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]


Presentation Overview

Objective

Effect of a Lateral Impulse

Basic Mathematical Model and Analytical Solution

Pairing the Impulses

Examples

Conclusions


Objective

To developed a well defined procedure for using paired impulses on fin- and spin-stabilized projectiles in order to achieve enhanced drift corrections


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)


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


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



( ) ( )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



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


∫∞

=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



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)


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)


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


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


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]


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)


Example: 105 mm Artillery Shell

J2

J1 JLJY

J2

J1

6-DOF computations

Analytical computations

6-DOF computations

Analytical computations

JLJY

(2/2)


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

Wey Corriveau

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