Adding Liquid Payloads Effects to the 6-DOF Trajectory 

of Spinning Projectiles 

by Gene R. Cooper 

ARL-TR-5118 March 2010 

Approved for public release; distribution is unlimited. 
NOTICES 

Disclaimers 

The findings in this report are not to be construed as an official Department of the Army position unless 

so designated by other authorized documents. 

Citation of manufacturer’s or trade names does not constitute an official endorsement or approval of the 

use thereof. 

Destroy this report when it is no longer needed. Do not return it to the originator. 
Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5069 

ARL-TR-5118 March 2010 

Adding Liquid Payloads Effects to the 6-DOF Trajectory 

of Spinning Projectiles 

Gene R. Cooper 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
ii 

REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering 

and maintaining the data needed, and completing and reviewing the collection information. Send comments regarding this burden estimate or any other aspect of this collection of information, 

including suggestions for reducing the burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson 

Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to 

comply with a collection of information if it does not display a currently valid OMB control number. 

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 

1. REPORT DATE (DD-MM-YYYY)

March 2010 

2. REPORT TYPE 

Final 

3. DATES COVERED (From - To) 

1 October 2006–30 September 2007 

4. TITLE AND SUBTITLE 

Adding Liquid Payloads Effects to the 6-DOF Trajectory of Spinning Projectiles 

5a. CONTRACT NUMBER 

5b. GRANT NUMBER 

5c. PROGRAM ELEMENT NUMBER 

6. AUTHOR(S) 

Gene R. Cooper 

5d. PROJECT NUMBER 

AH80 

5e. TASK NUMBER 

5f. WORK UNIT NUMBER 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

U.S. Army Research Laboratory 

ATTN: RDRL-WML-E 

Aberdeen Proving Ground, MD 21005-5069 

8. PERFORMING ORGANIZATION 

 REPORT NUMBER 

ARL-TR-5118 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 

11. SPONSOR/MONITOR'S REPORT 

 NUMBER(S) 

12. DISTRIBUTION/AVAILABILITY STATEMENT 

Approved for public release; distribution is unlimited. 

13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

This report outlines a relatively straightforward technique to incorporate the effect of liquid payloads on the numerical 

prediction of projectile trajectories. The key to this method is the addition of liquid moments imparted onto the projectile body. 

The first step to include the effects of liquid payloads is to model a low-viscosity liquid while undergoing typical projectile

translational and angular motions. Since most projectiles carrying liquid payloads are spin stabilized with high spin rates, the 

moment contribution of the liquid to the angular motion of the projectile can be found using linear theory. After developing an 

integrated dynamic model for a projectile with a liquid payload, example trajectory characteristics are shown that exhibit typical 

dynamic behavior of a projectile with a liquid payload. Results comparing solid projectiles with a similar liquid-filled 

projectile are presented as well as a discussion of flight stability caused by a liquid payload. 

15. SUBJECT TERMS

projectile dynamics, liquid payloads, rotation liquids, projectile stability, linear and nonlinear dynamics 

16. SECURITY CLASSIFICATION OF: 

17. LIMITATION 

OF ABSTRACT 

UU 

18. NUMBER 

OF PAGES 

26 

19a. NAME OF RESPONSIBLE PERSON

Gene R. Cooper 

a. REPORT 

Unclassified 

b. ABSTRACT 

Unclassified 

c. THIS PAGE 

Unclassified 

19b. TELEPHONE NUMBER (Include area code) 

410-306-0787 

Standard Form 298 (Rev. 8/98) 

Prescribed by ANSI Std. Z39.18 
iii 

Contents 

List of Figures iv

1. Introduction 1

2. Projectile Flight Dynamics 1

3. Liquid Payload Moments 5

4. Liquid Motion Model 8

5. Liquid Moments 10

6. Results 11

7. Conclusion 16

List of Symbols, Abbreviations, and Acronyms 17

Distribution List 19 
iv 

List of Figures 

Figure 1. Projectile position coordinate definitions. .......................................................................2

Figure 2. Projectile orientation definitions. ....................................................................................2

Figure 3. Coning frequency for the M483. ...................................................................................11

Figure 4. CLSM vs. coning frequency. ...........................................................................................12

Figure 5. Comparison of projectile roll moment for frozen and flowing liquid. ..........................13

Figure 6. Comparison of projectile spin rate for frozen and flowing liquid. ................................13

Figure 7. Time history of CLSM 1. ..................................................................................................14

Figure 8. Slow mode CLSM fast coning frequency T1. ..................................................................14

Figure 9. Roll rate vs. time for flowing and frozen liquid payloads. ............................................15

Figure 10. Comparison of pitch rates for flowing and frozen liquid payloads. ............................15

Figure 11. Angle of attack vs. time. ..............................................................................................16
1 

1. Introduction 

While the bulk of projectiles in use today behave as rigid bodies while in flight, a notable 

number of projectiles are purposefully designed to carry a liquid payload. For example, smoke 

screens delivered by artillery rounds consist of a typical spin-stabilized shell containing a 

canister filled with white phosphorous. White phosphorous is in a liquid state when hot. 

Another example is new less-than-lethal projectiles having a concentric cylindrical cavity filled 

with liquid that delivers this payload to a target upon impact. Finally, some projectiles are 

designed to be general payload delivery shells, including delivery of medical supplies such as 

intravenous fluid bags. 

Projectiles with fluid payloads have been heavily researched over the years, mainly due to the 

fact that these rounds can exhibit severe flight instabilities. Characteristics of this flight 

instability are sharp increases in projectile aerodynamic angle of attack accompanied by large 

changes in spin rate.1

Predictions of the instability induced by a liquid payload installed in a projectile have 

predominantly been analyzed by linear fluid dynamic theory subjected to a linear projectile 

coning motion. Unfortunately, these well-developed theories do not directly mesh with standard 

six-degree-of-freedom (6-DOF) projectile flight dynamic models. Hence, while a good deal of 

information is known about the effect of liquid payloads on projectiles, this information has not 

made its way into 6-DOF computer tools. The purpose of this report is to bridge the gap 

between the body of literature on effects of liquid payloads on projectiles and 6-DOF projectile 

flight dynamic modeling. The report begins with a review of rigid projectile flight dynamic 

modeling along with a description of modeling rotating liquid in a cavity. The two models are 

subsequently integrated such that a projectile flight dynamic model with a liquid payload results. 

This flight dynamic model is exercised on an example shell. Comparisons are made between a 

liquid-filled projectile and a similar solid projectile highlighting predictive capability of the new 

model. 

2. Projectile Flight Dynamics 

A 6-DOF rigid projectile model is employed to predict the dynamics of a projectile in flight. 

These equations assume a flat Earth. The 6-DOF comprises the three translational components 

describing the position of the projectile’s center of mass and the three Euler angles describing the 

orientation of the projectile with respect to the Earth. Figures 1 and 2 provide a visualization of 

the degrees of freedom. 

 1

Karpov, B. G. Experimental Observations of the Dynamic Behavior of Liquid-Filled Shell; BRL-TR-1171; U.S. Army 

Ballistics Research Laboratory: Aberdeen Proving Ground, MD, August 1961. 
2

Figure 1. Projectile position coordinate definitions. 

Figure 2. Projectile orientation definitions. 
3 

The equations of motion2 for the 6-DOF model, derived in the no-roll frame, are shown in 

equations 1–4. 

0

x cc s sc u

y cs c ss v

z s cw

  

  

 

                     

 

 

 

. (1)































































r

q

p

c

t

~

~

~

0 0 1

0 1 0

1 0

















. (2)

/

/

/

u X m rv qw

v Y m t rw ru

w Z m qu t rv





                   

           

     

    

     

. (3)

  1

0

0

0

p L rq p

q I M r rt I q

r N q rt r







                                     

     

    

    

    

. (4) 

The force acting on the projectile in equation 3 comprises the weight force (W), aerodynamic 

force, and liquid payload force (L). The aerodynamic force is split into a standard (A) and 

Magnus (M) aerodynamic force. The combination of forces is expressed in equation 5. 

W AM L

W AM L

W AM L

XX X X X

YY Y Y Y

ZZ Z Z Z

ìü ì ü ì ü ì ü ì ü ïï ï ï ï ï ï ï ï ï

í  = ++ + í  í  í  í  ïï ï ï ï ï ï ï ï ï

î  î  î  î  î 

   

   

   

. (5) 

Equation 6 provides the expression for the weight force in the no-roll coordinate system. 

0

W

W

W

X s

Y mg

Z c





           

      







. (6)

 2

Slegers, N.; Kyle, J.; Costello, M. Nonlinear Model Predictive Control Technique for Unmanned Air Vehicles. Journal of 

Guidance, Control, and Dynamics 2006, 29 (5), 1179–1188. 
4 

Equation 7 provides the expression for the aerodynamic force in the no-roll coordinate system. 

This force acts upon the projectile at the aerodynamic center of pressure. 

2 2

A X0 X2 X2

2 2

A A Y0 YB1

A Z0 ZA1

X C +C α + C β

π Y = ρV D C +C β 8

Z C +C α

           

     







. (7) 

Equation 8 provides the expression for the Magnus force in the no-roll coordinate system. The 

Magnus force acts upon the projectile at the Magnus force center of pressure. 

M

2 2 NPA

M A

A

M

NPA

A

0 X

π pDC α Y = ρV D

8 2V

Z

-pDC β

2V

   

   

      

          



 





. (8) 

Equations 7 and 8 are based on Mach-number-dependent coefficients, the aerodynamic angles of 

attack given in equations 9 and 10, and the total aerodynamic velocity given in equation 11. 

( ) -1 α = tan w u A A    . (9)

( ) -1

A A β = tan v u    . (10) 

2

A

2

A

2

A A w

~ v~ u~ V    . (11) 

The moment acting on the projectile in equation 4 comprises the moment due to the standard 

aerodynamic force (A), the moment due to the Magnus aerodynamic force (M), the unsteady 

aerodynamic moment (UA), and the liquid payload moment (L) as shown in equation 12. 

A M UA L

A M UA L

A M UA L

LL L L L

M=M +M +M +M

NN N N N

ìü ì üì üì üì ü ïï ï ïï ïï ïï ï

í  í  í  í  í  ïï ï ïï ïï ïï ï

î  î  î  î  î 

   

   

   

. (12) 
5 

The moment due to the aerodynamic force is expressed in equation 13. 

0

0

0

A CAZ CAY A

A CAZ CAX A

A CAY CAX A

L RR X

M R RY

N RR Z

 

 

 

               

        

 

 

 

. (13)

The moment due to the Magnus force is expressed in equation 14. 

0

0

0

M CMZ CMY M

M CMZ CMX M

M CMY CMX M

L RR X

MR R Y

N RR Z

 

 

 



 



            

       

 

 

 

. (14)

The unsteady aerodynamic moments acting on the projectile are expressed in equation 15. 

2 3

2

8 2

2

LP

DD

A

UA

MQ

UA A

A

UA

NR

A

pDC C

V

L qDC

M VD

V N

rDC

V









             

          



  





. (15)

The coefficients used in this model are projectile-specific functions of the Mach number of the 

projectile. For fin-stabilized projectiles, Magnus force and moment are typically ignored since 

their effect is rather small for slowing rolling projectiles. 

The dynamic equations of motion expressed in equations 1–15 are highly nonlinear. Due to this 

fact, numerical integration is commonly used to obtain solutions to this initial value problem. 

3. Liquid Payload Moments 

Angular motion of a projectile is altered by the internal motion of the liquid payload. The liquid 

dynamics in this report parallel the dynamics given by Murphy3 such that the linear projectile 

theory gives a second-order differential equation for     

~i ~ ~ written in terms of the lateral 

force and moment 

 3

Murphy, C. H. Angular Motion of a Spinning Projectile With a Viscous Liquid Payload; ARBRL-MR-3194; U.S. Army 

Ballistics Research Laboratory: Aberdeen Proving Ground, MD, August 1982. (See also Journal of Guidance, Control, and 

Dynamics 1983, 6, 280–286.) 
6 

  

 

 

2

A –2

NPA y MQ

2 3

A

MA

2

A –2

NPA x MPA

ξ+ H–iσφ ξ– M+iσφT ξ = 0

πρV D H = pC –k C 8m

πρV D M = C 8Iy

πρV D = pC +k pC 8m

σ = Ix Iy.



     



 

(16)

Solving this last equation gives the linear sum of two complex polar arms: 

1 2 iT P t iT P t

1 2 ξ = K e + K e  , (17)

where 

 

 

j j0 j j

j g

2 2

g 

j

j

j j

ln K K = ε T Pt ,

T = σ 2 1± 1–1/S ,

S = σ P 4M if H <<1, Σ<<1, and

T H–σ

ε =– .

2T –σ T P

   



Consider a projectile containing a liquid-filled cavity subjected to the angular motion of a freeflight projectile. This angular motion generates inertial waves in the contained liquid, which, in 

turn, impacts the projectile angular motion since liquid moments are now acting on the projectile. 

The assumption is made that liquid moments are small enough so the linear theory assumptions 

used to obtain equation 16 are not violated. 

X,Y,Z and Z

~ Y, ~ X, are respectively body and nonrolling coordinate systems, where the x-axis 

is the projectile’s symmetry axis. The earth-fixed axes E E ZE X ,Y , has XE initially along the 

velocity vector and ZE Z,

~ are initially downward. 

Moments due to the liquid payload are assumed to be caused by the coning motions; the linear 

response of these moments is modeled as3,4

2 2 i i 1 2

L L L 1 LM 1 2 LM 2 1 2

M iN m a P T C K e T C K e           . (18)

 4

Cooper, G. R. Moment Exerted on Coning Projectile by a Spinning Liquid in a Cylindrical Cavity Containing a Porous 

Medium; ARBRL-MR-3677; U.S. Army Ballistics Research Laboratory: Aberdeen Proving Ground, MD, June 1988. 
7 

LMj

C will depend on time,T , j j  , the shape of the cavity, Reynolds number, and the direction of 

spin. Note that the factor(s)T are introduced because the liquid moments will vanish when j

Tj = 0. Furthermore, LMj

C causes rotations in the plane of  j expi  as well as rotations out of the 

plane, so the following definition is introduced: 

LMj LSMj LIMj

C  C  iC , (19)

where LSMj

C and LIMjC are real and represent the liquid side moment and the liquid in-plane 

moment coefficients, respectively.3

Combining the payload moment of equation 19 with the aerodynamic force and moment 

produces a slightly more complicated differential equation3

 for 

~ . The projectile angular 

motion continues to be an epicycle of equation 16, which causes the frequency and damping to 

take the forms 

 

   

  

 

  

j

LIM j

j

LSM

2

j 0j 2j

j 2

0j j j g

2

j Lsm 0j

2j 2 2

0j 0j j

2

j L x LIM j

–1 2

j aj LSM L x j

C

1+ I C

T T TC

T = σ 2 f – –1 f –1/S 

f1CT

T = 

2T 2T f

f = m a

ε = ε +C m a I 2T σ–1



 

 

   



 

. (20)

The last equation shows that LSMj

C has the same effect on damping angular motion as does the 

aerodynamic damping moment. Since the coefficient LSMj

C is usually positive for the fast mode, 

this means thatC 0 LSMj

 and will hence undamp the motion. Similarly, negative LSMjC will 

undamp the slow mode. 

In addition to the two transverse liquid moments (which Murphy5

 has shown), the axial liquid 

moment L takes the form 

j

j j

22 2

L j j LRM

2

j

LRM LSM 2

L=m a TK C

Tε 4c C = –C + 1 – 2 3a

     

  

, (21)

 5

Murphy, C. H. Liquid Payload Roll Moment Induced by a Spinning and Coning Projectile; ARLBR-TR-02521; U.S. Army 

Ballistics Research Laboratory: Aberdeen Proving Ground, MD, September 1983. 
8 

thus the roll moment is obtained directly from the liquid side moment calculation. 

When analyzing the impact of inertial waves, it is convenient to consider only that part of the 

moment exerted by the liquid due to only one of the two projectile coning frequenciesT . For j

steady-state linear motion, this part will be the total liquid moment, but for unsteady or nonlinear 

liquid motion, this part will be an average contribution of the actual liquid moment. Thus, the 

total liquid moment will be the sum of two averaged liquid moments. 

4. Liquid Motion Model 

The linear liquid analysis used for this study is identical to the analysis given by Murphy3 and 

thus only a cursory review is presented here. Consider a projectile with a cylindrical payload 

cavity with radius, a , and height 2c . The cylinder’s axis is collinear with the projectile axis 

with its center located at the projectile’s center of mass. Linear theory is used to predict the 

liquid moment caused by coning motion, which has the form 

j i s

j ˆ ξ K e K e (j 1,2) ,      (22) 

where 

j

j0

i s

j

j j

i

j0

ˆ ξ K e Ke ,

s (ε i)T ,

t , and

K K e (j 1, 2) . ˆ

 



 

 

 



 





The liquid motion is approximated quasi-statically as the sum of two steady-state projectile 

coning motions. The liquid velocity components and liquid pressure have the same dependency 

on time and θ as the velocity components of points on the coning projectile.3

 Thus, four small 

dimensionless functions, v , w , u , p of r and x, are introduced: 

  

   

  

   

js t-iθ

r

jst-iθ

θ

jst-iθ

x

jst-iθ 2 22

L

V = R ve a

V = R we +r a

V =R ue a

P= R pe + r 2 ˆ ρ a



 



  

   

   











 



  , (23)
9 

for the liquid cylindrical velocity components , , , Vr V Vx and pressure P.ˆ These are placed in 

the linearized Navier-Stokes equations and the continuity equation to yield 

 

 

 

2 2

1 2

2 2

2 2

1 2

2 2

1 2

p a v 2a iw s i v 2 w a Re v

r rr

iap a w 2a iv s i w 2 v Re w

r rr

p s i u a Re u

x

rv u iw r 0 ,

r x







             

           

   



 

 

 







(24)

where 

2 2

2 2

2 22

1

a

r rr x r



              . 

These equations are solved by assuming the liquid has low viscosity so viscous effects can be 

ignored except for high-gradient boundary layers near the cylinder walls. This means the inverse 

Reynolds number is taken to be very small, Re 1 1   , so inertial wave solutions are modeled as 

the sum of inviscid, Re 0 1   , contributions plus viscous boundary layer contributions of order 

1/ Re . 

I v

I v

I v

I v

vv v

ww w

uu u

pp p .

 

 

 

  (25)

Order of magnitude arguments applied to equation 25 when substituted in equation 24 show the 

boundary layer solutions near r=a have the following form: 

   

   

 

a

a

r a –1 /δ

v I

r a –1 /δ

v I

v v

I

I

w = 1+is x aK – w e ˆ

ˆ u = i–s K+u e

2 p = w dr a

v x

v –a δa = i – s , at r = a .

r a

   

   







(26)
10 

Near the end walls one will find 

   

   

 

α 1 xc

vv II

β 1 xc

vv II

v

I

I c

w +iu = – w +iu e

c w –iu = w –iu 2 i+s K e ˆ a

p =0

u r ˆ u δ c = –i 1+ is K at x = c .

x a

 

   

  



 

 (27)

Note that w ,u ,p 0 vv v   exponentially with distance from the container walls; the details 

regarding these solutions are explained in an appendix in Murphy.3 

5. Liquid Moments 

The major components of the liquid moment are due to the pressure on the walls of the container, 

and the viscous shear produces smaller contributions to this moment. Therefore, liquid moment 

coefficient is written as the sum of four terms, 

TC =m +m +m +m LM pl pe vl ve   . (28)

The first two terms are the pressure contribution and the remaining two terms are the viscous 

shear contribution. The fluctuating part of the inviscid P, which is the primary contributor to the ˆ

inertial wave motion, is given by 

    2 2 s -iθ i s -iθ

LI p 2

rx Δ ˆ ˆ p=ρ a R p – K e R C e Ke

r

  p              

 . (29) 

Calculating the surface integral of this with the appropriate moment arm over the container wall 

yields the moment contribution due to pressure 

p c

p –c r=a

pl pe a x=c 2 i

p 2 p 0 x=–c

1 i x C e p dx– v 2a

m +m R 1 –i r C e dr

2Ca

i



                 

 





  . (30)
11 

The viscous shear moment along the lateral wall is calculated from the boundary layer solutions 

 –1 C v v

vl –-C r=a

w u

r r

m = 2 K Re ia +Cx dx ˆ  

        , (31) 

while the shear moment due to the end walls is given by 

  x=C

–1 a v v

0

x=-C

w –iv m = 2a K Re rdr ˆ ve x

             

    . (32)

6. Results 

The examples given here consider liquid payloads in the M483 Army projectile. The liquid 

cavity is a cylinder with aspect ratio Ca 3  completely filled with water. This projectile has a 

gun launch spin rate, = 1500/s  , which is large enough so large Reynolds number boundary 

layer analysis adequately governs the liquid physics along the entire trajectory. The range of T’s 

for a typical trajectory of the M483 with a frozen (solid) liquid payload is given in figure 3. 

Figure 3. Coning frequency for the M483. 

Applying steady-state liquid analysis to this liquid configuration reveals the liquid side moment 

coefficient CLSM for C/a = 3, for two Reynolds numbers 6 7 Re = 5 10 , 5 10   (see figure 4). 

These results depict typicalCLSM behavior where the peaks indicate large overturning liquid 

moments, possible resonances, generated by this contained liquid as a function of the fast-coning 

T1 rate. 

Coning Frequencies M483

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20 25 30

Time (s)

T1,2

Fast Frequency

Slow Frequency
12 

Figure 4. CLSM vs. coning frequency. 

This plot suggests that coning frequencies in the neighborhood of T = 0.072 will create 

significantly larger side moments than those found for frequencies outside this neighborhood. 

This result is characteristic of the root cause of projectile instabilities due to liquid payloads. 

The aspect ratio coupled with a large enough Reynolds number will force inertial waves in the 

coning fluid to generate large overturning liquid moments.2

 Increasing Re causes the liquid side 

moment to increase rapidly, resulting in a pronounced peak at a resonant frequency. 

The range of frequencies given in figure 4 is close to the fast mode T1 frequencies frozen liquid 

found in figure 3. This is one indicator that a flowing liquid could cause the fast-coning 

frequencies to change such that an instability occurs near T1 = 0.072. 

Figures 5 and 6 show results of spin rate and roll moment for the M483 comparing a frozen 

liquid to a flowing liquid payload. The parameters selected for these two plots are chosen 

because they are strong indicators of flight instability caused by liquid payloads.2

This example exhibits no flight instability caused by the liquid moments even with the increased 

magnitude of Mx. The decreased roll rate and roll moment due to the liquid is a result of the 

dominance of the fast mode liquid moment for which CLSM > 0. 

To force flight an instability, take into consideration figure 4 and let the liquid have a decreased 

viscosity so that the Reynolds numbers all increase during flight. Experience suggests that 

undamping j ε > 0.6 means the projectile has encountered flight instability due to liquid 

moments. An example for the new liquid that generates large overturning moments causing the 

projectile to go unstable is now discussed. Figure 7 shows the time dependence of the side 

moment coefficient, CLSM 1, rapidly increasing neart 11s » . 

Side Moment Resonance

Aspect Ratio c/a=3

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.060 0.065 0.070 0.075 0.080 0.085 0.090

Coning Frequency T

CLSM 104

Re=5. X 10(7)

Re=5.X 10(6)
13 

Figure 5. Comparison of projectile roll moment for frozen and flowing liquid. 

Figure 6. Comparison of projectile spin rate for frozen and flowing liquid. 

Roll Moment Comparison

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

0 5 10 15 20 25 30

Time (s)

Mx Lb-ft

Frozen

Flowing

Spin Rate Comparison

600

800

1000

1200

1400

1600

0 5 10 15 20 25 30

Time (s)

Roll Rate (1/s)

Frozen 

Flowing
14 

Figure 7. Time history of CLSM 1. 

These results again suggest that liquid-induced instability may exist close to t = 11 s, and the 

cross plot shown in figure 8 reveals a rapid increase in CLSM 1 in the neighborhood of the fast 

coning rate T1 = 0.072. 

Similar results are found for the roll moment coefficient CLRM 1, indicating that the fluid 

resonance found in figures 7 and 8 is responsible for the large overturning moment, roll moment, 

and undamping exhibited by the fast frequency coning motion. Figure 9 compares the roll rates 

of an unstable example, 1 ε > 0.6 ,which occurs at time t 11s,  and a frozen liquid payload. 

Figure 8. Slow mode CLSM fast coning frequency T1. 

Time History of Fast Mode CLSM 1

2

3

4

5

6

7

8

0 2 4 6 8 10 12

Time (s)

CLSM 104

Fast Mode CLSM 

Peak at T1=.072 at Time t=11s and 1>0.6

0.0

2.0

4.0

6.0

8.0

0.065 0.070 0.075 0.080 0.085

Fast Mode Frequency T1

CLSM 104
15

Figure 9. Roll rate vs. time for flowing and frozen liquid payloads. 

The rapid decrease in roll rate is caused by the low-viscosity liquid, which is forcing this 

projectile fast mode coning frequency to approach values close to fast mode resonance near 

T 0.072. 1  Figure 10 presents pitch rates of a projectile with both frozen and flowing lowviscosity liquid payloads. Again, the rapid increase in pitch rate takes place near time t 11s  in 

the neighborhood of the fast mode resonant frequency T1 = 0.072. Another indicator of liquidinduced instability is the rapid increase in the angle of attack given in figure 11. 

Figure 10. Comparison of pitch rates for flowing and frozen liquid payloads.

Roll Rate Comparisons

600

800

1000

1200

1400

1600

0 2 4 6 8 10 12

Time (s)

Roll Rate (1/s)

Unstable Liquid Payload

Frozen Liquid Payload

Pitch Rate Comparison

-20.0

-10.0

0.0

10.0

20.0

30.0

40.0

0 2 4 6 8 10 12

Time (s)

Pitch Rate (1/s)

Unstable Liquid Payload

Frozen Liquid Payload
16 

Figure 11. Angle of attack vs. time. 

The data displayed in the last five plots provide flight parameters that correspond to flight 

instability due to the side moment generated by the high Reynolds number liquid payload. In 

general, these kinds of results occur when the damping rate is 1 ε >0.6. 

7. Conclusion 

Liquid moments calculated from linear liquid motion undergoing two-mode steady-state coning 

motions are used as average moments when applied to a nonlinear 6-DOF time-dependent 

trajectory model. These quasi-static averages yield approximate predictions describing the 

motion of projectiles with low-viscosity liquid payloads filling a cylindrical cavity. The 

magnitudes of the liquid moments are dependent on the two projectile coning frequencies, two 

projectile coning undamping rates, payload aspect ratio, and the liquid Reynolds number. The 

nonlinear 6-DOF motion of a projectile is often well approximated as the sum of fast and slow 

coning motions. Thus, under these conditions the linear liquid analysis can predict when a 

projectile exhibits flight instability due to a liquid payload by tracking the coning frequencies 

during the 6-DOF numerical integration process. Even though such predications are based on 

steady-state theory, they at least serve as an indicator when liquid-induced flight instabilities are 

likely to exist. This hybrid numerical tool provides reasonable estimates of capturing the 

physical effects of liquid moments on free-flight projectiles. 

Angle of Attack

0

4

8

12

16

20

0 2 4 6 8 10 12

Time (s)

Degrees

Unstable Liquid Payload

Frozen Liquid Payload
17 

List of Symbols, Abbreviations, and Acronyms 

a radius of the cylinder containing fluid 

c half length of the cylinder containing N subcylinders 

f

j

 4

x LIM j 1+2 π a Cρ I C

mL liquid mass 2 m =2 L π a Cρ



 projectile angular velocity along the projectile axis 

Pˆ liquid pressure perturbation 

p pressure perturbation 

s complex coning frequency s (  ε  i)T

Tj T nondimensionalized coning frequency ,

Sg gyroscopic stability factor 





















w

v

u

 liquid perturbation velocity components 











































azimuthalvelocity

radialvelocity

axial velocity

w

v

u

K

J

 Cartesian unit vectors 

Kˆ coning angle 

Sg gyroscopic stability factor 

t time 

δa    1 2

1 2 1 Re i is 

      
18 

δc = 1 3

21 3 1

a c is is a i

is is is

  

 

 

           

j  ,  growth rate per cycle 

aj  growth rate per cycle due to aerodynamic moment 

 the dynamic viscosity of liquid 

j +T t j0 j  



 liquid mass density 



 variable in nonrolling system 
NO. OF 

COPIES ORGANIZATION 

19

 1 DEFENSE TECHNICAL 

 (PDF INFORMATION CTR 

 only) DTIC OCA 

 8725 JOHN J KINGMAN RD 

 STE 0944 

 FORT BELVOIR VA 22060-6218 

 1 DIRECTOR 

 US ARMY RESEARCH LAB 

 IMNE ALC HRR 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 1 DIRECTOR 

 US ARMY RESEARCH LAB 

 RDRL CIM L 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 1 DIRECTOR 

 US ARMY RESEARCH LAB 

 RDRL CIM P 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

ABERDEEN PROVING GROUND

 1 DIR USARL 

 RDRL CIM G (BLDG 4600) 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

20

 2 COMMANDER 

 ARDEC/RDEC

 AMSRD AAR AEM C 

 A READDY

 M CORZO 

 BLDG 94

 PICATINNY ARSENAL NJ 07806-5000 

 1 COMMANDER 

 ARDEC/RDEC

 AMSRD AAR AEM C 

 K CHUNG 

 BLDG 407

 PICATINNY ARSENAL NJ 07806-5000 

 5 COMMANDER 

 ARDEC/RDEC

 AMSRD AAR AEM A 

 B WONG 

 BLDG 94

 PICATINNY ARSENAL NJ 07806-5000 

 1 COMMANDER 

 ARDEC/RDEC

 AMSRD AAR AEM A 

 W KOENIG 

 BLDG 95

 PICATINNY ARSENAL NJ 07806-5000 

 1 M MILLER 

 504 HAVERHILL RD 

 JOPPA MD 21085-4319 

 1 SCHOOL OF AEROSPACE ENGRG 

 GEORGIA INST OF TECHLGY 

 M COSTELLO 

 ATLANTA GA 30332 

ABERDEEN PROVING GROUND

 1 COMMANDER 

 US ARMY ECBC 

 RDRL ECB RT D 

 WEBER BLDG E3516 

 D WEBER 

 APG MD 21010-5424 

8 DIR USARL 

 RDRL WM 

 J SMITH 

 RDRL WML 

 M ZOLTOSKI 

 RDRL WML F 

 D LYON 

 RDRL WML B 

 J MORRIS 

 RDRL WML E 

 F FRESCONI 

 G COOPER 

 B GUIDOS 

 RDRL WML D 

 M NUSCA