Projectile Aerodynamic Jump Due to Lateral Impulsives 

by Gene R. Cooper 

ARL-TR-3087 September 2003 

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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-3087 September 2003 

Projectile Aerodynamic Jump Due to Lateral Impulsives 

Gene R. Cooper 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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1. REPORT DATE (DD-MM-YYYY)

September 2003 

2. REPORT TYPE 

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November 2002–October 2003 

5a. CONTRACT NUMBER 

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4. TITLE AND SUBTITLE 

Projectile Aerodynamic Jump Due to Lateral Impulsives 

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1L1622618.H80 

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6. AUTHOR(S) 

Gene R. Cooper 

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U.S. Army Research Laboratory 

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Aberdeen Proving Ground, MD 21005-5066 

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

The linear theory for spinning projectiles is extended to account for the application of a simple lateral square impulse 

activated during free flight. Analytical results are shown to produce simple contributions to the familiar aerodynamic jump 

formulation. Inquiries regarding jump smearing caused by nonzero impulse length are addressed and answered. The 

formulation shows for sufficiently long-range target interception; lateral impulse trajectory response for a guided projectile is 

independent of when the impulse is activated during the yaw cycle. Simple mathematical limits show the presented results 

reducing to those previously found for a zero-spin projectile acted upon by a singular lateral impulse. 

15. SUBJECT TERMS

linear theory, aerodynamic jump, analytic solutions, square impulse 

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Gene R. Cooper 

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Standard Form 298 (Rev. 8/98) 

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iii

Contents 

List of Figures iv

1. Introduction 1

2. Projectile Dynamic Model 1

3. Projectile Linear Theory 5

4. Projectile Linear Theory Solution 7

5. Pulse Force and Moment Conditions 10

6. Lateral Pulse Smearing 15

7. Lateral Pulse Response Magnitude and Phase Angle 17

8. Target Interception 17

9. Conclusions 20

10. References 21

Appendix A. Flight Coefficients for a 40-mm Projectile 23

List of Symbols 25

Distribution List 27
iv

List of Figures 

Figure 1. Projectile position coordinate definition. ........................................................................2 

Figure 2. Projectile orientation definition.......................................................................................2 

Figure 3. Jump component-j due to lateral impulse vs. impulse duration. ...................................16 

Figure 4. Jump component-k due to lateral impulse vs. impulse duration. ..................................16 

Figure 5. Magnitude of jump due to lateral impulse vs. impulse duration. ..................................18 

Figure 6. Magnitude of jump due to lateral impulse vs. spin rate. ...............................................18 

Figure 7. Phase angle of jump due to lateral impulse vs. spin rate...............................................19 
1

1. Introduction 

The continuing development of microelectromechanical systems is pointing to the possibility of 

mounting complete sensor systems on medium- and small-caliber projectiles as part of an actively 

controlled smart munition. Two important technical challenges in achieving this goal are the 

development of small, rugged sensor suites, and control mechanisms. With regard to the 

development of control mechanisms, several concepts have emerged that produce controllable 

impulsive lateral forces on a projectile body. For example, Harkins and Brown (1) considered the 

use of a set of lateral pulse jets or squibs to reduce dispersion of a rocket by firing squibs to minimize 

projectile pitch and yaw rate. For the notional cases evaluated, dispersion was reduced by a factor of 

5. Jitpraphai and Costello (2) considered the same type of control mechanism and used a trajectory 

tracking flight control system to improve impact point performance of a direct fire rocket equipped 

with a ring of squibs. Amitay et al. (3) considered the use of synthetic jet actuators as a control 

mechanism on lifting bodies. For a spinning projectile incorporating a synthetic jet actuator for 

control, the synthetic jet actuator is activated over a small portion of a roll cycle leading to a train of 

lateral pulse forces acting on the projectile near the synthetic jet actuator cavity. 

The design of flight control systems for fin-stabilized configurations is well established in the missile 

community and the control response to force and moment inputs is reasonably well understood. 

Generally these configurations are treated largely in the same manner as airplanes. While the 

uncontrolled dynamics of spinning projectiles, both fin stabilized and spin stabilized, have been 

extensively studied in the ballistics community, issues with regard to control response have received 

considerably less attention due to the lack of practical application of control technology to spinning 

projectiles. Using projectile linear theory, this report analytically investigates several aspects of the 

response of a spinning projectile to lateral pulse forces including swerve response magnitude and 

phase angle, impulse force smearing, and yaw cycle pulse timing. The report begins with a 

discussion of the basic projectile dynamic model followed by judicious simplifications to these 

equations that result in the projectile linear theory equations. The solution of the projectile linear 

theory equations are used to shed light on intuitive and subtle factors that influence swerve response 

of a projectile exposed to lateral impulsive loads. 

2. Projectile Dynamic Model 

It is well known that the motion of most projectile configurations can be captured using a rigid 

body 6 degrees of freedom dynamic model (4, 5). The degrees of freedom include three position 

components of the mass center of the projectile as well as three Euler orientation angles of the 
2

body. Figures 1 and 2 show schematics of the dynamic model degrees of freedom. The 

equations of motion are provided in equations 1–4. 

I I

r

KI

r

I J

r

KB

r

B J

r

BI

r

x y

w

V

u

v

− z

Figure 1. Projectile position coordinate definition. 

I I

r

ψ

I J

r

I K

r

φ

θ

Figure 2. Projectile orientation definition. 
3









































−

+ −

− +

=





















θ φ θ φ θ

θ ψ φ θ ψ φ ψ φ θ ψ φ ψ

θ ψ φ θ ψ φ ψ φ θ ψ φ ψ

w

v

u

s s c c c

c s s s s c c c s s s c

c c s s c c s c s c s s

z

y

x

&

&

&

; (1) 









































=









































= −





















ψ

θ

φ

θ

θ

φ θ φ θ

φ φ

φ θ φ θ

r

~

q

~

p

0 0 1/ c

0 1 0

1 0 t

r

q

p

0 s / c c / c

0 c s

1 s t c t

&

&

&

; (2) 









































−

−

−

−





















=





















w

v

u

q p 0

r 0 p

0 r q

Z/ m

Y / m

X / m

w

v

u

&

&

&

; (3) 





























































−

−

−

−





















=



















 −

r

q

p

I

q p 0

r 0 p

0 r q

N

M

L

I

r

q

p

1

&

&

&

. (4) 

The applied forces in the body frame that appear in equation 5 contain contributions from weight 

(W), air loads (A), and lateral pulse forces (C). 





















+





















+





















=





















C

C

C

A

A

A

W

W

W

Z

Y

X

Z

Y

X

Z

Y

X

Z

Y

X

. (5) 

The weight force resolved into projectile body coordinates is given by equation 6, 



















−

=





















φ θ

φ θ

θ

c c

s c

s

mg

Z

Y

X

W

W

W

. (6) 

The air loads are split in two components, the standard aerodynamic forces and the Magnus 

forces, 





















+





















=





















AM

AM

AM

AS

AS

AS

A

A

A

Z

Y

X

Z

Y

X

Z

Y

X

. (7)
4

Equation 8 gives the standard air loads acting at the aerodynamic center of pressure. 

( )





























+

+

+

+

= −





















V C C w

V C C v

V

v w C C

q

Z

Y

X

Z0 NA

Y0 NA

2

2 2

X0 X2

a

AS

AS

AS

, (8) 

where 

2 2 2 2

a (u v w ) D

8

1

q = ρ + + π (9) 

and 

2 2 2 V = u + v + w . (10) 

The Magnus aerodynamic force acts at the Magnus center of pressure. 





































−

= −





























2V

pDC v

2V

pDC w

0

q

Z

Y

X

NPA

NPA

a

AM

AM

AM

. (11) 

The lateral pulse force is modeled as an impulse that acts on an arbitrary point on the body. 

d( )t

Z

Y

0

Z

X

X

I

I

C

C

C





















=





















. (12) 

In equation 12, YI and ZI represent the lateral components of the impulse. The total magnitude 

of the impulse on the projectile is 2

I

2 FI = YI + Z . The function d(t) is a constant positive value 

when the pulse is active and is zero otherwise. This function integrates to unity for a single 

pulse. 

The applied moments about the projectile mass center are due to aerodynamic forces and 

moments (A) as well as pulse forces (C). 





















+





















+





















=





















m

C

C

UA

UA

UA

SA

SA

SA

N

M

L

N

M

L

N

M

L

N

M

L

. (13)
5

The aerodynamic moments caused by standard and Magnus air loads are computed with a cross 

product between the distance vector from the mass center to the force application point and the 

force itself. An unsteady aerodynamic damping moment is also present, which provides a 

damping source for angular motion. 



































 +

=





















2V

rDC

2V

qDC

2V

pDC C

q D

N

M

L

MQ

MQ

LP

DD

a

UA

UA

UA

. (14) 

All aerodynamic coefficients and the center of pressures are a function of the Mach number of 

the projectile mass center. The dynamic model previously described is highly nonlinear due to 

both three-dimensional rotational kinematics expressions and the presence of complex 

aerodynamic forces. The applicability of the equations of motion previously shown, have been 

validated over the past 60 years at aeroballistic ranges throughout the world (5).

3. Projectile Linear Theory 

Pressed with the need to predict the trajectory and stability of a ballistic shell so that useful 

performance data could be generated with primitive computers, early ballisticians vigorously 

investigated mathematical simplifications to the equations of motion of a projectile. What 

emerged over time was a set of simplified and solvable, yet accurate, linear differential 

equations, which today is commonly termed “projectile linear theory.” 

The governing equations developed previously are expressed in the body reference frame. In 

projectile linear theory, the lateral translational and rotational velocity components are 

transformed to a nonrolling reference frame. The nonrolling frame or so-called fixed plane 

frame proceeds with only precession and nutation rotations from an inertial reference frame. 

Components of linear and angular body velocities in the fixed plane frame can be computed from 

the body frame components of the same vector through a single-axis rotational transformation. 

For example, the body frame components of the projectile mass center velocity are transformed 

to the fixed plane frame by 









































− φ φ

= φ φ





















w

v

u

0 sin cos

0 cos sin

1 0 0

w

~

v

~

u

~

. (15)
6

It should be noted that the ~ superscript indicates the vector components are described in the 

fixed plane reference frame. In projectile linear theory, a change of variables from station line 

velocity component, u , to total velocity,V , is performed. Equations 16 and 17 relate V and u

and their derivatives. 

2 2 2 2 2 2 w

~ v~ V = u + v + w = u + + ; (16) 

V

dt

w

~d

w

~

dt

v

~d

v

~

dt

du

u

V

dt

dw

w

dt

dv

v

dt

du

u

dt

dV + +

=

+ +

= . (17) 

A change of variables from time, t , to dimensionless arc length, s, is also made. Equation 18, 

as defined by Murphy (4), gives the dimensionless arc length, 

Vdt

D

1

s

t

0

∫ = . (18) 

Equations 19 and 20 relate time and arc length derivatives of a given quantity ζ . Dotted terms 

refer to time derivatives, and primed terms denote dimensionless arc length derivatives. 

ζ′



 



 ζ = V

& D , (19) 





 



 ζ′ ′  ζ′′ + 

 



 ζ = V

V

V

D 2

&& . (20) 

In projectile linear theory, several assumptions regarding the relative size of different quantities 

are made to simplify the analysis. Euler yaw and pitch angles are small so that: sin(θ ) ≈θ , 

cos(θ) ≈ 1, sin( , and 1 ψ) ≈ ψ cos(ψ) ≈ and the aerodynamic angle of attack is small so that 

w V ~ α = and v V ~ β = . The projectile is mass balanced such that I I I 0 XY = XZ = YZ = and 

ZZ YY YY ZZ Y I = I ⇒ I = I ≡ I . The projectile is aerodynamically symmetric such that 

CY0 = CZ0 = 0 . Quantities V and φ are large compared to v, θ, and ψ,q,r, w , such that 

products of small quantities and their derivatives are negligible. Application of these 

assumptions results in equations 21–29. 

x′ = D ; (21) 

′ = v + Dψ ~

V

D y ; (22) 

′ = w − Dθ ~

V

D z ; (23)
7

p V

D φ′ = ; (24) 

q

~

V

D θ′ = ; (25) 

r

~

V

D

ψ′ = ; (26) 

V

Dg V 2m

SDC V XO θ − ρ ′ = − ; (27) 

p 4I

SD C V

2I

SD C

p

X

LP

3

X

LDD

2 ρ + ρ ′ = ; (28) 





























+

+

























































−

−

−

− −

=





























′

′

′

′

I

I

I

I

N

M

Z G

Y

r

~

q

~

w

~

v

~

F E D

B

D

C

E F D

C

D

B

0 A D 0

A 0 0 D

r

~

q

~

w

~

v

~

; (29) 

( )

( )





























































−

−

=





































y 0

x

y

MQ

5

y

NA COP CG

4

y 0

YPA MAG CG

5

NA

3

I V

pDI

16I

π ρD C

8I

πρD C SL SL

16 I V

πρpD C SL SL

8 m

πρD C

F

E

C

B

A

. (30) 

These equations are a coupled set of linear differential equations except for the fact that the total 

velocity, V , appears in the coefficients of many of the dynamic equations. 

4. Projectile Linear Theory Solution 

Linear theory offers physical insight into the projectile flight dynamics because closed form 

solutions can be readily obtained (4). Using the assumption that V changes slowly with respect 
8

to the other variables, it is thus considered to be constant, V ≈ V0 , when it appears as a 

coefficient in all dynamic equations except its own. Moreover, pitch attitude of the projectile is 

regarded as constant in the velocity equation, uncoupling the velocity equation from the system. 

The angle-of-attack dynamics or epicyclic motion in equation 29, together with the roll dynamics 

in equation 28 are uncoupled and form a linear system of differential equations. In projectile 

linear theory, the Magnus force in equations 25 and 26 is typically regarded as small in 

comparison to the other aerodynamic forces and is shown only for completeness. In further 

manipulation of the equations, all Magnus forces will be dropped. Magnus moments will be 

retained however, due to the magnitude amplification resulting from the cross product between 

Magnus force and its respective moment arm. 

The solution to the differential equation equation 27, for the forward velocity, is 

( ) 



 



 = + − − −

e 1

a

b V s V e

2a s

V

V 2 2a s

0

V V

, (31) 

where 

2m

SDC

a X0

V

ρ = (32) 

and 

bV = gDθ0 . (33) 

When 0 θ0 = , the velocity solution reduces to the familiar exponential decay form (4). 

The roll dynamic equation is a nonhomogeneous linear differential equation with the following 

solution: 

( ) 0

p

b s p

0

p

p

0 V

b

a

V e

b

a

p s p p − 















= + , (34) 

where 

XX

LDD

2

p 2I

SD C

a ρ = (35) 

and 

XX

LP

3

p 4 I

SD C b ρ = . (36) 
9

In order to develop the swerve closed form solution, the epicyclic equations must first be solved 

because the lateral translation and rotational velocity components are contained in the attitude 

differential equations, and the attitudes are contained within the swerve differential equations. 

The epicyclic differential equations consist of a set of four coupled nonhomogeneous differential 

equations. The homogeneous solution is easily formed using the free vibration modes and mode 

shapes, and the results are given by the intrinsic complex expression, 

() () () ( )

() () () ( )



























− − ± + − + − + +

− + ± + − + + + + =





























E A iF E A F 4C 2i E A F 2B

E A iF E A F 4C 2i E A F 2B

s

s

s

s

2

2 2

2 2

s

f

s

f

. (37) 

The sum and product of f s and s s give the relations found in the next expression, 

( ) ( )



























− λ Φ + λ Φ

Φ Φ − λ λ

Φ + Φ

λ + λ

=





























− +

+

−

f s s f

f s f s

f s

f s

AF B

AE C

F

E A

. (38) 

Two more simplifications based on size are introduced. First, neglect the product of damping 

and secondly, the product AE is neglected because the density ratio is assumed small. A 

solution may now be obtained for both the fast and the slow damping factors and turning rates 

for the translational and rotational velocities. 

























































 



 −





 



 +

−

−

+





 



 − −

λ =

F A E

2A F 2B

1

F 4C

F 1 2

A E

2 f ; (39) 

















Φ = F + F − 4C 2

1 2

f ; (40) 

























































 



 −





 



 +

−

−

−





 



 − −

λ =

F A E

2A F 2B

1

F 4C

F 1 2

A E

2 s ; (41) 
10

















Φ = F − F − 4C 2

1 2

s . (42) 

5. Pulse Force and Moment Conditions 

The pulse force applied to the projectile is taken to be a lateral impulsive force, and this force is 

due to an actuator attached to the projectile body (equation 29). For this investigation, the force 

actuator is modeled as a scaled square wave pulse so that the resulting force and moment 

components in the nonrolling frame are as follows: 

n

B

0

0 n n n

I 2L

s V

Dp F V sgn s s sgn s s L cos

Y

















+ φ 













 



 



  − − −  



 −

=

∗

; (43) 

n

B

0

0 n n n

I 2L

s V

Dp F V sgn s s sgn s s L sin

Z

















+ φ 













 



 



  − − −  



 −

=

∗

; (44) 

n

B

0

0 n n n

I 2D L

s V

Dp M V sgn s s sgn s s L sin

M

















+ φ 













 



 



  − − −  



 −

= −

∗

; (45) 

n

B

0

0 n n n

I 2D L

s V

Dp M V sgn s s sgn s s L cos

N

















+ φ 













 



 



  − − −  



 −

=

∗

, (46) 

for a square wave pulse of length Ln that is initiated at n s represented as 

2

0

d

mV

DF F = ∗

 (47) 

and 
11

2

y 0

d r

2

I V

D F X M = ∗

. (48) 

Note that the last two expressions become equivalent to delta function impulses in the limit of 

L 0 n → . 

Once the simplified mode shapes of equation 29 are obtained, the initial conditions for 

r

~ q, and ~ w, ~ v, ~ are used to complete the solution. Equations 49 and 50 are the analytical solutions 

for the fixed plane translation velocities v

~ and w

~ , expressed in phase-amplitude form. 

( )

( ) ()( )

( ) ( ) ; 2

sgn s s L 1 F φ

2

sgn s s sgn s s L F φ

F Φ s,Φ s

v(s) V e cos(Φ s Θ ) V e sin(Φ s Θ ) ~

n n

3 B

n n n

2 B

1 f s

S V2

λ s

F V1 2

λ s

1 f s





 



 − − +

+





 



 − − − − +

+

= − + −

(49)

()( )

( ) , 2

sgn s s L 1

2

π F φ

2

sgn s s sgn s s L

2

π F φ

2

π , Φ s

2

π F Φ s

w(s) V e sin(Φ s Θ ) V e cos(Φ s Θ ) ~

n n

3 B

n n n

2 B

1 f s

S V2

λ s

F V1 2

λ s

1 f s





 



 − − + 



 



 + −





 



 − − − − 



 



 + −





 



 + − −

= − − −

 (50)

where 

2

f s f

f 0 f 0 0

2

f s f

f f 0 f 0 0

1

r D~ v E ~ w F ~ q D ~ w E ~ v

~ F

V





































 

 Φ Φ − Φ















  − − 

 

 Φ  − Φ

+





























 

 Φ Φ − Φ





 



 Φ + Φ −  



 − Φ

= (51) 
12

2

s s f

s 0 s 0 0

2

s s s

s 0 s 0 0 s

2

r D~ v E ~ w F ~ q D ~ w E ~ v

~ F

V





































 

 Φ Φ − Φ















  − − 

 

 Φ  − Φ

+





























 

 Φ Φ − Φ





 



 Φ + Φ −  



 − Φ

= ; (52)





































 



 Φ + Φ −  



 − Φ















  − − 

 

 Φ  − Φ

Θ = −

q D ~ w E ~ v

~ F

r D~ v E ~ w F ~

tan

f f 0 f 0 0

f 0 f 0 0

1

V1 ; (53) 















































  − − 

 

 Φ  − Φ





 



 Φ + Φ −  



 − Φ

Θ = −

r D~ v E ~ w F ~

q D ~ w E ~ v

~ F

tan

s 0 s 0 0

s 0 s 0 0 s 1

V2 ; (54) 

( )

(( ) ) ( )

( ) ()

( )( )

( ) ( ) ( )

( ) ()

( )( )

; Φ Φ

FG 

Φ Φ P Φ Φ

Φ P Esin Φ s

Φ P F Φ PΦ cos Φ s

G e

Φ P Φ Φ Φ

P Φ Esin Φ s

P Φ F PΦ Φ cos Φ s

G e

F Φ s,Φ s

f s

s s s f

s s

s s

2

s s λ s

f f s f

f f

f

2

f f f λ s

1 f s

s

f

−

− −

















+ −

− − +

−

− −

















+ −

− − +

=

 (55)
13

( ) ( )[( ) ( ) ]

( )( )

( ) [ ] ( ) ( )

( )( )

[ ] ( )( ) ( ) ( )

( )( ) ; P Φ P Φ L

V F P F sin Ps φ M EF cos Ps φ

Φ Φ P Φ L

V e Φ F F sinΩ M EF cosΩ

Φ Φ P Φ L

V e Φ F F sinΩ M EF cosΩ F φ

f s n

0 B B

s f s n

f s s

λ s s

0

s f f n

f f f

λ s s

0

2 B

s n

f n

− −

− + + + +

+

− −

− + + −

− −

− + + =

∗ ∗ ∗

− ∗ ∗ ∗

− ∗ ∗ ∗

 (56)

( )

( )

(( ) ( ) )

( ) ( )

( )( )

( )

( ) ( ) ( )

( ) ( )

( )( ) ; Φ Φ P Φ L

F Φ F sin∆ M EF cos∆

e M EF cosΩ F Φ F sinΩ

V e

Φ Φ P Φ L

F Φ F sin∆ M EF cos∆

e M EF cosΩ Φ F F sinΩ

V e

F φ

s f s n

f f f

s s s

λ L

λ s L s

0

s f f n

f f f

f f f

λ L

λ s L s

0

3 B

s n

s n n

f n

f n n

− −





















− − − +

+ + −

−

− −





















− − − +

+ + −

=

∗ ∗ ∗

∗ ∗ ∗

− −

∗ ∗ ∗

∗ ∗ ∗

− −

 (57)

V0

Dp P = ; (58) 

 s P s s , P s f f n B f s s n − φB − Φs 



 

  + φ + Φ Ω = − − Φ 

 

 Ω =  − Φ ; (59) 

f f n f s P f Ln s P L ,  − Ω 

 

  + Ω ∆ =  − Φ 

 

 ∆ =  − Φ . (60) 

Swerving motion is measured along the earth-fixed I J and KI axes. To an observer standing 

behind the gun tube, these axes are oriented such that positive I J is to the right and positive KI

is pointed downward. The swerving motion results from a combination of the normal 

aerodynamic forces, as the projectile pitches and yaws, plus the forces and moments due to the 

applied impulse. Differentiating equations 22 and 23 with respect to nondimensional arc length, 

generates the swerve equations such that 
14

n 0

n n n B

V

v

~A

2L

F sgn s s sgn s s L cos ps

D

y −





 



 + φ 













 



 



  − − −  



 −

=

′′

















∗

 (61) 

and 

( )( ) ()( )

n 0

n n n B

V

w

~ G A

2L

F sgn s s sgn s s L sin ps

D

z − + − − − − + φ =

′′















 ∗

. (62) 

Integrating these equations is straightforward, but the solutions are judged to be too long to be 

listed here. However, the asymptotic limit of these solutions, s → ∞ , is of special interest. 

For a stable projectile, the swerve caused by epicyclical vibration decays as the projectile 

progresses downrange and does not affect the long-term lateral motion of the projectile. When a 

lateral pulse is applied to the projectile at arc length n s , its effect on the target impact point is 

predominantly due to induced jump, provided the target distance is sufficient to allow the 

transients to decay. Projectile linear theory shows that the long-term center of mass solution, or 

swerve, contains terms that remain bounded with arc length s plus terms that are linear with s, 

and if gravity is included the solution will have even higher order diverging terms. These higher 

order terms are typically denoted as gravity drop. The linear terms are called jump terms, which 

are caused by initial conditions at the gun muzzle, lateral pulse forces, and aerodynamic 

characteristics. Mathematically, setting gravity to zero and subsequently evaluating the 

following limits formally defines total jump. 

K

s

J

s

Γ

Ds

z(s) Γ , and

Ds

y(s) lim = lim = →∞ →∞

. (63) 

The total jump vector Γ is expressed as the sum of two vectors. The first vector represents the 

jump attributable to muzzle conditions, and the lateral pulse is represented by the second, thus 

resulting in the following expression: 

( )( ) 















Π

Π

+ Λ

















− −

+ −

















+ +

+ − −





 





+ + +

= −

















Γ

Γ

sin

cos

w F v E r D

v F w E q D

AE C AF B

AF B AE C

AF B AE C V

A

0 0 0

0 0 0

0

2 2 K

J , (64) 

for which 
15

() ( )

( )( )

( ) ( )

( )( )

( ) Ps φ tan ( ) Z Y Ps φ as L 0, 2

PL Π tan Z Y

, AF B AE C

ABF ACE C B F A AE C M Z

, AF B AE C

A AF B M A BE CF F Y

n B n

1

n B

1 n

2 2

2 2

2 2

= − − − → − − →

+ + +

+ + + − + =

+ + +

+ + − =

− −

∗ ∗

∗ ∗

and 

( )

( ) Y Z as L 0.

PL 2

sin PL 2 Λ Y Z n

2 2

n

= 2 + 2 n → + → (65) 

The quantities Λ and Π are the magnitude and phase angle of the jump vector attributed to the 

lateral pulse. It is interesting to note that the jump terms induced by the lateral pulse are not 

dependent on the lateral state of the projectile ( ( ) ( ) ( ) r(s)) ~ q s , and ~ w s , ~ v s , ~ , thus this particular 

contribution to jump is not coupled to the projectile’s angle of attack. Equation 65 extends the 

work by Guidos and Cooper (6) who considered nonspinning projectiles subject to a singular 

delta function impulse. Limiting 0 Ln ,P  →

 



 in equations 64 and 65 produces expressions 

that agree with their previous predictions. 

6. Lateral Pulse Smearing 

In order to better understand the swerve response due to a lateral pulse, results for a 

representative spin-stabilized 40-mm projectile configuration are calculated and discussed in this 

and the next two sections. Nominal values for the aerodynamic coefficients, the projectile 

physical parameters, and flight characteristics are given in the Appendix. All results presented 

use these values unless specified differently. Diversions from these nominal values are clear 

from the context of the particular chart under examination. 

Plots showing the effect of smearing are given in Figures 3 and 4 for the applied force 

5 F 3.838 10 ∗ − = × and moment arms 



 



 = − → + 0.635 to Xr D 0.635 . Negative values of Xr

indicate that the application point of the pulse force is aft of the mass center while positive 

values indicate the pulse force is forward of the mass center. To illustrate the smearing effects, 

the roll position, φN , of the lateral impulse force is assumed to act primarily along the nonrolling 

Y-axis. This means the arc length, Ns , corresponding to the center of the pulse satisfies the 
16

-20

0

20

40

60

80

0 20 40 60 80 100 120 140

Ln

Λ Cos Π x 10 6

Xr/D = +0.635

Xr/D = +0.318

Xr/D = 0.0 

Xr/D = -0.318

Xr/D = -0.635

F 3.838 10 & tan Z Y 2 N 5 1= × − Π = π ∗ −

Figure 3. Jump component-j due to lateral impulse vs. impulse duration. 

-1.5

-1.0

-0.5

0.0

0.5

0 20 40 60 80 100 120 140

Ln

Λ Sin 

Π x 10 6

Xr/D = +0.635

Xr/D = +0.318

Xr/D = +0.0

Xr/D = -0.318

Xr/D = -0.635

F 3.838 10 & tan Z Y 2 N 5 1= × − Π = π ∗ −

Figure 4. Jump component-k due to lateral impulse vs. impulse duration. 
17

expression φN = φ′sN + φB = 2πN, N = 0,1,2L. To ensure this force is acting nominally along 

the nonrolling Y-axis, the activation point begins at sn = sN − Ln 2 so that the duration of the 

impulse brackets N φN = 2π . Increasing pulse length, Ln , causes the jump components to 

cyclically decay while its value at L 0 n = corresponds to a lateral impulse that is proportional to 

the delta function ( ) n δ s − s . Values of Ln where the jump is zero represent situations where the 

duration of the lateral pulse coincides with a roll cycle. This accounts for the post multiplier, 

















0

n

2V

pDL sin , in equation 65 having 2π zero crossings and causes the cyclical jump results in 

Figures 3 and 4. Notice for the cases presented the response due to a pulse in the I J direction is 

predominantly along I J ; however, a smaller out-of-phase component of swerve is also 

generated. 

7. Lateral Pulse Response Magnitude and Phase Angle 

Figure 5 has charts showing the absolute value of Λ as a function of the pulse length Ln for the 

previous parameter values 

5

F 3.838 10 ∗ −= × and 



 



 = − → + 0.635 to Xr D 0.635 . Here again 

the impulse is assumed to bracket N φN = 2π . The maximum value occurs at L 0 n = , which is 

the delta function result, as is justified from equation 65. Consideration of Λ as a function of 

spin rate p is displayed in Figure 6 for several pulse durations 



 



 = → 40 to Ln 0 . This figure 

shows that the rate at which the aerodynamic jump magnitude decreases, with spin rate, is 

strongly dependent on the pulse duration. Again, this is attributed to cancellation (i.e., smearing) 

effects, which become more pronounced with increasing values of Ln . 

Figure 7 gives the phase angle, Π , (equation 65) as a function of spin rate for the same values of 

the moment arm length Xr D previously discussed. Notice that the phase angle linearly 

increases with spin rate, indicating that spin-stabilized projectiles are most susceptible to an 

out-of-phase swerve response due to lateral pulse forces. 

8. Target Interception 

The last topic discussed in this report examines the relationship between a given change in 

aerodynamic jump, 













∆Γ

∆Γ

K

J , and the impulse needed to create this change. This may be desirable 
18

0

20

40

60

80

0 20 40 60 80 100 120 140

Ln

 Magnitude 

Λ x 10 6

Xr/D = +0.635

Xr/D = +0.318

Xr/D = 0.0 

Xr/D = -0.318

Xr/D = -0.635

F 3.838 10 & tan Z Y 2 N 5 1= × − Π = π ∗ −

Figure 5. Magnitude of jump due to lateral impulse vs. impulse duration. 

0

5

10

15

20

25

0 100 200 300 400 500 600

Spin Rate p (1/sec.)

Jump Magnitude Λ x 10 6

Ln = 0

Ln = 10

Ln = 

Ln = 30

Ln = 40

Spin Rate p (1/s)

F 3.838 10 & tan Z Y 2 N 5 1= × − Π = π ∗ −

Figure 6. Magnitude of jump due to lateral impulse vs. spin rate. 
19

-3

-2

-1

0

1

0 100 200 300 400 500 600

Spin Rate p (1/sec.)

Phase Angle 

Π (Degrees)

Xr/D = -0.636

Xr/D = -0.318

Xr/D = 0.0

Xr/D = +0.635

Xr/D = +0.318

F 3.838 10 & tan Z Y 2 N 5 1= × − Π = π ∗ −

Spin Rate p (1/s) 

Figure 7. Phase angle of jump due to lateral impulse vs. spin rate. 

when the activated impulse is strong enough to force a projectile to strike its target, located at a 

relative position 













∆Γ

∆Γ

K

J , with a single impulse. Hence, equation 64 requires 













Π

Π = Λ 











∆Γ

∆Γ

cos

sin

K

J . (66) 

Solving this system for Π and insisting that Π remains real valued yields the following two 

equations for target interception: 

( )

















Σ + Σ

∆Γ + ∆Γ =















2

K

2

0 J

n J K

0

n 2

2V

pDs

2V

pDs sin , (67) 

and 

tan 2 N

2V

pDL

V

pDs

J J K K

1 K J J K

0

n

B

0

n + π 















∆Γ Σ + ∆Γ Σ

∆Γ Σ − ∆Γ Σ

+ φ = − + − . (68) 
20

Using equation 65, the impulse force FI can be determined from equation 67, for a given 

moment arm Xr , and then equation 68 solved for φB will guarantee target impact, provided the 

relative target position, 













∆Γ

∆Γ

K

J , is known. 

9. Conclusions 

An analytical approach for quantifying the effect of a lateral square impulse disturbing a 

projectile during free flight has been presented. All of the analysis was based on projectile linear 

theory, which produces simple closed form solutions for the assumed square pulse disturbance. 

These solutions are then used to calculate the projectile swerving motion, so that the long-term 

effects of the lateral impulse are readily determined. The primary interest regarding target 

interception is the projectile’s aerodynamic jump. Changes to aerodynamic jump caused by the 

lateral impulse forces were shown to produce easy-to-understand additive contributions to the 

usual aerodynamic jump of a free-flight projectile with no applied impulse. 

The important question concerning smearing effects originating from a finite length pulse has 

been addressed where pulse-induced control authority is shown to diminish as Ln 1 . 

Calculations also show the additional aerodynamic jump magnitude, Σ , decreases with spin rate 

p when subjected to various pulse lengths Ln .
21

10. References 

1. Harkins, T. E.; Brown, T. G. Using Active Damping as a Precision-Enhancing Technology 

for 2.75-Inch Rockets; ARL-TR-1772; U.S. Army Research Laboratory: Aberdeen Proving 

Ground, MD, 1999. 

2. Jitpraphai, T.; Costello, M. Dispersion Reduction of a Direct Fire Rocket Using Lateral Pulse 

Jets. Journal of Spacecraft and Rockets 2001, 38 (6), 929–936. 

3. Amitay, M.; Smith, D.; Kibens, V.; Parekh, D.; Glezer, A. Aerodynamic Flow Control Over 

an Unconventional Airfoil Using Synthetic Jet Actuators. AIAA Journal 2001, 39 (3), 

361–370. 

4. Murphy, C. H. Free Flight Motion of Symmetric Missiles; BRL-TR-1216; U.S. Army 

Ballistic Research Laboratories: Aberdeen Proving Ground, MD, 1963. 

5. McCoy, R. L. Modern Exterior Ballistics: the Launch and Flight Dynamics of Symmetric 

Projectiles; Schiffer Publishing Ltd.: Atglen, PA, 1999. 

6. Guidos, B.; Cooper, G. Linearized Motion of a Fin-Stabilized Projectile Subjected to a 

Lateral Impulse. Journal of Spacecraft and Rockets 2002, 39 (3). 
22

INTENTIONALLY LEFT BLANK. 
23

Appendix A. Flight Coefficients for a 40-mm Projectile 

The numerical values used for the graphical presentations given in this report are shown in the 

following matrices: 

Aerodynamic coefficients: 





















































−

−

−

=





























































1.800

0.042

0.295

2.329

2.672

0.279

C

C

C

C

C

C

MQ

LP

YPA

NA

X2

X0

a . (A-1) 

Physical parameters: 





































































×

×

=





































































−

−

0.0713ft.

0.239ft.

0.237ft.

0.137 ft.

2.72 10 Slugft.

2.85 10 Slugft.

0.0116Slug

SL

SL

SL

D

I

I

m

5 2

5 2

CG

MAG

COP

Y

X

a . (A-2) 
24

Flight characteristics: 





































×

=



























 ρ

−

−

− −

1

1

3 3

0

399.7 sec.

250.0 ft. sec.

2.38 10 Slug ft.

p

a V . (A-3) 
25

List of Symbols 

Ci Projectile aerodynamic coefficients 

D Projectile characteristic length (diameter) 

F Dimensional impulse force d

∗ F Nondimensional impulse force 2

0

F = DFd mV ∗

g Gravitational constant 

G Scaled gravitational constant G = gD V0

Y

X

I

I

 Mass moments of inertia 

N

~

M

~

L

 Applied moments about projectile mass center expressed in the no-roll frame 

m Projectile mass 

r

~

q

~

p

 Angular velocity components vector of projectile in the no-roll frame 

S Surface area S D 4

2

= π

s Nondimensional arch length 

sgn(τ) 







τ =

τ τ τ ≠

0 0

0

w

v

u

 Mass center velocity components in the body reference frame 

V Forward velocity of projectile 0

w

~

v

~

u

 Mass center velocity components in the no-roll reference frame 
26

Xr Dimensional moment arm length 

I

I

Z

Y

 Applied force components in the no-roll reference frame 

















z

y

x

 Position vector of body center of mass in an inertial reference frame 

α Longitudinal aerodynamic angle of attack 

β Lateral aerodynamic angle of attack 

K

J

Λ

Λ K

J Components of aerodynamic jump due to lateral impulse 

Π Phase angle of the aerodynamic jump due to lateral impulse 

ψ

θ

φ

 Euler roll, pitch and yaw angles of the projectile 

φB Euler roll angle of the applied impulse 
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