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Transcript of Wey Corriveau
New Orleans 2008-09International Symposium on Ballistics 1
Trajectory Deflection of Fin- and Spin-Stabilized Projectiles
Using Paired Lateral Impulses
Pierre [email protected]
Daniel [email protected]
New Orleans 2008-09International Symposium on Ballistics 2
Presentation Overview
Objective
Effect of a Lateral Impulse
Basic Mathematical Model and Analytical Solution
Pairing the Impulses
Examples
Conclusions
New Orleans 2008-09International Symposium on Ballistics 3
Objective
To developed a well defined procedure for using paired impulses on fin- and spin-stabilized projectiles in order to achieve enhanced drift corrections
New Orleans 2008-09International Symposium on Ballistics 4
Effect of a Lateral Impulse
Dirac impulse J applied off the center of mass
Trajectory deflection: total lateral impulse = J + JL+ JY
Lift (α)
α total incidence angle
Lift (α) dt∫Additional velocity decrease: axial impulse = JD2
Drag (α2) dt∫
Drag (α2)Magnus (α, p)
Magnus (α) dt∫
More challenging to implement properly
(1/2)
New Orleans 2008-09International Symposium on Ballistics 5
Effect of a Lateral Impulse(2/2)
Basic idea: - J1 triggers the angular motion at s1
- J2 stops the angular motion at s2
Advantages: - the lateral correction is enhanced by the lift impulse resulting from the angular motion
- the range lost is minimized by stopping the angular motion
β
α
)( 21 s′ξ
J1
J2
s2
tota
l inc
iden
ce
s1)( 22 s′ξ
β
α
0ξ ′
JLJ < 0
Pairing lateral impulses off the center of mass
New Orleans 2008-09International Symposium on Ballistics 6
Linearized equation of the complex incidence motion:
)()()( 0 siPTMiPH δξξξξ ′=+−′−+′′
Dirac impulse
Basic Mathematical Model (1/4)
Damping factor+ vel. change ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛−−= mq
yDL C
Imd
CCmAd
H2
2 αρ
Gyroscopic effectVpd
II
Py
x=
Overturning moment (restoring or destabilizing) α
ρm
yC
IdA
M2
3=
Magnus moment + vel. change ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛+= αα
ρnp
xL C
Imd
CmAd
T2
2
New Orleans 2008-09International Symposium on Ballistics 7
Basic Mathematical Model (2/4)
( ) ( )siiS
siiF SSSFFF eeKeeK φλφφλφ
ξ ′+′+ += 00
00
β
α
ξ
KS
KF Fast arm
Slow arm
epicyclic motion
Fφ ′
Sφ ′
Motion = sum of two rotating arms:
⎟⎠⎞
⎜⎝⎛ −+=′ MPPF 4
21 2φ ⎟
⎠⎞
⎜⎝⎛ −−=′ MPPS 4
21 2φ
Angular frequencies
( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−−−=
MP
HTPHF
4
221
2λ
( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−+−=
MP
HTPHS
4
221
2λ
Damping factors
SF
iF
SF ieK
φφξφξφ
′−′
′+′−= 000
0SF
FiS
ieK S
φφξφξφ
′−′
′+′= 000
0
Initial arms
New Orleans 2008-09International Symposium on Ballistics 8
Basic Mathematical Model (3/4)
Initial conditions forced by the Dirac impulse:
mVJ
−=0ξ
VIdLJ
y
J2
0 =′ξ
main cause of angular motion(unless LJ → 0)
negligible in supersonic mode
β
α
0ξ ′
JLJ < 0
New Orleans 2008-09International Symposium on Ballistics 9
∫∞
=02
1dsCdVAJ LL ξρ α
SS
iS
FF
iF
i
eK
i
eKds
SF
φλφλξ
φφ
′+−
′+−=∫
∞ 00
00
0∫∞
=0
)( dttLJL
( ) ( ) ( ) ( )( ) ( )22
22
0
2 00000000sincos2
22 SFSF
SFSFSFSFSF
S
S
F
FKKKK
dsφφλλ
φφφφφφλλ
λλξ
′−′++
⎥⎦⎤
⎢⎣⎡ −′−′+−+
−−−=∫∞
∫∞
=0
2 )(2 dttDJD
Lateral impulses: Lift + Magnus
Axial impulse: additional Drag
∫∞
=0
222 2
1dsCdVAJ DD ξρα2
20 sinDDD CCC +=
LL
YpY J
Vdp
C
CiJ
α
α=∫∞
=0
dt)t(YJY
Basic Mathematical Model (4/4)
New Orleans 2008-09International Symposium on Ballistics 10
Pairing the Impulses: LocationGoal: maximizing J + JL
Fin-stabilized projectile
JL
0ξ ′J
LJ > 0
Spin-stabilized projectile
0ξ ′
JL
LJ < 0
J
φ
α
α Δ−= i
m
LJL e
CC
LJJRule #1: the lateral impulse must be applied
- ahead of the center of mass for fin-stab. shells- behind the center of mass for spin-stab. shells
(1/2)
New Orleans 2008-09International Symposium on Ballistics 11
Goal: maximizing J + JL for spin-stabilized projectile
0ξ ′
JL
L J<
0
J
Rule #1: the lateral impulse must be applied- ahead of the center of mass for fin-stab. shells- behind the center of mass for spin-stab. shells
β
α
)(0′ξ
JL
J β
α
JL
J
0ξ ′
JL
L J>
0
J
How to do it: How not to do it:
)(0′ξ
Pairing the Impulses: Location (2/2)
New Orleans 2008-09International Symposium on Ballistics 12
Pairing the Impulses: Orientation
- independent of (s2 - s1)- maximum if J1 and J2 are aligned
β
α
J2J1
J1L
J2LTotal lateral impulse = (J1 + J1L) + (J2 + J2L)
Rule #2: J1 and J2 must be triggered at the same roll angle
New Orleans 2008-09International Symposium on Ballistics 13
Pairing the Impulses: Timingα
β
)( 121 ss −′ξ
)(2 0′ξ
J1 J2
ξ1 ξ2
s1 s2
0)()(, 212 =+≥∀ ssss ξξ
21 ξξξ +=
Linearized equation of motion:
)( 121 ss −′ξ )(2 0′ξ= −
)( 121 ss −ξ )(2 0ξ 0==Motion strictly opposed if:
SFkss
φφπ
′−′=−
212
⎟⎟⎠
⎞⎜⎜⎝
⎛−
′
′′= 1
2)(sign
to integer nearest S
FSkφφφ
Rule #3:
J1
J2
0 s2 - s1
tota
l inc
iden
ce
SF φφπ
′−′2
k = 4
New Orleans 2008-09International Symposium on Ballistics 14
Example: GSP Shell
LJ 2.5 cal
J1 2.0 N.s
J2 1.4 N.s
s2 - s1 885 cal
⎯αmax 10.1°
d m Iy Mach p CD0 CLα CYpα Cmα Cmq30 mm 0.7 kg 5.04e-3 kgm2 0.193.5 22 Hz 7.6 ~ 0 -5.3 ~ -300
VJ 4.75 m/s
VL 16.63 m/s
Δφ 0.09°
VY 0 m/s
VD2 3.98 m/s
GAIN 3.48
(26.5 m or 22.3 ms)
(actual)
-1 -0.5 0 0.5 1-2
0
2
4
6
8
10
12JL
pitc
h [°
]
yaw [°]0 1000 2000 3000 4000 5000
0123456789
1011
J1
J2
tota
l inc
iden
ce [°
]
range [cal]
New Orleans 2008-09International Symposium on Ballistics 15
Example: 105 mm Artillery Shell
LJ -0.51 cal
J1 10 N.s
J2 7.83 N.s
s2 - s1 1293 cal
⎯αmax 1.5°
d m Iy Mach p CD0 CLα CYpα Cmα Cmq105 mm 15.05 kg 2.19e-1 kgm2 0.3751.5 310 Hz 2.12 -0.8 3.6 -17
VJ 1.18 m/s
VL 0.36 m/s
Δφ -3.5°
VY 0.05 m/s(135.6 m)
VJ1
VL
VJ2
(1/2)
New Orleans 2008-09International Symposium on Ballistics 16
Example: 105 mm Artillery Shell
J2
J1 JLJY
J2
J1
6-DOF computations
Analytical computations
6-DOF computations
Analytical computations
JLJY
(2/2)
New Orleans 2008-09International Symposium on Ballistics 17
Conclusions
An analytical model was developed to predict the angular motion of a projectile subjected to impulse thrustersThe analytical model predicts the projectile’s angular motion very wellA procedure to properly paired impulses in order to minimize the drag while maximizing the lateral velocity was developedThe gain in lateral velocity obtained from the induced angular motion is significant