Approved for public release; distribution is unlimited.

Improved Dispersion of a 

Fin-Stabilized Projectile Using 

a Passive Moveable Nose

by Mark Costello and Raditya Agarwalla

ARL-CR-464 April 2001

prepared by

Mark Costello and Raditya Agarwalla

Oregon State University

Department of Mechanical Engineering

Corvallis, Oregon 97331

under contract

DAAD17-00-P-0853
Approved for public release; distribution is unlimited.

Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5066

ARL-CR-464 April 2001

Improved Dispersion of a 

Fin-Stabilized Projectile Using 

a Passive Moveable Nose

Mark Costello and Raditya Agarwalla

Oregon State University
ii

Abstract

A key and often quoted metric associated with gun systems is impact point accuracy. 

The extent of impact dispersion is a complex function of a battery of parameters,

including gun geometry and tolerances, the fire control system, projectile manufacturing 

tolerances, etc. The work reported here investigates potential impact point accuracy 

improvement for a penetrator-type projectile realized by replacing the rigid nose cone 

wind screen with a passive gimballed nose. By comparing the impact point dispersion 

of a rigid projectile with a similar gimballed nose projectile, it is shown that impact point 

accuracy can be significantly improved. For the example penetrator projectile

considered, impact point dispersion is reduced by more than 50%. 
iii

Contents

List of Figures v

List of Tables vii

1. Introduction 9

2. Gimbal Nose Projectile Dynamic Model 10

3. Results 17

4. Conclusion 28

5. References 30

Report Documentation Page 31
iv

INTENTIONALLY LEFT BLANK.
v

List of Figures

Figure 1. Schematic of the gimballed nose projectile configuration.....................10

Figure 2. Range vs. time........................................................................................19

Figure 3. Cross range vs. time...............................................................................19

Figure 4. Altitude vs. time....................................................................................20

Figure 5. Total velocity vs. time............................................................................20

Figure 6. Roll rate vs. time. ...................................................................................21

Figure 7. Euler pitch angle vs. time......................................................................21

Figure 8. Euler yaw angle vs. time........................................................................22

Figure 9. Euler pitch angle of nose vs. time..........................................................22

Figure 10. Euler yaw angle of nose vs. time.........................................................23

Figure 11. Impact point dispersion at 1 km range................................................23

Figure 12. Impact point dispersion at 2 km range................................................24

Figure 13. Impact point dispersion at 3 km range................................................24

Figure 14. Impact point dispersion vs. nose lift coefficient..................................25

Figure 15. Impact point dispersion vs. forward-to-total mass ratio.....................26

Figure 16. Impact point dispersion vs. gimbal viscous friction coefficient..........27

Figure 17. Motion of nose tip with respect to the aft body (Cg = 0.00). ................27

Figure 18. Motion of nose tip with respect to the aft body (Cg = 1.00). ................28
vi

INTENTIONALLY LEFT BLANK.
vii

List of Tables

Table 1. Baseline configuration properties. ..........................................................17

Table 2. Nominal initial conditions.Table 3. Nominal initial conditions.............18
viii

INTENTIONALLY LEFT BLANK.
9

1. Introduction

The merit of a penetrator is often assessed by a relatively short list of metrics that 

typically includes parameters such as terminal velocity, penetrator weight, cost, system 

accuracy, etc. Of these parameters, system accuracy is usually near the top of the list in 

terms of importance, and it is of significant concern during weapon system

development. Given two identical weapon systems with the exception of accuracy, the 

system with superior accuracy enjoys a distinct advantage on the battlefield. A system 

with improved accuracy can engage targets at a greater range and obtain the same 

probability of hit, providing the tank commander with increased flexibility during an 

engagement. Alternatively, a system with better accuracy will register more first-volley 

hits at the same range, reducing the counter fire threat. Furthermore, a gun system with 

superior accuracy ultimately requires fewer shots to achieve mission objectives, hence 

inducing less burden on the logistics pipeline.

The initial state of a projectile as it exits the gun muzzle and enters free flight can be 

viewed as a random process. The random nature of the initial free flight state stems 

from many effects, but perhaps most notably from gun tube and projectile

manufacturing tolerances combined with the resulting gun tube and projectile vibration. 

As the projectile flies down range, these uncertainties, along with aerodynamic

disturbances along the trajectory, map into dispersion at the target. Designers can take 

two basic approaches toward improving accuracy; they can reduce the variability of 

projectile initial free flight conditions or reduce the sensitivity of the projectile trajectory 

to initial free flight conditions. One way to attack this problem using the latter approach 

is to replace the rigid wind screen with a passive gimballed nose. If the pivot point of 

the nose section is forward of the nose aerodynamic center, then the nose will tend to 

rotate into the relative wind and subsequently reduce aerodynamic jump caused by

projectile normal force. A passive gimballed nose projectile is an attractive design 

modification because it is a relatively simple mechanism that requires no active

electronic controls. Furthermore, for many penetrator designs, the nose cone is empty 

and could easily house the gimbal joint. 

Early in the development of controlled rockets, the notion of utilizing a moveable nose 

to actively control the trajectory of a projectile was established [1]. Goddard obtained a 

patent titled “An Apparatus for Steering Aircraft” which outlined the basic concept. 

More recently, Barrett and Stutts [2] further developed this concept and subsequently 

developed and tested a gun-launched, actively controlled nose. The moveable nose 

concept has also been investigated in unguided projectile applications as well. Krantz [3] 

obtained a patent for a telescopic passive nose on a high velocity aerodynamic body. 

Schmidt and Donovan [4] developed a simple closed form solution for an effective CLa

and C Ma for a moveable nose projectile configuration that is based on projectile linear 

theory [5]. A limited number of prototype projectiles were fired, and range data was 

reduced to estimate aerodynamic coefficients. The work reported herein extends the 

previous work mentioned by simulating the exterior ballistics of a gimballed nose
10

projectile in atmospheric flight and subsequently comparing impact point dispersion 

statistics with a similarly-sized rigid projectile. The gimbaled nose projectile dynamic 

model includes the typical six degrees of freedom for the main body plus an additional 

three degrees of freedom for the rotation of the nose with respect to the main body. 

Impact point dispersion statistics are generated through Monte Carlo simulation of the 

initial pitch and yaw rates of the projectile. 

2. Gimbal Nose Projectile Dynamic Model

A schematic of the gimballed nose projectile configuration is shown in Figure 1. The 

gimballed nose projectile consists of forward and aft projectile sections. The

configuration possesses three position degrees of freedom, which are the inertial

position components of the mass center of the composite body described in an inertial 

reference frame. 

Figure 1. Schematic of the gimballed nose projectile configuration.

O I I I

r xi yj zk

r r r r

→⊕ = + + . (1)

A total of six degrees of freedom describe the aft and forward body orientation. The 

orientation of the aft projectile is obtained through a sequence of three

body-fixed rotations. Starting from an inertial coordinate system, the aft body is

successively rotated through Euler yaw, pitch, and roll angles to arrive at its final

orientation in space. The forward body orientation is also obtained by a sequence of 

three body-fixed rotations. Starting from the aft body reference frame, the forward body 

is successively rotated through Euler yaw, pitch, and roll angles to arrive at its final 

orientation in space. With these definitions, a rigid projectile configuration is realized 

XA d

r

A

F

Gimbal Joint

F

ρ

r

XF d

r

F

β

r

A

β

r

∗

r

r

A

ρ

r

Mass Center of

Composite Body

⊄

⊄
11

when the forward-body Euler angles are zero, f F =q F =y F = 0.

As shown in equation 2, the velocity vector components of the mass center of the 

composite body are defined in the aft body reference frame.

I A A A

v ui vj wk

r r r r

⊕/ = + + . (2)

With the definitions given in equations 1 and 2, the resulting translational kinematic 

differential equations are given by equation 3.

[ ]









































=

w

v

u

A

T

z

y

x

&

&

&

. (3)

Equation 3 contains the transformation matrix from the aft body reference frame to the 

inertial reference frame, which is provided as equation 4. As shown in equation 5, the 

transformation from the forward body reference frame to the aft body reference frame 

takes on the same form as equation 4, except the angles are the nose angles.





















−

+ −

− +

=

A A A A A

A A A A A A A A A A A A

A A A A A A A A A A A A

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

A

T

q f q f q

q y f q y f y f q y f y

q y f q y f y f q y f y

. (4)





















−

+ −

− +

=

F F F F F

F F F F F F F F F F F F

F F F F F F F F F F F F

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

F

T

q f q f q

q y f q y f y f q y f y

q y f q y f y f q y f y

. (5)

The angular velocity vector expressions for the aft and forward bodies with respect to an 

inertial reference frame are provided in equations 6 and 7, respectively.

A I A A A A A A p i q j r k

r r r

w / = + + . (6)

F I F F F F F F p i q j r k

r r r

w / = + + . (7)

With these definitions, the rotational kinematic differential equations are given by

equations 8 and 9.









































−





















=

A

A

A

A

A

A

r

q

p

s c c c

c s

s t c t

A A A A

A A

A A A A

f q f q

f f

f q f q

y

q

f

0 / /

0

1

&

&

&

. (8)
12









































−









































−





















= −

A

A

A

F

F

F

F

F

F

r

q

p

c t s t

s c

c c s c

r

q

p

s c c c

c s

s t c t

F F F F

F F

F F F F

F F F F

F F

F F F F

1

0

/ / 0

0 / /

0

1

y q y q

y y

y q y q

f q f q

f f

f q f q

y

q

f

&

&

&

. (9)

The translation dynamic equations for the mass center of the composite body are given 

by equation 10.

























































−

−

−





















+

= +

Z

Y

X

w

v

u

q p

r p

r q

w

v

u

F

m

A

m

A A

A A

A A

1

0

0

0

&

&

&

. (10)

In equation 10, the first term on the right hand side utilizes the aft body angular velocity 

components in the cross product operator since the composite body mass center velocity 

components are defined in the aft body reference frame. The total applied force vector 

components are given in the aft body reference frame.

The rotational dynamics of the forward and aft projectile sections are derived by first 

splitting the system at the gimbal joint, which exposes the constraint forces and

moments at the joint. As shown in equation 11, by subtracting the force balance of both 

bodies, the components of the constraint force in the aft body coordinate system can be 

written in terms of the rotational state variables and their derivatives.

{ AF}

F

F

F

F

A

A

A

A

C

C

C

B

r

q

p

A

r

q

p

A

Z

Y

X

= + +





























































&

&

&

&

&

&

, (11)

where 

A

S

m m

m m

A F

A F

A

A

+ r

= , (12)

F

S

m m

F

T

F

m

A

m

A F

F

A

+ r

= , (13)
13

, 

ZFYFXF

F

T

Z

A

Y

A

X

A

AA

F

 

F

Z

F

Y

F

X

F

T

F

m

A

m

A

m

A

Z

AYAX

F

m

A

m

F

m

AF

B

FFA

A

SSS

S

F

mm

mm  

−  

+  

+

−  

=

+

ρρρ

ωω

ρρρ

+

ωω

(14)  

=

−

−

−

0

0

0

X

A

Y

A

X

A

ZA

YAZ

A

A

S

rr

rr

rr

r

, (15)  

=

−

−

−

0

0

0

X

F

YF

X

F

ZF

YFZF

F

S

rr

rr

rr

r

, (16)  

=−

−

−

0

0

0~

~

~~

~~

~

X

F

YF

X

F

ZF

YFZF

F

S

rr

rr

rr

r

, (17)
14





















−

−

−

=

0

0

0

A A

A A

A A

q p

r p

r q

SwA, (18)

and





















−

−

−

=

0

0

0

F F

F F

F F

q p

r p

r q

SwF. (19)

The constraint moment at the gimbal joint is generated by friction in the joint and 

potentially by geometric interference between the nose and main projectile body when 

the total angle between the nose and projectile axes of symmetry exceeds a specific 

value. Gimbal joint friction is modeled as viscous damping. The gimbal joint friction 

constraint moment is proportional to the difference in angular velocity between the two 

projectile body components. Geometric interference is modeled as a stiff linear torsion 

spring with dead band. The dead band region corresponds to the rotational envelope of 

the nose with respect to the main projectile body. The gimbal nose geometric

interference constraint moment magnitude is proportional to the angle between Ai

r

and 

F

i

r

, denoted as ag, and is computed using equation 20. 

cos (cos cos )

1

ag q F y F

−

= . (20)

The direction of this moment is perpendicular to the plane formed by Ai

r

 and Fi

r

. 

Equation 21 provides an expression for the gimbal joint constraint moment.

 









































−

−





















+

+

=

r

q

p

g

C

g

M

N

M

L

F F

F

C

C

C

F F F

g

K

d

d

d

q y

q

q q y cos( )sin( )

sin( )

0

( )

2

( )sin 2( ) cos 2sin

, (21)

where 

and

[ ]





























































= −

F

F

F

F

A

A

A

r

q

p

T

r

q

p

r

q

p

d

d

d

. (23)

The first term in equation 21 is the geometric interference constraint moment, while the 

second term represents the friction constraint moment. Notice that when Ai

r

 and Fi

r

 are 

aligned ( = 0 ag), the interference constraint moment is singular. Fortunately, this

0 if αg ≤ α*

Kg (αg - α*) if αg > α* , 

Mg

= 

(22)
15

singularity is avoided since the gimbal joint interference constraint moment is zero in 

this case. 

The rotational dynamic equations of the aft and forward projectile bodies are given by 

equations 24 and 25, respectively. 





















+





















+





















= −





















+





















C

C

C

A

A

A

C

C

C

A

A

A

A

A

A

A

A

Z

Y

X

S

N

M

L

N

M

L

r

q

p

S I

r

q

p

I

A A

w r

&

&

&

. (24)





















−





















+





















=





















+





















C

C

C

F

F

F

F

F

C

C

C

F

F

F

F F

F

F

F

F F

Z

Y

X

T S

N

M

L

T

N

M

L

r

q

p

T S I

r

q

p

T I

F F

w r

~

&

&

&

. (25)

By substituting the constraint force and moment expressions into equations 24 and 25, 

the final form of the rotational dynamic equations is obtained and expressed in equation 

26.

























































































































+

=

− −

Z

Y

X

Z

Y

X

F

F

F

A

A

A

F

F

F

A

A

A

F

F

A F F F

F

F

F

A

A

A

A

g

g

g

g

g

g

r

q

p

r

q

p

T S A T I T S A

I S A S A

&

&

&

&

&

&

r r

r r

~ ~

, (26)

where 

{ } { } AF

A

A

A

C

C

C

A

A

A

A S B

N

M

L

N

M

L

r

q

p

g S I

A

A

A

w + r





















+





















−





















= − , (27)

and

{ } { } F AF

F

F

F

F

C

C

C

F

F

F

F F F T S B

N

M

L

T

N

M

L

r

q

p

g T S I

F F

w r

− ~





















+





















+





















= − . (28)

Collectively, equations 3, 8, 9, 10, and 26 constitute the gimbal nose projectile dynamic 

model. 

The total external load acting on the composite body is due to weight and steady

aerodynamic forces on both the forward and aft body projectile components. The
16

weight force components in the aft body reference frame is given by equation 29. 





















= +



















 −

A A

A A

A

c c

c s

s

m m g

Z

Y

X

A F

G

G

G

q f

q f

q

( ) . (29)

The steady aerodynamic force on the aft body is provided by equation 30.

( )





































+ +

+ +

+ +

















+ +

+

+

















= −





















2 2 2

2 2 2

2 2 2

2 2 2

2 2

2

2

2 4

1

C w u v w

C v u v w

u v w

u v w

v w

C C

D

Z

Y

X

A

N

A

N

A

X

A

XO

A

A

A

a

a

p

r . (30)

Expressions for the forward body aerodynamic forces take on the same form as equation 

30. Aerodynamic coefficients in equation 30 depend on local Mach number at the 

projectile mass center and are computed using linear interpolation from tabulated data. 

The right-hand side terms in equation 26 contain the external moments acting on each 

section of the projectile. These equations contain contributions from steady and

unsteady aerodynamics. The steady aerodynamic moments are computed for each 

individual body with a cross product between the steady body aerodynamic force vector 

and the distance vector from the center of gravity to the center of pressure. The

unsteady body aerodynamic moments provide a damping source for projectile angular 

motion and are given for the forward body by equation 31.





































+

=





















V

r DC

V

q DC

V

p DC C

q D

N

M

L

F

F NR

F

F MQ

F

F F LP

DD

a

F

UA

F

UA

F

UA

2

2

2

~ , (31)

where 

( )

2 2 2 2

a

u v w D

8

1

q

~ = ρ + + π .

The expression for the aft section takes on similar form. Air density is computed using 

the center of gravity position of the projectile in concert with the standard atmosphere 

[6]. Finally, the total aerodynamic angle of attack of the aft section is defined in equation 

32.















 +

=

−

u

v w

A

2 2

1 a tan . (32)
17

The aerodynamic angle of attack of the forward body is computed in the same manner, 

except the velocity components are first converted to the forward body reference frame. 

3. Results

The equations of motion for the nine degree-of-freedom gimbal nose projectile model 

discussed, and a six degree-of-freedom rigid projectile model [6] were numerically

integrated to obtain simulated impact points at 1 km, 2 km, and

3 km. The equations of motion were integrated using a fourth-order Runge-Kutta 

scheme with a time step of 0.000001 s. The physical properties of the projectile are 

provided in Table 1. Table 2 shows the nominal launch conditions of the projectile. 

Table 1. Baseline configuration properties.

Physical Parameter Value

Total Projectile Mass 3.16688 Kg

Aft-to-total mass ratio 0.99

Aft Roll Inertia 0.00065 slug.ft2 (1.36 Kg.m2)

Aft Pitch and Yaw Inertia 0.033 slug.ft2 (0.045 Kg.m2)

Aft Length Dimension 0.34 m

Aft Reference Diameter 0.037 m

Forward Roll Inertia 4.2e-6 slug.ft2 (5.69e-6 Kg.m2)

Forward Pitch and Yaw Inertia 2.5e-5 slug.ft2 (3.39e-5 Kg.m2)

Forward Length Dimension 0.13 m

Forward Reference Diameter 0.04 m

Torsional Spring Constant 99,000 lbf.ft/rad

Torsional Spring Damper 0.00 N.sec

F CNa

2.00

Gimbal Joint Location 0.38 m

Figure 2, 3, and 4 show position traces vs. time under baseline launch conditions. This 

trajectory is typical for a cannon-launched tank projectile. Both the rigid and gimballed 

nose projectiles follow a similar path with small differences not notable when viewing 

the entire trajectory. Figure 5 plots the velocity of the mass center of the rigid and 

gimbal nose projectiles over the baseline trajectory. The total velocity decays from a 

launch speed of 5,590 ft/s to a speed of 2,470 ft/s at 3-km range. The roll rate of the aft 

section of the gimbal nose projectile and the roll rate of the rigid projectile are shown in 

Figure 6. Because the aft body of the gimbal nose projectile has slightly lower roll inertia 

than the rigid projectile and the gimbal friction is zero, the rigid projectile roll rate is 

slightly less than the aft section of the gimbal nose projectile. The Euler pitch and yaw 

angles of the aft main projectile body are compared to the rigid projectile Euler pitch and 

yaw angles in Figures 7 and 8, while the Euler pitch and yaw angles of the nose section 

are plotted in Figures 9 and 10. Because the nose section inertia properties are smaller 

than the main projectile body, it oscillates at a notably higher frequency. Both the nose 
18

and main projectile sections angular motion is well behaved, with maximum oscillation 

under 1°. 

Dispersion at the target was created through Monte Carlo simulation of the initial pitch 

and yaw rate of the projectile. The initial pitch and yaw rates were modeled as

independent Gaussian random variables with a mean of zero and standard deviation

Table 2. Nominal initial conditions.Table 3. Nominal initial conditions.

Parameter Value

x 5.4 m

y 6.6e-6 m

z -0.001 m

u 1703.6 m/s

v 0.13 m/s

w -0.9 m/s

y A

3.0°

q A

5.0°

f A

0.0°

pA

0.0 deg/s

q A

60.7°/s

A

r

-0.8°/s

y F

0.0°

q F

0.0°

fF

0.0°

pF

0.0°/s

qF

60.7°/s

F

r

-0.8°/s

of 3 rad/s. A sample size of 50 simulations was used in computing impact point 

dispersion statistics. Figures 11–13 show the Monte Carlo simulation impact points for 

the baseline rigid and gimbal nose projectile configurations at a range of 1 km, 2 km, and 

3 km, respectively. In all the charts, the large circles correspond to a region such that 

66% of the shot impacts fall within the circle. The large dashed circle corresponds to the 

rigid projectile, while the small solid circle corresponds to the gimbal nose

configuration. The dispersion circle radii for the rigid projectile at 1 km, 2 km, and 3 km 

is 1.1 m, 2.2 m, and 3.3 m, while the dispersion circle radii for the gimbal nose projectile 

at 1 km, 2 km, and 3 km is 0.5 m, 0.9 m, and 1.4 m, respectively. Notice that the mean 

impact point of the two projectile configurations is different. The ratio of the dispersion 

circle radius for the gimbal nose to rigid projectile configuration is 0.43, and it is

independent of range. Thus, for the example penetrator projectile equipped with a 

gimbal nose, dispersion at any range can be reduced by a factor of 0.43. 
19

Figure 2. Range vs. time.

Figure 3. Cross range vs. time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

250

500

750

1000

1250

1500

Time (sec)

Rigid 

Gimbal Nose

Range (m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

10

20

30

40

50

60

70

80

Time (sec)

Cross Range (m)

Rigid 

Gimbal Nose

Cross Range (m)
20

Figure 4. Altitude vs. time.

Figure 5. Total velocity vs. time.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-140

-120

-100

-80

-60

-40

-20

0

Time (sec)

Altitude (m, positive down)

Rigid 

Gimbal Nose

Altitude (m, positive down)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1250

1300

1350

1400

1450

1500

1550

1600

1650

1700

1750

Time (sec)

Total Projectile Velocity (m/sec)

Rigid 

Gimbal Nose

Total Projectile Velocity (m/sec)
21

Figure 6. Roll rate vs. time.

Figure 7. Euler pitch angle vs. time.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

2

2.5

3

3.5

4

x 104

Time (sec)

Roll Rate (deg/sec)

Rigid 

Gimbal Nose (Aft Body)

Roll Rate (deg/sec)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

4.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

5.5

5.6

Time (sec)

Euler Pitch Angle (deg)

Rigid 

Gimbal Nose (Aft Body)

Euler Pitch Angle (deg)
22

Figure 8. Euler yaw angle vs. time.

Figure 9. Euler pitch angle of nose vs. time.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2.94

2.96

2.98

3

3.02

3.04

3.06

3.08

Time (sec)

Euler Yaw Angle (deg)

Rigid 

Gimbal Nose (Aft Body)

Euler Yaw Angle (deg)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time (sec)

Euler Pitch Angle of Nose (deg) Euler Pitch Angle of Nose (deg)
23

Figure 10. Euler yaw angle of nose vs. time.

Figure 11. Impact point dispersion at 1 km range.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time (sec)

Euler Yaw Angle of Nose (deg) Euler Yaw Angle of Nose (deg)

50 50.5 51 51.5 52 52.5 53 53.5 54

-88

-87.5

-87

-86.5

-86

-85.5

-85

-84.5

-84

 y (m) 

 ----- Rigid (*)

 Gimbal Nose (o)

z (m)
24

Figure 12. Impact point dispersion at 2 km range.

Figure 13. Impact point dispersion at 3 km range.

101 102 103 104 105 106 107 108 109

-172

-171

-170

-169

-168

-167

-166

-165

-164

 y (m) 

 ----- Rigid (*)

 Gimbal Nose 

(o)

 z (m)

150 152 154 156 158 160 162 164

-248

-246

-244

-242

-240

-238

-236

-234

 y (m) 

 ----- Rigid (*)

 Gimbal Nose 

 z (m)
25

The impact point charts shown in Figures 11–13 are for a nominal configuration. 

As shown in Table 1, the baseline gimbal nose configuration has a mass ratio of 

0.99, and the gimbal joint is frictionless. Figure 14 investigates how the nose 

normal force coefficient effects the dispersion radii previously discussed. When 

the lift coefficient of the nose is zero, the aerodynamic normal load on the

projectile is only from the aft body. The impact statistics approach the rigid 

projectile case, which are shown as diamonds on the chart. A steady decrease in 

the impact dispersion is realized as the nose normal force coefficient F CNA is 

increased according to slender body theory F CNA = 2. Figure 15 plots the effect of 

the mass ratio between the forward and aft projectiles section on impact point 

dispersion radii. Within practical design limit, the mass ratio between the

forward and aft projectile sections does not effect impact point dispersion. 

Figure 14. Impact point dispersion vs. nose lift coefficient.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

1.5

2

2.5

3

3.5

Lift Coefficient of Forward Section (CNA

F )

1 km

2 km

3 km

Dispersion Radius (m)
26

Figure 15. Impact point dispersion vs. forward-to-total mass ratio.

Figure 16 shows the impact point dispersion radii at 1 km, 2 km, and 3 km range 

as a function of the friction constant Cg. At 1 km and 2 km range, the dispersion 

radii is an essential constant for all values of Cg. However, at range of 3 km, the 

dispersion rapidly increases. Since projectile velocity exponentially decreases

with range, a critical combination of projectile velocity and damper constant 

combined to induce large impact point dispersion, which is in fact a much larger 

dispersion than a similar rigid projectile. The root of this problem is shown in 

Figures 17 and 18, which plot the position of the tip of the nose of the projectile 

with respect to the main projectile body. In the case where the gimbal joint is 

frictionless (Figure 17), the nose initially rotates with relatively large angles and 

progresses toward a steady state limit cycle of low amplitude. In the case where 

Cg = 1.0 (Figure 18), the nose angle continuously increases as it approaches the 

interference limit of total nose deflection of ag = 5

o

. Hence, the rotational

dynamics of the nose are unstable in this case. 

0 0.05 0.1 0.15 0.2 0.25

0

0.5

1

1.5

2

2.5

3

3.5

Forward-to-Total Projectile Mass Ratio

1 Km

2 Km

3 Km

 Dispersion Radius (m)
27

Figure 16. Impact point dispersion vs. gimbal viscous friction coefficient.

Figure 17. Motion of nose tip with respect to the aft body (Cg = 0.00).

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

10

0

0

1

2

3

4

5

6

7

8

Damper Coefficient (Cg)

1 Km

2 Km

3 Km

 Dispersion Radius (m)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

x 10-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

x 10-3

j

A

-axis (m)

kA-axis (m)
28

Figure 18. Motion of nose tip with respect to the aft body (Cg = 1.00).

4. Conclusion

A penetrator projectile equipped with a gimbal nose wind screen has the

potential to drastically reduce impact point dispersion. By mounting the gimbal 

joint forward of the nose aerodynamic center, the nose tends to turn into the 

wind, reducing the sensitivity of the trajectory to launch disturbances. In the 

example case considered, impact point dispersion was reduced by more than 

50%. The mean impact point of the rigid and gimbal nose projectile

configurations are different. This difference will require fire control system logic 

to be modified, depending on the particular projectile configuration being

launched. Gimbal joint friction is an important design parameter that influences 

the effectiveness of the gimbal joint to reduce impact point dispersion. For 

sufficiently large friction in the gimbal joint, the impact point dispersion

increases well beyond the dispersion encountered with a rigid body projectile 

because the nose rotational dynamics are unstable. Hence, the gimbal joint must 

be designed such that the joint does not degrade as the round sits in long-term 

storage. Impact point dispersion steadily increases as the nose aerodynamic 

normal coefficient decreases. Also, dispersion is essentially independent of the 

mass ratio of the nose and main projectile sections. 

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

j

A

-axis (m)

Ak A-axis (m)
29

INTENTIONALLY LEFT BLANK.
30

5. References

1. Goddard, R. H. “Apparatus for Steering Aircraft.” U.S. Patent 2594766, April 

1952.

2. Barrett, R., and J. Stutts. “Modeling, Design, and Testing of a BarrelLaunched Adaptive Munition.” Proceedings of the 4th Annual Society of 

Photo-Optical Engineers Symposium on Smart Structures, San Diego, CA, 

Society of Photo-Optical Engineers, New York, March 1997.

3. Krantz, W. “High Velocity Aerodynamic Body Having Telescopic Nose Tip.” 

U.S. Patent 4756492, July 1988.

4. Schmidt, E., and W. Donovan. “Technique to Reduce Yaw and Jump of FinStabilized Projectiles.” Journal of Spacecraft and Rockets, vol. 35, no. 1,

pp. 110–111, 1998.

5. Murphy, C. “Free Flight Motion of Symmetric Missiles.” U.S. Army Ballistic 

Research Laboratories, BRLR-1216, Aberdeen Proving Ground, MD, 

July 1963. 

6. Costello, M. F., and D. P. Anderson. “Effect of Internal Mass Unbalance on 

the Terminal Accuracy and Stability of a Field Artillery Projectile.” 

Proceedings of the 1996 AIAA Atmospheric Flight Mechanics Conference, 

San Diego, CA, 1996.
SYMBOLS

x, y, z Components of the position vector of center of mass of the composite body 

in an inertial reference frame. 

u, v, w Components of the velocity vector of the mass center of the composite body in the aft body 

reference frame.

f A q A y A, , Aft body Euler roll, pitch, and yaw angles.

f F q F y F, , Forward body Euler roll, pitch, and yaw angles. 

A A A p ,q ,r Components of the angular velocity vector of the aft body in the aft body reference frame. 

F F F p ,q ,r Components of the angular velocity vector of the forward body in the forward body 

reference frame. 

X,Y,Z Total external force components on the composite body in the aft body reference frame.

X A YA Z A, , Total external force components on the aft body in the aft body reference frame.

XF YF ZF, , Total external force components on the forward body in the forward body reference frame.

XC YC ZC, , Gimbal joint constraint force components in the aft body reference frame.

ii
L,M ,N Total external moment components on the composite body in the aft body reference frame.

LA M A N A, , Total external moment components on the aft body in the aft body reference frame.

LF MF NF, , Total external force components on the forward body in the forward body reference frame.

mA Aft body mass.

mFForward body mass.

A

I Mass moment of inertia matrix of the aft body about the aft body mass.

F

I Mass moment of inertia matrix of the forward body about the forward body mass center.

AX AY AZ

r ,r ,r Components of the position vector from the gimbal joint to the mass center of the aft body 

in the aft body reference frame.

FX FY FZ

r , r , r Components of the position vector from the gimbal joint to the mass center of the forward 

body in the forward body reference frame.

FX FY FZ

r r r

~

,

~

,

~

Components of the position vector from the gimbal joint to the mass center of the forward 

body in the aft body reference frame.

Cg Gimbal joint viscous damping constant.

iii
Kg Gimbal joint spring constant.

SA Aerodynamic reference area for aft projectile section.

SF Aerodynamic reference area for forward projectile section.

r Air density.

ag Angle between forward body centerline and aft body center line.

∗ a Dead band angular interference limit between forward and aft section. 
31

REPORT DOCUMENT A 110N PAGE Form Approved 

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Pllbllo noDO ng bunl•n tlr lcooll.a on oflntlnna on lc• ma d awon~ga 111our~~ar noq~onGII,Inoludlng • m• tlrno lfa•:.olng In o on.,_ '2'anllllng nlc ng Cia aJUnl,.,_ 

gatl•nng •nd malnta lnlng tie 11a ta n"CifCI, and oomDI•tn 1 and no ~•·:.olng tl• ooll .a ton otln llnnatan. ••nd oomm•ntc nooanllng til c bunl•n • dma" or an~· otltr a cDtotottll c 

ooll.aton otlntl nnaton, lnol udl no QIOif don c fbr noCiuolng til c IIUnlfn, tlll\ll ~ lngbn NnCiouarUr-c 8tr~oo .,_ 01 noobn~" llr In llnnalon 0Dfnltlon c and AIDOrk, 12 16 Je•ra~n 

0.\lc 111 n•:.''" 8ul" 120 Arlin bn VA 22202~02 :tnd b tl• Ofto• otUan:t •m•nhnCI QICI • AtCiuoton Av o 704.0 1U ll\ll~ln tin 002060S. 

1. AGENCY USE ONLY(Luve blank) 

4. TITLE A NO SUBTITLE 

U.S. Army Research Laboratory 

1\TTN: AMSRL-WM-BC 

lo.berdeen Proving Ground, MD 21005-5066 

11. SUPPLEMENTARY NOTES 

*Oregon State University 

13. A BSTRA CT(Maximum 200 words) 

A key and often quoted metric associated with gun systems is impact point accuracy. The extent of impact 

dispersion is a complex function of a battery of parameters, including gun geom e1ry and tolerances, the fire control 

system, projectile manufacturing tolerances, etc. The work reported here inveftigates potential impact point accuracJii 

improvement for a penetrator-type projectile realized by replacing the rigid nose cone wind screen with a passive 

gimballed nose. By com paring the impact point dispersion of a rigid projectile with a similar gimballed nose projectile, 

it is shown that impact point accuracy can be significantly improved. For the ex ample penetrator projectile considere~ 

impact point dispersion is reduced by more than 50%. 

14. SUBJECT TERMS 

dispersion, fin-stabilized projectile, nose control 

17. SECURITY CLASSIFICATION 

OF REPORT 

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1 US MILITARY ACADEMY

MATH SCI CTR OF 

EXCELLENCE

DEPT OF MATHEMATICAL SCI

MDN A MAJ DON ENGEN

THAYER HALL

WEST POINT NY 10996-1786

3 DIR

SNL

A HODAPP

W OBERKAMPF

F BLOTTNER

DIVISION 1631

ALBUQUERQUE NM 87185

3 ALLIANT TECH SYSTEMS

C CANDLAND

R BURETTA

R BECKER

7225 NORTHLAND DR

BROOKLYN PARK MN 55428

3 DIR USARL

AMSRL SE RM

H WALLACE

AMSRL SS SM

J EIKE

A LADAS

2800 POWDER MILL RD

ADELPHI MD 20783-1145

1 OFC OF ASST SECY OF ARMY

FOR R&D

SARD TR

W MORRISON

2115 JEFFERSON DAVIS HWY

ARLINGTON VA 22202-3911

2 CDR USARDEC

AMSTA FSP A

S DEFEO

R SICIGNANO

PICATINNY ARESENAL NJ

07806-5000

2 CDR USARDEC

AMSTA AR CCH A

M PALATHINGAL

R CARR

PICATINNY ARESENAL NJ

07806-5000
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

4

5 TACOM ARDEC

AMSTA AR FSA

K CHIEFA

AMSTA AR FS 

A WARNASCH

AMSTA AR FSF

W RYBA

AMSTA AR FSP

S PEARCY

J HEDDERICH

PICATINNY ARESENAL NJ

07806-5000

5 CDR US ARMY MICOM

AMSMI RD

P JACOBS

P RUFFIN

AMSMI RD MG GA

C LEWIS

AMSMI RD MG NC

C ROBERTS

AMSMI RD ST GD

D DAVIS

RSA AL 35898-5247

3 CDR US ARMY AVN TRP CMD

DIRECTORATE FOR ENGRNG

AMSATR ESW

M MAMOUD

M JOHNSON

J OBERMARK

RSA AL 35898-5247

1 DIR US ARMY RTTC

STERT TE F TD

R EPPS

BLDG 7855

REDSTONE ARSENAL AL

38598-8052

2 STRICOM

AMFTI EL

D SCHNEIDER

R COLANGELO

12350 RESEARCH PKWY

ORLANDO FL 32826-3276

1 CDR OFFICE OF NAVAL RES

J GOLDWASSER CODE 333

800 N QUINCY ST RM 507

ARLINGTON VA 22217-5660

1 CDR US ARMY RES OFFICE

AMXRO RT IP TECH LIB

PO BOX 12211

RESEARCH TRIANGLE PARK NJ

27709-2211

4 CDR US ARMY AVN TRP CMD

AVIATION APPLIED TECH DIR

AMSATR TI 

R BARLOW

E BERCHER

T CONDON

B TENNEY

FT EUSTIS VA 23604-5577

3 CDR NAWC

WEAPONS DIV

CODE 543400D

S MEYERS

CODE C2744

T MUNSINGER

CODE C3904

D SCOFIELD

CHINA LAKE CA 93555-6100

1 CDR NSWC

CRANE DIVISION

CODE 4024

J SKOMP

300 HIGHWAY 361

CRANE IN 47522-5000

1 CDR NSWC

DAHLGREN DIV

CODE 40D

J BLANKENSHIP

6703 WEST HWY 98

PANAMA CITY FL 32407-7001

1 CDR NSWC

J FRAYSEE

D HAGEN

17320 DAHLGREN RD

DAHLGREN VA 22448-5000
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5

5 CDR NSWC

INDIAN HEAD DIV

D GARVICK CODE 40D

L FAN CODE 4110C

V CARLSON CODE 4120

H LAST CODE 4140E

T GRIFFIN CODE 450D

101 STRAUSS AVE

INDIAN HEAD MD 20640-5000

1 CDR NSWC

INDIAN HEAD DIV

LIBRARY CODE 8530

BLDG 299

101 STRAUSS AVE

INDIAN HEAD MD 20640

2 US MILITARY ACADEMY

MATH SCI CTR OF 

EXCELLENCE

DEPT OF MATHEMATICAL SCI

MDN A 

MAJ D ENGEN

R MARCHAND

THAYER HALL

WEST POINT NY 10996-1786

3 CDR US ARMY YUMA PG

STEYP MT AT A

A HOOPER

STEYP MT EA

YUMA AZ 85365-9110

6 CDR NSWC

INDIAN HEAD DIV

CODE 570D J BOKSER

CODE 5710 L EAGLES

J FERSUSON

CODE 57 C PARIS

CODE 5710G S KIM

CODE 5710E S JAGO

101 STRAUSS AVE ELY BLDG

INDIAN HEAD MD 20640-5035

1 BRUCE KIM

MICHIGAN STATE UNIVERSITY

2120 ENGINEERING BLDG

EAST LANSING MI 48824-1226

2 INDUSTRIAL OPERATION CMD

AMFIO PM RO

W MCKELVIN

MAJ BATEMAN

ROCK ISLAND IL 61299-6000

3 PROGRAM EXECUTIVE 

OFFICER

TACTICAL AIRCRAFT 

PROGRAMS

PMA 242 1

MAJ KIRBY R242

PMA 242 33

R KEISER (2 CPS)

1421 JEFFERSON DAVIS HWY

ARLINGTON VA 22243-1276

1 CDR NAVAL AIR SYSTEMS CMD

CODE AIR 471

A NAKAS

1421 JEFFERSON DAVIS HWY

ARLINGTON VA 22243-1276

4 ARROW TECH ASSOCIATES INC

R WHYTE

A HATHAWAY

H STEINHOFF

1233 SHELBOURNE RD SUITE D8

SOUTH BURLINGTON VT 05403

3 US ARMY AVIATION CTR

DIR OF COMBAT 

DEVELOPMENT

ATZQ CDM C

B NELSON

ATZQ CDC C 

T HUNDLEY

ATZQ CD

G HARRISON

FORT RUCKER AL 36362
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6

ABERDEEN PROVING GROUND

3 CDR

USA ARDEC

AMSTA AR FSF T

R LIESKE

J WHITESIDE

J MATTS

BLDG 120

1 CDR

USA TECOM

AMSTE CT

T J SCHNELL

RYAN BLDG

3 CDR

USA AMSAA

AMXSY EV

G CASTLEBURY

R MIRABELLE

AMXSY EF

S MCKEY

49 DIR USARL

AMSRL WM

I MAY

T ROSENBERGER

ABERDEEN PROVING GROUND

AMSRL WM BA

W HORST JR

W CIEPELLA

F BRANDON

T BROWN (5 CPS)

L BURKE

J CONDON

B DAVIS

T HARKINS (5 CPS)

D HEPNER

V LEITZKE

M HOLLIS

A THOMPSON

G BROWN

ABERDEEN PROVING GROUND

AMSRL WM BB

B HAUG

OS

J SAHU

M BUNDY

K SOENCKSEN

D LYON

AMSRL WM BC

P PLOSTINS (4 CPS) 

G COOPER

B GUID

AMSRL WM BC

J BENDER

J NEWILL

J GARNER

V OSKAY

S WILKERSON

W DRYSDALE

R COATES

A MIKHAL

J WALL

AMSRL WM BD

B FORCH

AMSRL WM BE

M SCHMIDT

AMSRL WM BF

J LACETERA

P HILL

AMSRL WM BR

C SHOEMAKER

J BORNSTEIN
7