1). The<br />
remaining derivation proceeds in the same fashion as for the first derivative (given in Eqs. (24) through (27)).<br />
α T<br />
= α sinα cosβ + β cosα sin β + α 2 + β 2 2<br />
( − α T )cosα T<br />
− 2 α β sinα sin β<br />
(28)<br />
sinα T<br />
Δ α =<br />
Δ β =<br />
V<br />
wθ<br />
= 1<br />
α = θ + Δ α (29.1)<br />
β = ψ + Δ β (29.2)<br />
( )( V wθ ) − ( V tθ<br />
V wθ<br />
−V Vwθ<br />
<br />
tθ )( 2V Vwθ<br />
<br />
wθ )<br />
# V V −V 2<br />
V<br />
% t θ wθ tθ wθ<br />
sinγ<br />
4 θ<br />
+<br />
1 %<br />
V wθ<br />
%<br />
cosΔα % V<br />
2 tθ<br />
V wθ<br />
−V tθ<br />
V wθ<br />
γ<br />
2 θ<br />
cosγ θ<br />
+ V t θ<br />
%<br />
γ 2 θ<br />
sinγ θ<br />
+ γ θ<br />
cosγ θ<br />
$ V wθ<br />
V wθ<br />
# V V −V 2<br />
( V t ψ wψ t ψ wψ )( V wψ ) − (<br />
V tψ<br />
V wψ<br />
−V tψ<br />
V wψ ) 2V wψ<br />
V<br />
%<br />
( wψ )<br />
sinγ<br />
4 ψ<br />
+<br />
1 %<br />
V wψ<br />
%<br />
cosΔβ % V tψ<br />
V wψ<br />
−V Vwψ<br />
<br />
tψ<br />
2<br />
γ<br />
2<br />
ψ<br />
cosγ ψ<br />
+ V t ψ<br />
%<br />
γ 2 ψ<br />
sinγ ψ<br />
+ γ ψ<br />
cosγ ψ<br />
$ % V wψ<br />
V wψ<br />
V wθ<br />
V<br />
wψ<br />
= 1<br />
V wψ<br />
"<br />
#<br />
Vtθ 2 + V tθ<br />
V tθ<br />
−V c<br />
cosγ θ<br />
"<br />
Vtψ<br />
<br />
#<br />
2 + <br />
V tψ<br />
&<br />
(<br />
(<br />
(<br />
(30.1)<br />
( ) + Δ α 2 (<br />
sinΔα(<br />
'<br />
&<br />
(<br />
(<br />
(<br />
(30.2)<br />
( ) + Δ (<br />
β 2 sinΔβ(<br />
'(<br />
V<br />
t θ<br />
= R <br />
cp<br />
θ (31.1.1)<br />
γ θ<br />
= θ (31.1.2)<br />
( ) + 2 γ θ<br />
V <br />
cVtθ sinγ θ<br />
+V c<br />
V tθ ( γ θ<br />
sinγ θ<br />
+ γ 2 θ<br />
cosγ θ ) −V 2<br />
wθ<br />
$<br />
% (31.1.3)<br />
V<br />
t ψ<br />
= R cp<br />
ψ (31.2.1)<br />
γ ψ<br />
= ψ (31.2.2)<br />
( V tψ<br />
−V c<br />
cosγ ψ ) + 2 γ ψ<br />
V <br />
cVtψ sinγ ψ<br />
+V c<br />
V tψ<br />
γ ψ<br />
sinγ ψ<br />
+ γ 2 ψ<br />
cosγ ψ<br />
( ) −V 2$<br />
wψ<br />
% (31.2.3)<br />
18<br />
American Institute <strong>of</strong> Aeronautics <strong>and</strong> Astronautics<br />
This material is declared a work <strong>of</strong> the U.S. Government <strong>and</strong> is not subject to copyright protection in the United States.
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