Aerodynamic Stability and Performance of Next-Generation ...

C. Local Wind Velocity at the Canopy
The parachute center of pressure can be expressed in the inertial frame via the transformation matrix in Eq. (1).
The inertial coordinates of the center of pressure are found in Eq. (32).
"
$
R cp
= R cp $
# $
cosθ cosψ %
'
sinψ '
−sinθ cosψ
&'
(32)
The inertial angular velocity vector (Ω) of the canopy can be determined by rotating the Euler angle rates back to
the inertial frame, as in Eq. (33).
Downloaded by GEORGIA INST OF TECHNOLOGY on April 1, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-1356
"
$
Ω = $
$
#
cosθ 0 sinθ
0 1 0
−sinθ 0 cosθ
%"
' $
' $
' $
&#
0
θ
0
% "
' $
' + $
' $
& #
cosθ 0 sinθ
0 1 0
−sinθ 0 cosθ
!
#
Ω = #
#
"
#
ψ sinθ
$
θ
&
&
& ψ cosθ
%
&
%"
cosψ −sinψ 0 %"
' $
' $
' $ sinψ cosψ 0 ' $
'
&#
$ 0 0 1 $
&'
#
0
0
ψ
%
'
'
'
&
(33.1)
(33.2)
Knowing the inertial coordinates of the parachute center of pressure and the inertial angular velocity, the
tangential velocity vector (V t ) of the canopy can be determined via Eq. (34).
!
#
V t
= #
#
"
#
x cp
y cp
z cp
V t
= Ω × R cp
(34.1)
$ !
&
−
$
#
θ sinθ cosψ − ψ cosθ sinψ
&
& = R cp
# ψ cosψ &
& #
%
&
"
−θ
&
# cosθ cosψ + ψ sinθ sinψ
%
&
(34.2)
The total wind velocity at the canopy is the sum of the blockage-corrected wind velocity and the wind velocity
due to tangential motion of the canopy. The actual wind velocity vector and magnitude are given in Eq. (35).
!
#
V w
= #
#
"#
V c
0
0
$ !
& #
& −#
& #
%&
"
#
x cp
y cp
z cp
$ !
& #
& = #
& #
%
& #
"
V w
=
V c
+ R cp
( θ sinθ cosψ + ψ cosθ sinψ)
−R cp
ψ cosψ
R cp
( θ cosθ cosψ − ψ sinθ sinψ)
( V c
− x cp ) 2 + y 2 2
cp
+ z cp
$
&
&
&
&
%
(35.1)
(35.2)
Acknowledgments
This work was directed by the Jet Propulsion Laboratory, California Institute of Technology, under a contract
with the National Aeronautics and Space Administration. The authors would like to thank Mark Schoenenberger and
Juan Cruz. Their insight on the data reduction improvements and final results was invaluable.
References
1 Tanner, C., Clark, I., Gallon, J., Rivellini, T., and Witkowski, A., “Aerodynamic Characterization of New Parachute
Configurations for Low-Density Deceleration,” 22 nd AIAA Aerodynamic Decelerator Systems Conference, Daytona Beach, CA,
March 2013.
2 Cruz, J. R., Mineck, R., Keller, D., and Bobskill, M., “Wind Tunnel Testing of Various Disk-Gap-Band Parachutes,”
19
American Institute of Aeronautics and Astronautics
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.