Moments on a Coning M864 by a Liquid Payload: The 

Candlestick Problem and Porous Media 

by Gene R. Cooper 

ARL-TR-3837 July 2006 

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. 

DESTRUCTION NOTICE⎯Destroy this report when it is no longer needed. Do not return it to 

the originator. 
Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-3837 July 2006 

Moments on a Coning M864 by a Liquid Payload: The 

Candlestick Problem and Porous Media 

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)

 July 2006

2. REPORT TYPE 

Final 

3. DATES COVERED (From - To) 

 May 2004 to December 2004 

5a. CONTRACT NUMBER 

5b. GRANT NUMBER 

4. TITLE AND SUBTITLE 

 Moments on a Coning M864 by a Liquid Payload: The Candlestick Problem 

 and Porous Media 

5c. PROGRAM ELEMENT NUMBER 

5d. PROJECT NUMBER 

 622618.H8099 

5e. TASK NUMBER 

6. AUTHOR(S) 

 Gene R. Cooper (ARL) 

5f. WORK UNIT NUMBER 

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

U.S. Army Research Laboratory 

Weapons and Materials Research Directorate 

Aberdeen Proving Ground, MD 21005-5069

8. PERFORMING ORGANIZATION 

 REPORT NUMBER 

ARL-TR-3837

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 

Moments that are caused by a payload assumed to be an inviscid liquid flowing in a coning projectile are predicted. This 

payload is contained in a uniform sequence of end-to-end cylinders stacked in columns along and off the symmetry axis of the 

projectile. A theoretical model is used to analyze inertial waves passing through the liquid, which are generated by the coning 

motion. This analysis continues by considering a single column along the projectile symmetry axis, which is filled with an 

inviscid fluid saturating porous media. Eigen frequencies and their impact on liquid moments are discussed concerning the 

flight stability of the projectile for a wide range of payload configurations and porosities when the projectile is subjected to 

various coning frequencies. 

15. SUBJECT TERMS

 bessel functions; eigensolutions; liquid moments; projectile flight dynamics; rotating liquids 

16. SECURITY CLASSIFICATION OF:

19a. NAME OF RESPONSIBLE PERSON

 Gene R. Cooper 

a. REPORT 

Unclassified

b. ABSTRACT 

Unclassified

c. THIS PAGE 

Unclassified 

17. LIMITATION

OF ABSTRACT 

SAR

18. NUMBER 

OF PAGES 

25 19b. TELEPHONE NUMBER (Include area code) 

 410-278-3684

Standard Form 298 (Rev. 8/98) 

 Prescribed by ANSI Std. Z39.18
iii

Contents 

List of Figures iv

Acknowledgments v

1. Introduction 1

2. Equations of Motion for the Off-axis Candlestick Configurations 1

3. Candlestick(s) Liquid Moments 5

4. Equations of Motion for the Symmetry Axis Porous Media Configuration 9

5. Porous Media Liquid Moments 11

6. Calculation Method 14

7. Conclusions 14

8. References 16

Distribution List 18 
iv

List of Figures 

Figure 1. Coordinate systems of configuration.............................................................................. 2 

Figure 2. Details of payload configuration. ................................................................................... 3 

Figure 3. Free flight coning frequencies of the M864 projectile. .................................................. 7 

Figure 4. Values of CLSM showing Eigen-frequencies as a function of T and N . ...................... 8 

Figure 5. Values of CLIM showing Eigen-frequencies as a function of T and N . ...................... 8 

Figure 6. Values of CLSM showing a possible existence of an instability due to an Eigenfrequency <0.1. ....................................................................................................................... 9 

Figure 7. Values of CLSM as a function of T for increasing Ct and Cx ..................................... 13 

Figure 8. Values of CLIM function of T for increasing Ct and Cx . ........................................... 13 

Figure 9. Values of ΔCLSM showing possible flight instabilities (Eigen frequencies) as a 

function of T ........................................................................................................................ 14 
v

Acknowledgments 

The author wishes to acknowledge Kok Y. Chung, of the U.S. Army Research, Development, 

and Engineering Center (ARDEC), for his guidance, suggestions and technical assistance which 

had a great impact on this work. Mr. Chung kept the author informed of new design and 

geometric configurations, thus causing the investigation to closely address the Army concerns 

regarding liquid non-lethal payloads. This work was funded by ARDEC. 
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

Predicting the moment attributable to a liquid payload in a spinning and coning projectile is a 

problem of considerable interest to the Army. Stewartson (1) was first to consider inviscid payloads contained in a right circular cylinder, and his results gave coning Eigen frequencies that can 

possibly cause catastrophic yaw for a liquid-carrying projectile. First order viscous boundary layer 

corrections of the Stewartson theory were made by Wedemeyer (2) and Murphy (3). A method for 

calculating the linear liquid moment using the full linear viscous equations with boundary layer 

corrections confined only to the end caps (candlesticks) was presented by Hall, Sedney, and 

Gerber (4, 5). 

A further interest to the army is to consider a series of uniform circular cylinders stacked end to end 

separated by impenetrable end caps. These candlesticks may be situated along the symmetry axis 

or offset from this axis but parallel to the symmetry axis of the projectile. Coning motion-induced 

liquid moments are considered here for a number of candlestick configurations. The Eigen 

frequencies for such configurations are shown to be identical to those found by Stewartson (1). 

Liquid payloads contained in a highly permeable material have been of interest to the U.S. Army 

for some time. Laboratory tests and flight tests have shown that a highly permeable medium can 

significantly reduce the spin-up time of a liquid payload (6, 7, 8). Flight stability for liquidsaturated permeable payloads has also been examined by D’Amico (9, 10). An investigation by 

Cooper (11) considered inertial waves in a coning projectile with a saturated porous media payload 

so that the media are homogenous and isotropic. The present work extends the Stewartson and 

Cooper problems by considering a cylindrical cavity filled with a permeable medium that is 

impregnated with an inviscid liquid, but this medium is not necessarily isotropic. Following 

Cooper, we introduced a further modification by segmenting the cavity along the symmetry axes 

into a sequence of equal length cylinders. Each of these cylinders is separated by impermeable end 

caps. The porous media are modeled by a drag term, which is proportional to the liquid velocity 

relative to the assumed ridge porous media that are added to the linearized Euler equations. This 

analysis examines the induced liquid moment as a function of parameters found by Stewartson (1) 

plus parameters describing the porous media and the number of segments in the cylindrical cavity. 

2. Equations of Motion for the Off-Axis Candlestick Configurations 

Figure 1 shows the X′,Y′,Z′ axes rotating uniformly about X′ with angular velocity P = (P, 0, 0), 

and figure 2 presents details of the internal configuration of the candlestick payloads. The liquid is 
2

assumed to be initially rotating as a rigid body with the same angular speed P so the velocity V′ of 

the liquid inside the cylinders is 

V′ = P × ( ) x, R0 cosB + r cosθ, R0 sin B + rsin θ (1) 

The unperturbed state for equation 1 satisfies the Euler equation: 

( ) ( )

ρ

P P x, R cosB r cos , R sin B rsin s

x x 0 0

2 e × e × + θ + θ = −∇ (2) 

for which P is the unperturbed liquid pressure. s Following Stewartson and letting the position 

vector have the Cartesian form (x, R cosB r cosθ, R sin B rsin θ) R = 0 + 0 + allows equation 2 to 

be integrated as a scalar potential so that 

P P R cos(B ) P r 2 2 2

0

2

S ρ = − θ − (3) 

Figure 1. Coordinate systems of configuration. 

Δ

K0

X′

X
3

Figure 2. Details of payload configuration. 

Now we excite wave motion in the rigidly rotating liquid by perturbing the angular velocity of 

the projectile for small angular components, y z ω ,ω , so that the total angular velocity takes the 

form 

( ) y ωz Ω = P + 0, ω , . (4) 

The projectile is assumed to undergo small angle coning motion about the Z X′,Y′, ′ frame and is 

related to the projectile body axes, Z X, , by the following transformation ( Y, 3): 

() ()

() ()

0

0

10 0 1 0 0 1 0

0 cos sin 1 0 0 cos 1 sin 1

0 01 0 sin cos 0 sin 1 cos 1

TPt

TPt

X K e X

Y tPT PT K e tP T tP T Y

Z Z tPT tPT tP T tP T

ε

ε

⎡ ⎤′ ⎡ ⎤⎡ ⎤ ⎡ ⎤ − ⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − − − ′ ⎡⎤ ⎡⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎣ ⎦ ⎣ ⎦⎥ ⎢ ⎣ ⎦ ⎣ ⎦⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ′ ⎢ ⎥ ⎡⎤ ⎡⎤ ⎡ ⎤ ⎡ ⎤ −− − ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎢ ⎥⎢ ⎥ ⎣⎦ ⎣⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

 (5) 

where the coning damping rate is ε the coning frequency is T and K is the magnitude of the 0

small coning angle. Equation 5 shows that the angular velocity, to first order in K , is written as 0

a column vector with body frame components 

[ ] ( ) () ( ) ( )

[ ] ( ) () ( ) ( ) ⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

− − − ε −

− ε − + −

ε

ε

PTt

PTt

K0PT sin t P T 1 cos t P T 1 e

K0PT sin t P T 1 cos t P T 1 e

P

 (6) 

a 

R0

Liquid/Porous Media 

Payload 

Δ 

2a 

TOP 

SIDE 

Spin X Axis 

Z 

Y 

B 

X=C 
4

This suggests that the velocity of the liquid payload should be given by 

V P ( ) x, R cosB r cos , R sin B rsin vˆ = × 0 + θ 0 + θ + (7) 

for which the components of vˆ and the pressure perturbation pˆ all have the magnitudes of order K . 0

The liquid payload is assumed to have low viscosity and the magnitude P is assumed to be large 

so that the fluid motion is adequately described by Euler’s equations. (The M864 projectile in 

free flight with a water payload has a Reynolds number on the order of 6 10 .) Neglecting all 

higher order terms in small K makes the Euler equations take the following form: 0

0

d x

1 d pˆ

d t

Uˆ d

0

d

d pˆ

r

1 PVˆ 2

d t

Wˆ d

0

d r

1 d pˆ P Wˆ 2

d t

Vˆ d

= ρ+

= ρ θ+ +

= ρ − +

 (8) 

in which ( U) ˆ W, ˆ V, ˆ are cylindrical components of the perturbed velocity for the perturbed 

pressure, pˆ . 

Normal boundary conditions at the solid wall satisfy 

(Vˆ, Wˆ , Uˆ )•n = Ω×Rs •n (9) 

in which n is an outward unit vector on the wall and Rs a point on any cylinder wall. Equation 6 

when substituted into equation 9 causes the normal boundary conditions to become 

( )

( ) ( )

i 1, S ( )i T

U e K P e R r e Se ˆ

V e x K PSe ˆ

i P S t iPt i

0

iB

0

P S t iPt i

0

= − ≡ ε +

= −R +

= R

θ − − θ

− − θ

. (10) 

These show that separable solutions to equation 8 are obtained by 

− − θ

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

=

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

PSt iPT i e

p

u

w

v

pˆ

Uˆ

Wˆ

Vˆ

 (11) 

and solving for the velocity components yields 
5

( )

( )( )

( )

( )( )

P( ) S i

d x

d p

u

r P S 3i S i

d r

d p S i 2r

d

d p

w

r P S 3i S i

d

d p r S i 2

d r

d p

v

ρ − = −

ρ − +

− − θ = −

ρ − +

θ − +

= −

 (12) 

Using the continuity equation, ∇ •( ) v, w, u = 0 , produces the following equation for the 

pressure p : 

( )( )

( )2

2

2

2

2 2

2

2

2

2

2

S i

S 3i S i

0

d x

d p r

d

d p

d r

d p r

d r

d p r

−

− +

σ = −

− σ = θ+ +

 (13) 

At this point in the analysis, it is useful to consider each cylindrical candlestick ranging from 

− C ≤ x ≤ C to consist of an end-to-end sequence of N equal length, Δ , cylinders with 

impenetrable end caps so that Δ = 2C N . Applying equation 10 to each of the sub-cylinders 

gives the following Fourier-Bessel series for the pressure in the given candlestick: 

( ( ) ) ( )

( )

() ( )

( ) ( )( ) ( )( )

( )

( )

( )

0 n N,

F e R S i S a

Na 2 D 2C N 2n 1 S

E S i S

zz a k N 2C

a k N J zz S i zz J zz S i

16 1 CS S 3i A

r x E a r D a e Fx

e A cos k C x J k r

p K P a

0

iB

1 0

2

n 2

k

2 i

k odd

k 1

i

2 2

0

≤ ≤

= −

= − −

= −

= π σ

π + − −

− − = −

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ + +

π + Δ π σ Δ

= ρ

θ

∈

− θ

∑

 (14) 

3. Candlestick(s) Liquid Moments 

The moment induced by the liquid contained in the segmented cavity is calculated from the time 

derivative of the angular momentum field. Non-dimensionalizing the moment with 4 2 2 πρ a CP

makes it convenient to write the side moment components, Y MZ M , , as (3): 
6

( )

() () J K

J K CLM

C C T, N,C C T, N,C

M M T 2 a CP K e

LM LSM LIM

PSt iPt

0

4 2

Y Z

= +

+ = πρ −

 (15) 

Therefore, the induced moment on the entire candlestick consisting of N end-to-end sub-cylinders 

of length Δ is given by 

( )

Ω = π Δ

⎪

⎪

⎪

⎪

⎪

⎭

⎪

⎪

⎪

⎪

⎪

⎬

⎫

⎪

⎪

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎪

⎪

⎨

⎧

Ω

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −

πρ = −∑R × ∫ −

=

2

N 1

n 0

4 2

2 a

d

d t

Uˆ d

P Vˆ 2

d t

Wˆ d

P Wˆ 2

d t

Vˆ d

T 2 a CP

x

θ

r

LM

e

e

e

C R

 (16) 

Note that symmetry attributable to the geometry of each candlestick and equation 14 causes the 

axial component of CLM to integrate identically to zero. 

The calculation of equation 16 is tedious. The details are not given here (3, 11), but the result is 

the following expression: 

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

[ ] ( )( ) ( )( )

zz a k N 2C,

J zz S i zz J zz S i k

J zz

a N S i

128C S S 3i

3a N S i

2C S S i

12a N

12i cosBe N R 4C 3a N

12cosBe N R i 4C 3a N 8iC S

iS

T M iM

k odd

4

1 0

1

4 2 2

2 3

2 2

2 2

2 2

2 2 2 2

0

iB 2

2 2 2 2 2

0

iB 2

Y Z

= π σ

π + + − −

−

+

+

− +

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− − −

+ + −

+ =

∑∈

 (17) 

The summation terms of equation 17 show that large over-turning moments can result if 

( )( ) ( )( )

zz a k N 2C,k odd

J1 zz S i zz J0 zz S i 0

= π σ ∈

+ − − = (18) 

This is the same criterion first found by Stewartson (1) for the symmetrically located liquid 

payload, i.e., at R = 0 . This says that the Stewartson tables can be used to determine such Eigen 

frequencies where the aspect ratio is taken to be 2C a N . 

One further comment regarding equation 17 is that CLM approaches the value produced by a 

frozen liquid as N becomes large, provided that Eigen frequencies are sufficiently removed 

from the region of interest. In fact, the rate of approach to the frozen limit is very rapid since it 

goes as 2 1 . Typical coning frequencies of the M864 N are presented in figure 3. Figures 3 and 4 
7

give examples of moment coefficients, C , for the M864 projectile. The value for the LSM ,CLIM

axes’ offset is taken to be R = 2a and the orientation angle is B = π 3 . In these plots, the sharp 

spikes correspond to the increased over-turning moment, C , that is encountered when the LSM

projectile’s coning frequency passes a Stewartson Eigen frequency. These results clearly show 

that even for small values of N , the moment coefficients are very close to the frozen limit 

values. The reason for this is the aspect ratio for these examples is relatively large. Some 

mathematical manipulations of equation 17 show that for large aspect ratios, the values 

C approach their frozen lim LSM ,CLIM it values for all values of N as long as equation 18 is not 

satisfied. Figure 3 shows that coning rates are in the range of 0 ≤ T ≤ 0.1, and figures 4 and 5 

indicate that large over-turning moments caused by the liquid payload should not result in flight 

instabilities for the parameters examined here. However, if the parameters and particularly, the 

aspect ratio, were to change, instabilities could result (see figure 6 as an example). 

Figure 3. Free flight coning frequencies of the M864 projectile. 

M864 Trajectory Simulations

QE = 735 mils

Nondimensional Coning Frequencies 

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0 20 40 60 80 100

Time (s)

τ

Zone 1 Zone 3 Zone 4 Zone 5
8

Aspect Ratio (C/a)=23 R0/a=2

M864

-35

-30

-25

-20

-15

-10

-5

0

0.0 0.2 0.4 0.6 0.8 1.0

T

CLSM

N=5

N=10

N=19

Frozen Liquid

Figure 4. Values of CLSM showing Eigen frequencies as a function of T and N . 

Aspect Ratio (C/a)=23 R0/a=2

 M864

0

100

200

300

400

500

0.0 0.2 0.4 0.6 0.8 1.0

T

CLIM

N=5

N=10

N=19

Frozen Liquid

Figure 5. Values of CLIM showing Eigen frequencies as a function of T and N . 
9

However, if the parameters and particularly the aspect ratio were to change, instabilities could 

result. Figure 6 is one such example. 

Aspect Ratio (C/a)=23 R0/a=2

 M864

-10

-8

-6

-4

-2

0

0.0 0.2 0.4 0.6 0.8 1.0

T

CLSM

N=23

Frozen Liquid

Figure 6. Values of CLSM showing a possible existence of an instability attributable to 

an Eigen frequency <0.1. 

4. Equations of Motion for the Symmetry Axis Porous Media Configuration 

For this problem, the moment arm 0 R0 = so the position vector becomes R = ( ) x, r cos θ, rsin θ

since the candlestick axis is the symmetry axis of the projectile. When the liquid payload axis is 

coincidental with the projectile symmetry axis, the analysis is simplified if we move to the 

X′,Y′,Z′ frame rather than the body frame used before. This modifies the transformation given 

by equation 5 so that 

( )

( )

0

0

10 0 1 0 0 1 0

0 cos sin 1 0 0 cos 1 sin

0 01 0 sin cos 0 sin 1 cos

TPt

TPt

X K e X

Y tPT PT K e tP T tPT Y

Z tPT tPT tP T tPT Z

ε

ε

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ − ⎡ ⎤ ′ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥= − − ′ ⎡⎤ ⎡⎤ ⎡ ⎤ ⎡⎤ ⎢ ⎥

⎢ ⎥ ⎢ ⎣ ⎦ ⎣ ⎦⎥ ⎢ ⎣ ⎦ ⎣ ⎦⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥′ ⎢ ⎥ ⎣ ⎦ ⎡⎤ ⎡⎤ ⎡ ⎤ ⎡⎤ − − ⎢ ⎥⎢ ⎥ ⎣⎦ ⎣⎦ ⎣ ⎦ ⎣⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

%

%

%

 (19) 

and Z~ Y,

~ X,~ are coordinates in the non-rolling reference frame. The velocity of the media obtained 

from the rotation kinematics is now given by 
10

( )

( )

( ) − θ

− θ

θ

− θ

′ = −

′ = + −

′ = − −

PSt i

x 0

PSt i

PSt i

r 0

V rK P S i e

V P r ixK0P S i e

V xK P S i e

 (20) 

and x r,θ, are cylindrical coordinates in the non-rolling frame. The Euler equations are now 

modified to account for an inviscid liquid flowing through porous media. This modification 

assumes that flow in porous media can be represented by additional terms that are proportional to 

the liquid velocity relative to the media which is taken to be rigidly attached to the coning 

projectile. Using the factor, PSt−iθ e , for each independent variable allows the modified Euler 

equations now written in cylindrical components to have the following form: 

( ) ()

( ) ()

( ) () 0

d x

d p

P

1 S Cx i u rC P S i

0

P r

i p S Ct i w 2 v i xC P S i

0

d r

d p

P

1 S Ct i v 2w xCt P S i

x

t

= ρ+ − − − +

= ρ+ − + − − −

= ρ+ − − + − +

 (21) 

In this equation, u v are the x , w , r,θ, components of the fluid velocity and p is the perturbation 

pressure. Note the terms that are proportional to the constants t Cx C represent drag that the ,

porous media exerts on flow passing through these media. Motivation for this model stems from 

Darcy’s Law (12) which says that the drag force caused by porous media, Dr , is given by 

2

dynamic viscosity

porosity (dimensions of length ( ))

velocity of fluid relative to porous media

liquid density

pressure gradient induced by the porous media on the fluid 

P aCr

P a

2

μ ρ

κ

μ

κ

ρ

=− =

→

→

→

→

→

Vr Dr Vr -

Vr

Dr flow

Cr dimensionless constant which is a measure of the pressure gradient

so that Cr .

P

μ

ρ κ

→

=

Solving equation 21 leads to the velocity field expressions: 
11

( )

( )( )

( )

( )

( )( )

( )

( )

( )

S C i

rC P S i

P S i

dx

dp

u

S C i

ixC P S i

r P S 3i S i

dr

dp i S i p 2r

w

S C i

xC P S i

r P S 3i S i

S i r

dr

dp 2ip

v

x

x

t

t

t

t

+ −

− +

ρ − = −

+ +

− +

ρ − +

− +

=

+ +

− − ρ − +

− −

= −

 (22) 

Invoking the fact that the flowing liquid is incompressible causes the continuity equation to give 

the following expression for the perturbation pressure p 

( )( )

( )( ) S C i S C i

S C 3i S C i

0

d x

d p p r

d r

d p r

d r

d p r

t x

2 t t

2

2

2 2

2

2

2

+ − + −

+ − + +

σ = −

+ − − σ =

. (23) 

Equations 13 and 23 show when 0 Ct = 0,Cx = causes 2 2 σ = σ . Using the demands of equation 20 

modifies the coefficients for the Fourier-Bessel series as follows: 

( ) ( )( )

( ) ( )( ) ( )( )

( )

( )( )

0 n N

F 0

D 2C N 2n 1 S i S Na

E S i

zz a k N 2C

a k N J zz S C i zz J zz S C i

16 1 CS S i S C 3i A

2

1 t 0 t

2 2

t

n

j

≤ ≤

=

= − + + −

= − −

= π σ

π + + − + −

− − + − =

 (24) 

5. Porous Media Liquid Moments 

The procedure (equation 16) for calculating liquid side moments gives the following expression 

for liquid-saturated porous media in N cylinders, separated by impermeable end caps, located 

along the projectile symmetry axis: 
12

( ) ( )(( ) ( ) ( ) )

( )

( ) ( )

( )( )

( )

( )

[ ] ( )( ) ( )( )

zz k N 2C

J zz S C i zz J zz S C i k

J zz

N a S C i

128C S S i S C 3i

4

S i

12a

12C S 1 4C 3a

3a N S C i

4C S i N 1 S C 2i N S iC 1 N T C iC

k odd

4

1 t 0 t

1

t

4 2 2

t

2 2

2

2

2 2 2 2

t

2 2

2

t

2

t

2 2 2

LSM LIM

≡ π σ

π + + + + − + −

− + − −

+ − + − − − +

+

+ +

− − + + + − + =

∑∈

. (25) 

Taking the limits of this expression for large N or large C gives the frozen limit values of t

C given by ( LSM and CLIM 3) 

T( )1

12

4C a 3

2

1 C

3

4C a 3 1 T

2

C

2

2 2

LIM _ F

2 2

LSM _ F

ε − + = +

⎥

⎦

⎤ ⎢

⎣

⎡ + − ε =

 (26) 

whenever T is not in the neighborhood of an Eigen frequency S = (i + ε)T which by equation 25 

satisfies 

J1(zz)(S +Ct+i)− zz J0(zz)(S +Ct−i) = 0 (27) 

The definition of σ given in equation 23 shows that the Stewartson tables (1, 3) can again be 

used to find Eigen frequencies, provided that Cx = Ct and S by S + Ct while these is replaced

tables are used. In general, these frequencies S will have complex values, ε ≠ 0 , thus indicating 

that damping/undamping will occur. The cases when Cx ≠ Ct also result in ε ≠ 0 but numerical 

methods need to be used to find such Eigen frequencies. 

Scheidegger (14) has tabulated permeability’s 5 2 1.0 10 cm − κ ≅ × , and assuming that the media 

studied here for the M864 have similar permeabilities, which average 6 C ,C 0. t x ≅ . Sample plots 

showing that C again approach the values for a frozen liquid as LSM and CLIM C becomes large t

(see figures 7 and 8), which are in agreement with equation 26 when ε = 0. 

Typical plots showing C changing with increasing values of LSM N are given in figure 9. For 

this plot, the vertical axis assumes values defined as the difference between C and the frozen LSM

values given by equation 26: 

ΔCLSM ≡ CLSM −CLSM _ F (28) 

The sharp peaks indicate Eigen frequencies which are now complex since Cx = Ct = 1 3 and 

therefore ε ≠ 0 . Evidently, C approaches the values LSM CLSM _ F as N gets larger for values of 
13

T sufficiently removed from any Eigen frequencies. In general, such Eigen frequencies are 

found with a numerical search routine in the complex plane. 

Aspect Ratio (C/a)=23 (R0/a)=0 N=2 ε=0 

M864

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.0 0.2 0.4 0.6 0.8 1.0 T

CLSM

Ct=1/3 Cx=1/2 Ct

Ct=3 Cx=1/2 Ct

Ct=5 Cx=3/2 Ct

Figure 7. Values of CLSM as a function of T for increasing Ct and Cx . 

Aspect Ratio (C/a)=23 (R0/a)=0 N=2 ε=0 

 M864

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

0.0 0.2 0.4 0.6 0.8 1.0

T

CLIM

Ct=1/3 Cx=1/2 Ct

Ct=3 Cx=1/2 Ct

Ct = 5 Cx=1/2 Ct

Ct=Cx=Infinity

Figure 8. Values of CLIM function of T for increasing Ct and Cx . 
14

Aspect Ratio (C/a)=23 (R0/a)=0 Ct=1/3 Cx=1/2

 M864

-10

-5

0

5

10

15

20

0.0 0.2 0.4 0.6 0.8 1.0

T

Δ

CLSM

N=1

N=2

N=5

Figure 9. Values of ΔCLSM showing possible flight instabilities (Eigen frequencies) as a function 

of T . (As N increases, the payload behaves like a frozen [liquid] payload when T is 

far from Eigen frequencies.) 

6. Calculation Method 

The equations of the last sections need to be calculated for a wide range of flight, geometry, and 

porous media parameters all of which require values of Bessel functions. For small values of zz , 

simply using power series expansions of each Bessel function works very well. Bessel functions 

at large values of zz were obtained by asymptotic expansions (13). Generally, calculating Bessel 

functions for complex arguments for intermediate values of zz is a non-trivial problem and the 

methods used here employ Gaussian continued fractions. This author has judged that a further 

discussion of these methods is not appropriate for this article but the reader should be aware of the 

numerical difficulties associated with calculating complex Bessel functions. 

7. Conclusions 

The off-axis candlestick problem has been shown to be equivalent to the inviscid Stewartson (1) 

problem whenever porous media are not present or can be ignored. Resonant frequencies are 

independent of the candlestick off-axis position and can be found with tabulations that have 

already been found (1, 3). Values not found in such tables can readily be obtained by simple 

numerical root-finding methods. The design configurations examined here show that flight 
15

instabilities should not occur for candlesticks with a low viscosity liquid and as N increases, the 

liquid behaves more like a frozen liquid. 

Cases when the candlestick contains saturated porous media that are located along the symmetry 

axis of the projectile can possibly force resonances. In these cases, the Eigen S frequencies 

assume complex values so that damping or un-damping can occur. For the particular case when 

CT = CX , it is possible to find the resonances from the Stewartson tables, but if CT ≠ CX , then a 

numerical search for resonances in the complex plane for the Eigen frequencies is required. The 

current analysis is based on assumed values of the porosities, T CX C , and the results indicate 

that a flight instability of the M864 is unlikely to transpire. However, if the actual porosity of the 

proposed media differs significantly from the assumptions used here, catastrophic flight 

instabilities could result. A thorough search for “problem Eigen frequencies” requires 

experimental measurements of T CX C and in order to gain a better idea where to search, in the 

complex plane, for such frequencies (7, 8, 10). In all cases, the liquid moments approach the 

values for a frozen liquid with increasing values of N if Eigen frequencies are not present. 

Similar results also apply for increasing values of CT . 
16

8. References 

1. Stewartson, K. on the Stability of a Spinning Top Containing Liquid. Journal of Fluid 

Mechanics 1959, 5, Part 4, 577–592. 

2. Wedemeyer, E.H. Viscous Correction to Stewartson’s Stability Criterion; BRL Report 

1325; US Army Ballistic Research Laboratory: Aberdeen Proving Ground, Maryland, June 

1966. (AD 489687) 

3. Murphy, C.H. Angular Motion of a Spinning Projectile with a Viscous Liquid Payload; 

BRL-MR-3194; U.S. Army Ballistic Research Laboratory: Aberdeen Proving Ground, MD, 

August 1982. (AD A118676) (See also Journal of Guidance, Control, and Dynamics, Vol. 6. 

pp.280-286, July –August 1983.) 

4. Gerber, N.; Sedney, R. Moment on a Liquid-Filled Spinning and Nutating Projectile: Solid 

Body Rotation; BRL-TR-02470; U.S. Army Ballistic Research Laboratory: Aberdeen 

Proving Ground, MD, February 1983. (AD A125332) 

5. Hall, P.; Sedney, R.; Gerber, N. Fluid Motion in a Spinning, Coning Cylinder via Spatial 

Eigen function Expansion; BRL Technical Report; U.S. Army Ballistic Research Laboratory: 

Aberdeen Proving Ground, MD, August 1997 (AD A190758) 

6. D’Amico, W.P.; Mark, A. The Application of a Highly Permeable Medium to Reduce SpinUp Time and to Stabilize a Liquid-Filled Shell; BRL-MR-02851; U.S. Army Ballistic 

Research Laboratory: Aberdeen Proving Ground, MD, July 1978. (AD A058595) 

7. Miller, M.C.; Molnar, J.W. Laboratory Studies to Improve Flight Stability of the M825 

Projectile; CRDEC-TR-153; U.S. Army Chemical Research, Development and Engineering 

Center: Aberdeen Proving Ground, MD, June 1990. 

8. Miller, M.C.; Molnar, J.W. Laboratory Flight Stability Evaluation of M825A1 Production 

Payloads; CRDEC-TR-240; US Army Chemical Research, Development and Engineering 

Center: Aberdeen Proving Ground, MD, October 1990. 

9. D’Amico, William P.; Soencksen, Keith P. Aeroballistic Testing of the M825 Projectile: 

Phase VII-Larger Radius Felt Wedge Payloads; ARBRL-MR-3586; US Ballistic Research 

Laboratory: Aberdeen Proving Ground, Maryland, April 1987. 

10. D’Amico, W.P.; Clay, W.H. Flight Tests for Prototype Felt Wedge/White Phosphorous 

Improved Smoke Concept; BRL-MR-02824; U.S. Army Ballistic Research Laboratory: 

Aberdeen Proving Ground, MD, April 1978. (AD A054643) 
17

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

Cavity Containing a Porous Medium; BRL-MR-3677; U.S. Army Ballistic Research 

Laboratory: Aberdeen Proving Ground, MD, June 1988. (AD A195291) 

12. Darcy, H. Les Fontaines Publiques de la Ville de Dijon, 1886. 

13. McLachlan, N.W. Bessel functions for Engineers; Oxford University Press, London, 1955. 

14. Scheidegger, A. E. The Physics of Flow Through Porous Media; University of Toronto 

Press, Canada, 1960. 
18

NO. OF 

COPIES ORGANIZATION

 1 DEFENSE TECHNICAL 

 (PDF INFORMATION CTR 

 ONLY) DTIC OCA 

 8725 JOHN J KINGMAN RD 

 STE 0944 

 FORT BELVOIR VA 22060-6218 

 1 US ARMY RSRCH DEV & ENGRG CMD 

 SYSTEMS OF SYSTEMS 

 INTEGRATION 

 AMSRD SS T 

 6000 6TH ST STE 100 

 FORT BELVOIR VA 22060-5608 

 1 INST FOR ADVNCD TCHNLGY 

 THE UNIV OF TEXAS AT AUSTIN 

 3925 W BRAKER LN STE 400 

 AUSTIN TX 78759-5316 

 1 DIRECTOR 

 US ARMY RESEARCH LAB 

 IMNE ALC IMS 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 1 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL CI OK TL 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 2 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL CI OK T 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 1 SCHOOL OF AEROSPACE ENGINEERING 

 GEORGIA INST OF TECHNOLOGY 

 ATTN DR M COSTELLO 

 ATLANTA GA 30332 

 10 CDR ARDEC 

 ATTN AMSRD AAR AEM C K CHUNG 

 R LEE A READDY M CORZO 

 AMSRD AAR AEM A W KOENIG 

 B WONG (5 CYS) 

 PICATINNY ARSENAL NJ 07806-5000 

 1 MR MILES MILLER 

 504 HAVERHILL ROAD 

 JOPPA MD 21085-4319 

NO. OF 

COPIES ORGANIZATION

 1 ROY KLINE 

 27 FREDON GREENDELL RD 

 NEWTON NJ 07860-5213 

 ABERDEEN PROVING GROUND

 1 DIRECTOR 

 US ARMY RSCH LABORATORY 

 ATTN AMSRD ARL CI OK (TECH LIB) 

 BLDG 4600 

 2 COMMANDER 

 US ARMY ECBC 

 ATTN AMSRD ECB RT D WEBER 

 BLDG E3516 

 APG EA 

 1 DIRECTOR 

 US ARMY RSCH LABORATORY 

 ATTN AMSRD ARL WM J SMITH 

 BLDG 4600 

 2 DIR USARL 

 ATTN AMSRD AR WM B M ZOLTOSKI 

 BLDG 4600 

 1 DIR USARL 

 ATTN AMSRD AR WM BA D LYON 

 BLDG 4600 

 1 DIR USARL 

 ATTN AMSRD ARL WM BD B FORCH 

 BLDG 4600 

 4 DIR USARL 

 ATTN AMSRD ARL WM BC P PLOSTINS 

 G COOPER B GUIDOS 

 P WEINACHT 

 BLDG 390 

 1 DIR USARL 

 ATTN AMSRD ARL WM BD M NUSCA 

 BLDG 390