Prediction of the Pitch-Damping Coefficients Using Sacks’ 

Relations 

by Paul Weinacht and James E. Danberg 

ARL-TR-3231 June 2004 

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Army Research Laboratory 

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ARL-TR-3231 June 2004 

Prediction of the Pitch-Damping Coefficients Using Sacks’ 

Relations 

Paul Weinacht and James E. Danberg 

Weapons and Materials Research Directorate, ARL 

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Prediction of the Pitch-Damping Coefficients Using Sacks’ Relations 

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14. ABSTRACT 

Throughout its development, slender body theory has been generalized to predict a large variety of aerodynamic coefficients for

a wide class of flight bodies. For most applications, slender body theory provides only a qualitative predictive capability. There 

is, however, a set of slender body relationships which has been previously derived by Sacks that allows the individual pitchdamping coefficients and the pitch-damping coefficient sums to be related to each other. Until recently, it has been difficult to 

assess the accuracy of these relationships due to the lack of high-quality pitch-damping data or the lack of a higher order theory. 

The current work applies a recently developed computational fluid dynamics capability for predicting all three pitch-damping 

coefficients. From this analysis, the accuracy of these relationships has been assessed and their engineering significance 

demonstrated. One important result is that the pitch-damping relations developed by Sacks allow the individual pitch-damping 

coefficients to be determined from the pitch-damping coefficient sum with a high degree of accuracy. 

15. SUBJECT TERMS

aerodynamics, projectile flight mechanics, pitch-damping 

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iii

Contents 

List of Figures iv

1. Introduction 1

1.1 Helical Motions ...............................................................................................................3 

1.2 Coning Motion ................................................................................................................6 

2. Computational Technique 7

3. Results 9

3.1 Ogive-Cylinder Results ...................................................................................................9 

3.2 Flared Projectile Results................................................................................................19

4. Summary 23

5. References 25

List of Symbols, Abbreviations, and Acronyms 27

Distribution List 30 
iv

List of Figures 

Figure 1. Helical motion with nonzero α and zero q . ..................................................................4 

Figure 2. Helical motion with zero α and nonzero q . ..................................................................5 

Figure 3. Schematic of coning motion............................................................................................6 

Figure 4. Schematic of ANSR configuration................................................................................10 

Figure 5. Comparison of the three computed deviations ∆1 , ∆2 , and ∆3 vs. body 

length, ANSR, and Mach 2.5...................................................................................................11

Figure 6. Comparison of the three computed deviations ∆1 , ∆2 , and ∆3 relative to the 

damping force coefficients, ANSR, Mach 2.5, = 0.5598 cg x /L , and length-to-diameter 

(L/D) ratio = 9..........................................................................................................................12 

Figure 7. Comparison of the three computed deviations ∆1 , ∆2 , and ∆3 relative to the 

damping moment coefficients, ANSR, Mach 2.5, = 0.5598 cg x /L , and L/D = 9....................13 

Figure 8. Comparison of the damping moment coefficients predicted from Sacks’ 

relations with CFD predictions, ANSR, Mach 2.5, = 0.5598 cg x /L , and L/D = 9. .................14 

Figure 9. Comparison of the damping moment coefficients predicted from Sacks’ 

relations with CFD predictions, ANSR, Mach 1.8, = 0.5598 cg x /L , and L/D = 9. .................15 

Figure 10. Comparison of the damping moment coefficients predicted from Sacks’ 

relations with CFD predictions, ANSR, Mach 3.5, = 0.5598 cg x /L , and L/D = 9. .................15 

Figure 11. Comparison of the damping moment coefficients predicted from Sacks’ 

relations with CFD predictions, ANSR, Mach 4.5, = 0.5598 cg x /L , and L/D = 9. .................16 

Figure 12. Comparison of the damping force coefficient predicted from Sacks’ relations 

with CFD predictions, ANSR, Mach 2.5, = 0.5598 cg x /L , and L/D = 9.................................16 

Figure 13. Comparison of the predicted damping moment coefficients with CFD and 

slender body theory results, ANSR, Mach 2.5, and L/D = 9. .................................................17 

Figure 14. Comparison of the damping moment coefficients derived from experimental 

data or CFD using Sacks’ relations with direct CFD prediction, ANSR, Mach 2.5, 

and L/D = 9. ............................................................................................................................18 

Figure 15. Comparison of the pitch-damping force coefficient Nq C obtained from 

direction CFD predictions, predictions using Sacks’ relation (equation 28) and 

equation 2, ANSR, Mach 2.5, = 0.5598 cg x /L , and L/D = 9...................................................19 

Figure 16. Comparison of the pitch-damping moment coefficient sum predicted 

from Bryson’s relation (equation 1) with CFD predictions, ANSR, Mach 2.5, 

= 0.5598 cg x /L , and L/D = 9. ..................................................................................................20 

Figure 17. Schematic of the flared projectile geometry................................................................20 

Figure 18. Comparison of the three computed deviations, ∆1 , ∆2 , and ∆3 relative to 

the damping moment coefficients, flared projectile, and Mach 2............................................21 
v

Figure 19. Comparison of the damping moment coefficient predicted from Sacks’ relations 

with CFD predictions, flared projectile, and Mach 2...............................................................22 

Figure 20. Comparison of the damping moment coefficient predicted from Sacks’ relations 

with CFD predictions, flared projectile, and Mach 5...............................................................22 

Figure 21. Comparison of the damping moment coefficient predicted from Sacks’ relations 

with CFD predictions, 15° flared projectile, and Mach 5........................................................23 
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

The pitch-damping moment coefficients mq

C (due to body transverse angular rate) and mα C

(due to angular rate associated with angle of attack) play an important role in the performance 

and dynamic stability of flight bodies. The pitch-damping moment coefficient sum mq

C + mα C

is of most practical importance, although the individual damping coefficients are often required 

in aerodynamic analyses. Throughout the past several decades, a variety of techniques and 

theories have been developed for predicting the pitch-damping coefficients (1–10). These 

techniques vary in their ease of use as well as their ability to accurately predict the pitchdamping coefficients. 

During the course of its development, slender body theory was generalized to predict a large 

variety of aerodynamic coefficients including the pitch-damping coefficients (1–2). In general, 

direct application of these methods provides only qualitative results for the aerodynamic 

coefficients. However, elements of slender body theory have been incorporated into current 

engineering methods. These methods (3–5) have evolved considerably, although their 

implementation is fairly complex. Apart from implementation issues, modern engineering 

methods, once embodied into a computer code, are relatively easy to use and provide fast and 

reasonably accurate aerodynamic predictions for a large variety of flight geometries. 

From slender body theory, some important relationships between the various aerodynamic 

coefficients can be derived, although these relationships only hold rigorously within the context 

of theories from which they were obtained. Bryson (2) derived the relatively well-known slender 

body result that relates the pitch-damping moment coefficient sum to the normal force 

coefficient, shown in equation 1: 

α Nα

2

cg

mq m C

D

L x

C C ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − + = −  . (1) 

Sacks (1), using the Blasius method for calculating the forces and moments on slender bodies 

from the cross-flow potential, found that many of the aerodynamic coefficients were related to 

each other. Sacks obtained expressions that directly related the individual pitch-damping 

coefficients to the normal force coefficient, such as the pitch-damping force coefficient shown in 

equation 2. These expressions have a form similar to Bryson’s result shown in equation 1: 

Nα

cg

Nq

C

D

L x

C ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − = . (2) 

In practice, these relations that directly relate the damping coefficients to the normal force 

coefficient do not perform particularly well, even when the slender body evaluation of the 
2

normal force coefficient is replaced with a more accurate evaluation of the normal force 

coefficient from sources such as experimental measurement or computational fluid dynamics 

(CFD) (6). However, these relations can be combined with empirical corrections to yield more 

reliable results (6, 7). 

Sacks also found expressions that related the individual damping coefficients to each other, 

including the following relationship between the pitch-damping force coefficients: 

Nq Nα mα C = C − C  . (3) 

Sacks’ explicitly derived relation shown in equation 3 can be easily generalized using his theory 

to the individual pitch-damping force and moment coefficients, and the pitch-damping force and 

moment sums as shown in equations 4–9. For the purposes of this report, these relationships will 

be referred to as Sacks’ relations: 

Nq Nα mα C = C − C  , (4) 

mq mα 2mα C = C − C  , (5) 

[ ] Nq Nα Nα mα C + C = 2C − C   , (6) 

[ ] mq mα mα 2mα C + C = 2C − C   , (7) 

[ ] Nq Nα Nq mα C + C = 2C + C  , (8) 

and 

[ ] mq mα mq 2mα C + C = 2C + C  . (9) 

Sacks’ contribution is the recognition that these relationships exist, although the same relations 

are implicitly contained in other theories such as that presented by Bryson (2). 

For slender bodies, these relations are independent of configuration and are applicable to both 

winged and wingless bodies. The importance of these relationships is that if just one of the three 

pitch-damping force (or moment) coefficients can be determined ( N N q C ,C α or N N q

C C α

⎡ ⎤ + ⎢ ⎥ ⎣ ⎦ ), 

the other two damping coefficients can be obtained using simple closed-form expressions. This, 

of course, assumes that the pitching moment coefficient mα C (first moment of the normal force) 

and the second moment of normal force 2mα C can be obtained as well. Both of these 

coefficients can be obtained if the normal force distribution is known as shown in equations 10 

and 11: 
3

∫ − =

x

0

cg N

m dx

dx

dC

D

(x x) C α

α , (10) 

and 

∫ − =

x

0

N

2

2

cg

2m dx

dx

dC

D

(x x) C α

α . (11) 

Predictive methods for the static normal force and static pitching moment are well established, 

even for fast-design methods. 

Because Sacks’ relations were derived using simple approximate theories, it remains to be shown 

whether the validity of the relations shown in equations 4–9 exists only within the context of the 

theories from which they were derived or whether they are universally valid for slender bodies, 

or perhaps, more importantly, whether they are of general engineering significance. It is 

important to note that the relations shown in equations 4–9 differ somewhat from the results in 

equations 1 and 2 because they relate the pitch-damping coefficients to each other rather than 

purely to the static normal force coefficient. This suggests that Sacks’ relations may more 

properly represent the physics compared to their counterparts which directly relate the damping 

coefficients to the normal force coefficient. Because of this, there is reason to investigate the 

validity and accuracy of Sacks’ relations. 

Recently, a computational approach for predicting all three of the pitch-damping coefficients has 

been developed (8, 9). The approach solves the three-dimensional (3-D) thin-layer NavierStokes equations for three different imposed motions that allow the three pitch-damping 

coefficients to be predicted independently. The pitch-damping force and moment coefficient 

sums are determined from the computation of a body undergoing an imposed coning motion. 

The individual pitch-damping coefficients are obtained from computations of a body undergoing 

two specific types of imposed helical motions. Each of these motions is described in the 

following sections. One of the key components of this method is that steady flow techniques can 

be employed to predict aerodynamic derivatives normally associated with time-dependent 

motions. 

1.1 Helical Motions 

Forces and moments related to the two individual rates q and α can be excited independently 

using two types of motion where the center of gravity (CG) of the flight vehicle traverses a 

helical flight path. The first motion requires the vehicle’s longitudinal axis to be oriented in the 

same direction as the center of rotation of the helix but displaced by a constant distance. 

Figure 1 shows a 3-D view of the motion. 

This particular motion produces no rotation of the body-fixed nonrolling coordinate frame 

relative to an earth-fixed coordinate frame, and hence, the transverse angular velocity of the body 

is zero. The angle of attack and its angular rate vary continuously, producing moment 
4

Figure 1. Helical motion with nonzero α and zero q . 

components associated with the coefficients mα C and mα C , respectively. This motion is 

referred to as “q = 0 helical motion” because the angular rates associated with the damping 

coefficient mq

C are zero. 

For the second motion, the longitudinal axis of the flight vehicle remains tangent to the helical 

flight path at each point along the trajectory. Figure 2 shows a 3-D view of this motion. The 

angle of attack of the incident airstream is zero because both the longitudinal axis of the body 

and the free-stream velocity vector are tangent to the flight path. The resulting yawing rate is 

also zero because the angle of attack is constant. The angular orientation of the flight body 

changes continuously with respect to an earth-fixed reference frame, producing a nonzero 

transverse angular rate. 

As a result, moment components associated with the damping moment mq C are produced. This 

motion is referred to as “α = 0 helical motion” because the angular rates associated with the 

damping coefficient mα C are zero. 

For each of the helical motions, the transverse aerodynamic moment in the nonrolling frame will 

be periodic in time, which also indicates that the flowfield will be periodic in time when viewed 

from the nonrolling coordinate frame. The time dependency is removed by transforming to an 

orthogonal right-handed coordinate system that has its x-axis aligned with the longitudinal axis 

of the body and its z-axis along a line between the body’s CG and the axis of rotation of the 

helix. 
5

Figure 2. Helical motion with zero α and nonzero q . 

For each of the helical motions previously described, the spin rate of the body has not been 

defined. To eliminate any contributions to the aerodynamic forces and moments from the 

Magnus forces and moments, the spin rate is fixed to zero (see references [8] and [9] for details). 

The resulting in-plane moment ( ) Cm and side moment ( ) Cn coefficients in the transformed 

coordinate system for both type of helical motions are shown in equations 12 and 13: 

Zero spin q 0 helical motion − =

V

R iC

V

R

V

D C iC C o m o m n m

Ω

+

Ω ⎟

⎠

⎞ ⎜

⎝

⎛ Ω

+ = − α α , (12) 
6

Zero spin 0 helical motion − = α

V

R

V

D C iC C o m n mq

Ω ⎟

⎠

⎞ ⎜

⎝

⎛ Ω

+ = . (13) 

Here, Ω is the angular velocity of the body about the helix axis, R is the perpendicular o

distance between the helix axis and the body CG, and V is the total linear velocity of the CG. 

Similar expressions for the individual damping force coefficients can be developed using the 

same approach as applied for the moment coefficients. 

1.2 Coning Motion 

To predict the pitch-damping coefficient sum, coning motion is employed. In steady coning 

motion, the longitudinal axis of the flight body performs a rotation at a constant angular velocity 

about a line parallel to the free-stream velocity vector and coincident with the body’s CG, while 

oriented at a constant angle with respect to the free-stream velocity vector. This is shown 

schematically in figure 3. In the context of this report, coning motion also requires the CG to 

traverse a rectilinear path at constant velocity such that the free-stream velocity vector has a 

fixed orientation with the inertial frame. 

Figure 3. Schematic of coning motion. 

Relative to a nonrolling coordinate frame, both the transverse angular rate of the body and the 

angular rate associated with the angle of attack vary in a periodic manner thereby exciting the 

aerodynamic forces and moments associated with both of the individual pitch-damping 

coefficients. Here, a specific form of coning motion, described as zero-spin coning motion, is 

employed. In zero-spin coning motion, the total angular velocity of the body along the 

longitudinal axis (the spin rate) is zero. By imposing zero spin rate on the body, the 

contributions from the Magnus forces and moments are eliminated. 

The time dependency is removed by transforming the body-fixed nonrolling coordinate frame to 

an orthogonal right-handed coordinate system that has its x-axis aligned with the longitudinal 

axis of the body and its z-axis in the pitch-plane of the body. Within this transformed coordinate 

frame, in-plane moment ( ) Cm and side moment ( ) Cn coefficients have the following form:
7

Zero-spin coning motion 

δ [ γ ] δ m n mq mα mα C C C

V

D C iC i ⎟ + + ⎠

⎞ ⎜

⎝

⎛ Ω

+ =  . (14) 

Here, the side-moment is proportional to the pitch-damping moment coefficient sum and varies 

linearly with the coning rate Ω and sine of the total angle of attack δ . For small angles of 

attack, the cosine of the total angle of attack, , γ can be assumed to be one. 

The pitch-damping coefficient sum can also be determined by simply adding the individual 

damping coefficients. In practice, there is very little difference in the two results. However, for 

the current study, directly predicting the pitch-damping coefficient sum using coning motion 

provides an alternative prediction of the pitch-damping sum and additional confirmation of the 

predictions of the individual coefficients. 

2. Computational Technique 

In the previous section, several types of steady motion were presented that produce aerodynamic 

forces and moments from which the various pitch-damping coefficients can be obtained. One 

unique feature of these motions is that they are steady. The advantage of a steady motion over 

an unsteady motion is that a potentially time-independent flowfield can be produced by a steady 

motion, permitting analysis using steady flow CFD techniques. Such techniques can be 

computationally less expensive than time-dependent CFD approaches. To fully exploit the 

steady character of the flow, special body-fixed coordinate systems have been employed to 

capture the steady flowfield. One feature of these coordinate frames is that they are rotating at a 

constant rate with respect to an inertial frame. Because of this, the governing equations of fluid 

motion must be modified to take into account the centrifugal and Coriolis force terms associated 

with the noninertial rotating frame. 

The steady thin-layer Navier-Stokes equations are shown in equation 15: 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

∂

∂

+ + = ∂

∂

+

∂

∂

+

∂

∂ c Sc

ˆ S

ˆ

Re

1 G Hˆ Hˆ ˆ Eˆ Fˆ

ξ η ζ ζ . (15) 

The inviscid flux vectors, Eˆ , Fˆ , and Gˆ , the viscous term S

ˆ , the inviscid and viscous source 

terms due to the cylindrical coordinate formulation, Hc ˆ and Sc

ˆ , and the source term H, ˆ

containing the Coriolis and centrifugal force terms that result from the rotating coordinate frame, 

are functions of the dependent variables represented by the vector ) q ( , u, v, w,e T = ρ ρ ρ ρ , where 

ρ and e are the density and the total energy per unit volume, and u , v , and w are the velocity 

components in axial, circumferential, and normal directions. The inviscid flux vectors and the 
8

source term are shown in equation 16. Details of the thin-layer viscous term are available in the 

literature (11): 

 

( ) ⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

+

+

=

e p U

wU

vU

uU p

U

J

1 Eˆ x

ρ

ρ

ρ ξ

ρ

 

( ) ⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

+

+

+

+

=

e p V

wV p

vV p / r

uV p

V

J

1 Fˆ

r

x

ρ η

ρ η

ρ η

ρ

φ

 (16) 

( ) ⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

+

+

+

+

=

e p W

wW p

vW p / r

uW p

W

J

1 Gˆ

r

x

ρ ζ

ρ ζ

ρ ζ

ρ

φ 

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

+ +

=

x r

r

x

u f v f w f

f

f

f

0

J

1 Hˆ

ρ ρ ρ

ρ

ρ

ρ

φ

φ . 

The pressure, p, can be related to the dependent variables by applying the ideal gas law: 

( ) ( ) 22 2 p 1e u v w 2

ρ γ

⎡ ⎤ =− − ++ ⎢ ⎥ ⎣ ⎦

. (17) 

The turbulent viscosity, , μt which appears in the viscous matrices, was computed using the 

Baldwin-Lomax turbulence model (12). 

The Coriolis and centrifugal acceleration terms due to the rotating coordinate system, which are 

contained in the source term, ˆH, are shown in equation 18: 

f2 u R = Ω × +Ω × Ω× ( ) G G G GG G . (18) 

The Coriolis acceleration is a function of the angular velocity of the coordinate frame with 

respect to the inertial frame, Ω,

G and the fluid velocity vector, u,

G which can be represented by the 

velocity components, u , v , and w . The centripetal acceleration is a function of the angular 

velocity of the rotating frame, Ω,

G and the displacement vector, R,

G between the axis of rotation 

and the local position in the flowfield. The acceleration vector, f,

G

 can be written in terms of its 

components along the x , φ, and r axes, fx , φ f , and r f . 

The steady thin-layer equations are solved using the parabolized Navier-Stokes (PNS) technique 

of Schiff and Steger (11). This “space-marching” approach integrates the governing equations 

from the nose of the flight body to the tail. Following the approach of Schiff and Steger, the 

governing equations, which have been modified here to include the Coriolis and centrifugal force 

terms, are solved using a conservative, approximately factored, implicit finite-difference 
9

numerical algorithm as formulated by Beam and Warming (13). Details of the implementation 

of the source term that contains the Coriolis and centrifugal force terms are given in references 

(8) and (9). 

The technique has been validated with available experimental data where possible, and excellent 

agreement is found (8, 14). Grid resolution studies were also performed in the original studies to 

ensure grid-independent solutions (8, 9). 

3. Results 

In the current context, this computational procedure allows the general applicability of Sacks’ 

pitch-damping relations to be examined. Arguably, up to now, it has not been possible to assess 

the validity or accuracy of these relationships due to the uncertainty associated with 

experimentally derived pitch-damping data and the lack of a higher order theory. In the current 

research effort, the technique has been applied to two axisymmetric body geometries: an ogivecylinder configuration and a cone-cylinder body with a flared afterbody. The accuracy of Sacks’ 

relations is examined for each of these body geometries in supersonic flight. 

3.1 Ogive-Cylinder Results 

The computational approach was applied to a secant-ogive cylinder body (designated as the 

Army Navy Spinner Rocket [ANSR]) shown in figure 4. Results for sea-level atmospheric 

conditions ( 5 Re 4.53 10 D = × M∞ ) are shown as an example, although other Reynolds numbers 

and nose geometries were considered during the course of the study. Baseline results are for a 

flight velocity of Mach 2.5 with additional results at Mach 1.8, 3.5, and 4.5. The results 

presented here are representative of the other flight conditions and nose geometries examined. 

For the force coefficients, the deviation or “error” in the application of Sacks’ relations is defined 

in equation 19. The deviation ∆1 represents the difference between the right-hand and left-hand 

sides of equation 4: 

1 N m Nq

∆ = C − C − C ≡ ∆ α α . (19) 

Note that by simple algebraic manipulations, the following relations that are algebraically 

equivalent to equation 19 are found: 

[ ] Nq N m CNq 2 ∆ = C + C − C − 2 ≡ ∆ α α , (20) 

and 

[ ] CN Cm CNq

CN 3 ∆ = 2 − − + ≡ ∆ α α α . (21) 
10

Figure 4. Schematic of ANSR configuration. 

Essentially, these deviations are a measure of the accuracy of the Sacks’ relations shown in 

equations 4, 6, and 8. It could be argued that the differences 1 ∆ , 2 ∆ , and ∆3 are numerical 

errors that are not representative of physical phenomena. However, it should be noted that the 

deviation ∆ can be computed independently in three different ways. The deviation ∆1 is 

computed using the individual damping coefficients Nq

C and Nα C obtained independently 

from two different types of helical motion, while ∆2 and ∆3 are computed from the pitchdamping coefficient sum Nq Nα C + C and one of the two individual damping coefficients Nq

C

and , CNα respectively. The importance of having three independent methods for determining ∆

is to demonstrate that the deviation is not due to an “error” in the prediction of any one of the 

three damping coefficients. Similar expressions for the deviation of the moment coefficients ∆

can also be obtained and are shown in equations 22–24: 

∆ = Cmα − C2mα − Cmq ≡ ∆1 , (22) 

[ ] 2 m mq m 2 mq

∆ = C + C − C − 2C ≡ ∆ α α , (23) 

and 

[ ] m 3 m mq

∆ = 2Cm − C2 − C + C ≡ ∆ α α α . (24) 
11

Figure 5 shows the computed deviation for the force coefficient as a function of body length at 

Mach 2.5. Here, the body has been lengthened to 20 calibers to more clearly illustrate the 

variation of the deviation with body length. There is very good correlation between the three 

different methods of computing ∆, suggesting that the deviation is representative of a physical 

effect rather than simply numerical error. The deviation is very small over the nose and begins 

to grow in a linear fashion a couple of body diameters aft of the nose. The deviation (for the 

force) can be shown to be independent of CG position using the CG translation relations for the 

individual force coefficients. This was also confirmed by varying the CG position in the 

computations as well. The computed deviation was also found to be somewhat dependent on 

nose length and Mach number, although the results shown are representative of the trends 

observed for the other flight conditions. 

0

1

2

3

4

5

6

0 4 8 12 16 20

X/D

Deviations

∆1

∆2

∆3

Figure 5. Comparison of the three computed deviations ∆1 , ∆2 , and ∆3 vs. body 

length, ANSR, and Mach 2.5. 

Figure 6 shows the computed deviations for the force, 1 ∆ , 2 ∆ , and 3 ∆ , compared with the 

individual pitch-damping coefficients and the pitch-damping sum. Relative to Nα C , and the 

pitch-damping force sum, the deviation is quite small. The distribution of the deviation is also 

small compared with the pitch-damping force coefficient Nq

C , except near the end of the body 

where Nq C itself is nearly zero. Equation 4 shows that according to Sacks’ relation, the 

difference between Nα C and Nq C is the pitching moment coefficient . Cmα This is also 

graphically shown in figure 6. Near the nose of the body, the difference between the two 

damping coefficients, Nα C and Nq C , increases at nearly the same rate as the pitching moment 
12

-10

-5

0

5

10

15

20

0369

X/D

Coefficients

mα

C

Nq

C

Nq Nα C + C

Nα C

1 2 3 ∆ ,∆ ,∆

Figure 6. Comparison of the three computed deviations ∆1 , ∆2 , and ∆3 relative to the 

damping force coefficients, ANSR, Mach 2.5, = 0.5598 cg x /L , and length-todiameter (L/D) ratio = 9. 

coefficient. On the aft of the body, the pitching moment coefficient is nearly constant, and the 

difference between Nα C and Nq C becomes relatively constant. 

Figure 7 shows the computed deviation for the moment relative to the three damping coefficients 

of interest. Again, there is very good correlation between the three methods for computing the 

deviation. The computed deviation is small compared to individual damping moment 

coefficients and to the pitch-damping moment sum. Sacks’ relation for the moment (equation 5) 

shows that the difference between mα C and mq C is the pitching second-moment coefficient 

2mα C . This also can be seen in figure 7. In a sense, figures 6 and 7 provide some measure of 

the expected error in applying Sacks’ relations to obtain the various damping moment 

coefficients given that normal force distribution (or mα C and 2mα C ) and one of the pitchdamping coefficients is known. 

The most likely application of Sacks’ relations in practical situations is to compute the individual 

damping coefficients from the pitch-damping coefficient sum because the pitch-damping 

moment sum is much easier to measure. Through simple algebraic manipulations of equations 

7 and 9, the following form of the Sacks’ relations can be obtained: 
13

C ( [C C ] C m )/ 2 mα mq mα 2 α = + +   , (25) 

and 

C ( [C C ] C m )/ 2 mq mq mα 2 α = + −  . (26) 

-80

-60

-40

-20

0

20

40

60

0369

X/D

Coefficients

1 2 3 ∆ ,∆ ,∆

2mα

C

mα C

mq

C

mq mα C +C

Figure 7. Comparison of the three computed deviations ∆1 , ∆2 , and ∆3 relative to the 

damping moment coefficients, ANSR, Mach 2.5, = 0.5598 cg x /L , and L/D = 9.

Figure 8 shows a comparison of the pitch-damping moment coefficients mq

C and mα C

obtained by applying the Sacks’ relationships using the pitch-damping moment coefficient sum 

Cmq

Cm ⎡ ⎤ + ⎣ ⎦ α and the second-moment coefficient of the normal force 2mα C . Here, CFD has 

been used to compute both Cmq

Cm ⎡ ⎤ + ⎣ ⎦ α and 2mα C . The results from the application of these 

relations are compared with the CFD predictions of the individual damping coefficients. The 

predictions were obtained at a flight velocity of Mach 2.5. The coefficients mq

C and mα C are 

overpredicted and underpredicted by ~5% and 12%, respectively, although the absolute error is 

similar for both coefficients. The distribution of the damping coefficients over the body is also 

very well predicted. 
14

-60

-50

-40

-30

-20

-10

0

10

0369

X/D

Pitch-Damping Coefficients

Direct CFD Prediction

Derived from Sacks' Relations

mq

C

mα C

Figure 8. Comparison of the damping moment coefficients predicted from Sacks’ 

relations with CFD predictions, ANSR, Mach 2.5, = 0.5598 cg x /L , and 

L/D = 9.

Figures 9–11 show similar comparisons for flight velocities of Mach 1.8, 3.5, and 4.5. The 

relative comparisons are very similar to the predictions at Mach 2.5 shown in figure 8. The 

largest Mach number effect appears to occur on the cylindrical portion of the body rather than on 

the nose. The relative Mach number effect, though relatively small, is captured in the application 

of the Sacks’ relations. 

Similar analysis can be performed for the force coefficients. Figure 12 shows the comparison of 

the individual pitch-damping moment coefficients Nq

C and Nα C predicted using the form of 

Sacks’ relations in equations 27 and 28 with direct CFD predictions. The distribution of the 

force coefficients along the body is very well predicted using Sacks’ relations: 

C ( [C CN ] Cm )/ 2 Nα Nq α α = + +   , (27) 

and 

C ( [C CN ] Cm )/ 2 Nq Nq α α = + −  . (28) 
15

-60

-50

-40

-30

-20

-10

0

10

0369

X/D

Pitch-Damping Coefficients

Direct CFD Prediction

Derived from Sacks' Relations

mq

C

mα C

Figure 9. Comparison of the damping moment coefficients predicted from Sacks’ 

relations with CFD predictions, ANSR, Mach 1.8, = 0.5598 cg x /L , and 

L/D 9. =

-60

-50

-40

-30

-20

-10

0

10

0369

X/D

Pitch-Damping Coefficients

Direct CFD Prediction

Derived from Sacks' Relations

mq

C

mα C

Figure 10. Comparison of the damping moment coefficients predicted from Sacks’ 

relations with CFD predictions, ANSR, Mach 3.5, = 0.5598 cg x /L , and 

L/D 9. =
16

-60

-50

-40

-30

-20

-10

0

10

0369

X/D

Pitch-Damping Coefficients

Direct CFD Prediction

Derived from Sacks' Relations

mq

C

mα C

Figure 11. Comparison of the damping moment coefficients predicted from Sacks’ 

relations with CFD predictions, ANSR, Mach 4.5, = 0.5598 cg x /L , and 

L/D = 9. 

-10

-5

0

5

10

15

20

0369

X/D

Pitch-Damping Coefficients

Direct CFD Prediction

Derived from Sacks' Relations

Nq

C

Nα C

Figure 12. Comparison of the damping force coefficient predicted from Sacks’ relations 

with CFD predictions, ANSR, Mach 2.5, = 0.5598 cg x /L , and L/D 9. =
17

Figure 13 shows the results for the individual pitch-damping coefficients for various CG 

positions for the body with the L/D ratio of 9 at Mach 2.5 obtained by applying equations 27 and 

28. The results are compared with CFD results and with direct slender body theory results. 

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

34567

Xcg/D

Pitch-Damping Coefficients

CFD Prediction

Slender Body Theory

Sacks' Relations

mq

C

mα C

Figure 13. Comparison of the predicted damping moment coefficients with CFD and 

slender body theory results, ANSR, Mach 2.5, and L/D 9. =

There is very good correlation of the results obtained with Sacks’ relations and the CFD results, 

and the results are a significant improvement over the slender body theory. 

For the results shown in figures 8–12, comparisons have been made between the force and 

moment distributions along the body. This was done because these distributions are easily 

extracted from the computational results, and they allow detailed examination of the 

performance of Sacks’ relations. It must be emphasized that it is not necessary to obtain the 

pitch-damping force and moment distributions in order to apply Sacks’ relations because they are 

applicable for the global force and moment coefficients as well. This is particularly important to 

consider when the source of the pitch-damping coefficient data is from experiment where only 

global coefficients are typically available. However, Sacks’ relations do require the normal force 

distribution to determine 2mα C because generally only the normal force and pitching moment 

are available as global coefficients from many sources. In lieu of more sophisticated 

computational methods for determining the normal force distribution, fast-design aeroprediction 

codes (3) should provide acceptable accuracy for a variety of flight vehicle geometries. 
18

To demonstrate the use of Sacks’ relations when only global forces and moments are available, 

pitch-damping moment coefficient sum data from range firings of the ANSR have been used to 

estimate the individual pitch-damping moment coefficients using Sacks’ relations. The normal 

force distribution from the fast-design aeroprediction code AP02 (3) has been used to predict 

2mα C . Figure 14 shows the predicted pitch-damping coefficients obtained from Sacks’ relations 

using experimental measurements of the pitch-damping coefficient sum and 2mα C obtained 

from AP02. The results are compared with direct CFD predictions of the individual damping 

coefficients as well as the results obtained from Sacks’ relation using CFD predictions of the 

pitch-damping coefficient sum and 2mα C shown previously in figure 10. The experimentally 

derived values of the individual pitch-damping coefficients compare well with the predicted 

results. The results also indicate that the biggest source of error is produced by the uncertainty in 

the experimentally derived pitch-damping moment coefficient rather than the error in Sacks’ 

relations. (The pitch-damping data used here are from a highly regarded data set and are 

representative of the expected accuracy for the pitch-damping coefficient sum from range 

firings.) 

-60

-50

-40

-30

-20

-10

0

10

3.5 4 4.5 5 5.5 6

Xcg/D

Damping Coefficient

Direct CFD

Direct CFD

Sacks w/Exp and AP02

Sacks w/Exp and AP02

Sacks w/CFD

Sacks w/CFD mα C

mq

C

Figure 14. Comparison of the damping moment coefficients derived from experimental 

data or CFD using Sacks’ relations with direct CFD prediction, ANSR, Mach 

2.5, and L/D 9. =

Both Sacks (1) and Bryson (2) obtained relations between the pitch-damping coefficients and the 

normal force coefficient shown previously in equations 1 and 2. Relatively speaking, equations 

1 and 2 do not perform particularly well, especially when compared to Sacks’ relations 
19

(equations 4–9). Figure 15 shows a comparison of the CFD prediction of the pitch-damping 

coefficient Nq C with results obtained using equation 2 and Sacks’ relation (equation 28). As 

before, CFD results were used to obtain the aerodynamic coefficients required in equations 2 and 

28. The results obtained with equation 2 provide only a qualitative prediction of the distribution 

of Nq C compared with the result obtained using Sacks’ relation (equation 28), which is in much 

better agreement with the CFD predictions. Instead of relating the pitch-damping coefficients to 

the normal force coefficient as in equation 2, Sacks’ relations relate the pitch-damping 

coefficient to each other. These results seem to imply the pitch-damping coefficients are more 

closely related to each other than they are to the normal force coefficient. Figure 16 shows a 

comparison of the pitch-damping moment coefficient sum mq mα C + C from direct CFD 

prediction with the results obtained using Bryson’s relation (equation 1), which like equation 2, 

relates the damping coefficient to the normal force coefficient. The distribution of mq mα C + C

is relatively poorly predicted compared with the CFD result, although total moment for entire 

configuration differs by only 25%. 

-10

-5

0

5

10

15

0369

X/D

Direct CFD Prediction

Derived from Sacks' Relations

Eq. 2

Nq

C

Figure 15. Comparison of the pitch-damping force coefficient Nq C obtained from 

direction CFD predictions, predictions using Sacks’ relation (equation 28) 

and equation 2, ANSR, Mach 2.5, = 0.5598 cg x /L , and L/D 9. =

3.2 Flared Projectile Results 

The performance of Sacks’ relations was also examined for a flared projectile geometry shown in 

figure 17. Predictions of the pitch-damping coefficient sum have been validated previously for 
20

-70

-60

-50

-40

-30

-20

-10

0

10

0369

X/D

Pitch-Damping Coefficient Sum 

Direct CFD Prediction

Derived from Bryson's Relation, Eq. 1

Figure 16. Comparison of the pitch-damping moment coefficient sum predicted from 

Bryson’s relation (equation 1) with CFD predictions, ANSR, Mach 2.5, 

= 0.5598 cg x /L , and L/D 9. =

Figure 17. Schematic of the flared projectile geometry. 

this configuration (14). For the current report, additional predictions of the individual pitchdamping coefficients were also performed for flight velocities between Mach 2 and Mach 5. 
21

Figure 18 shows the predicted deviations of the moment compared with the individual pitchdamping coefficients at Mach 2. Again, the deviation over the body is small compared to the 

pitch-damping coefficients even on the afterbody where both the geometry and moment 

coefficients are changing significantly. Similar results were found at the other flight velocities. 

-300

-200

-100

0

100

200

300

0 4 8 12 16

X/D

Coefficients

mq

C

1 2 3 ∆ ,∆ ,∆

2mα

C

mα C

Figure 18. Comparison of the three computed deviations, ∆1 , ∆2 , and ∆3 relative to 

the damping moment coefficients, flared projectile, and Mach 2. 

Similar to the results for the secant ogive/cylinder configuration, the form of Sacks’ relations 

embodied in equations 25 and 26 was applied to determine the individual pitch-damping 

coefficients from the pitch-damping moment coefficient sum and the pitching second-moment 

coefficient for the flared projectile geometry. Comparisons were made with direct CFD 

predictions. 

Figures 19 and 20 show comparisons of the individual pitch-damping moment coefficients at 

Mach 2 and 5, respectively, for the flare projectile geometry. Very good agreement between the 

results obtained by applying Sacks’ relations and direct CFD predictions are found. Additional 

results for the same flare projectile with the 6° flare replaced by a 15° flare were also obtained 

and are shown in figure 21. Despite a nearly doubling of the pitch-damping coefficient due to 

the larger flare, the Sacks’ relations results are in very good agreement with the direct CFD 

predictions.
22

-300

-250

-200

-150

-100

-50

0

50

0 4 8 12 16

X/D

Pitch-Damping Coefficients

Direct CFD Prediction

Derived from Sacks' Relations

mq

C

mα C

Figure 19. Comparison of the damping moment coefficient predicted from Sacks’ 

relations with CFD predictions, flared projectile, and Mach 2. 

-300

-250

-200

-150

-100

-50

0

50

0 4 8 12 16

X/D

Pitch-Damping Coefficients

Direct CFD Prediction

Derived from Sacks' Relations

mq

C

mα C

Figure 20. Comparison of the damping moment coefficient predicted from Sacks’ 

relations with CFD predictions, flared projectile, and Mach 5.
23

-700

-600

-500

-400

-300

-200

-100

0

100

0 4 8 12 16

X/D

Pitch-Damping Coefficients

Direct CFD Prediction

Derived from Sacks' Relations

mq

C

mα C

Figure 21. Comparison of the damping moment coefficient predicted from Sacks’ 

relations with CFD predictions, 15° flared projectile, and Mach 5. 

4. Summary 

In conclusion, the results presented here indicate that Sacks’ pitch-damping relations are only 

strictly valid under the context of the theory from which they were originally developed. They 

do, however, provide a reasonably good means of estimating the pitch-damping coefficients 

when one of the three pitch-damping coefficients can be determined. The most likely practical 

use of these relations might be to provide estimates of the individual pitch-damping coefficients 

using values of the pitch-damping coefficient sum determined from some other source, such as 

experimental data and engineering estimation approaches or when the additional expense of the 

separate CFD computations of the individual pitch-damping coefficients is not justified. In some 

cases, it appears that the error in applying these relationships is smaller than the error associated 

with generating the initial pitch-damping coefficient (such as with engineering estimation 

approaches) from which the other two damping coefficients are derived using Sacks’ relations. 

Applying Sacks’ relations to determine the individual pitch-damping coefficients from the pitchdamping sum represents only one possible application of Sacks’ relations. These relations could 

also benefit theoretical developments because theories for predicting the pitch-damping 

coefficients need only focus on a single damping coefficient. The other damping coefficients 
24

could then be obtained from Sacks’ relations. Such an approach has been already used as an 

estimation procedure for the damping coefficients (6). In this work, the distribution of 

Nα C along the body is predicted using slender body theory with empirically based corrections. 

The damping coefficient Nq

C can then be obtained from Sacks’ relations. Once the damping 

force distributions are known, the damping moments can be easily obtained by integration of the 

force loadings. Improvements in the estimates of Nq

C from Sacks’ relations can also be 

obtained by correlating the error ∆ as well – an approach used in reference (6) to further 

improve the estimates of Nq

C . 

Finally, in the currently report, only axisymmetric configurations in supersonic flight have been 

considered. The theory from which Sacks derived the relations considered here is applicable to 

both wingless and winged vehicles. Further research is still required to assess the performance 

of Sacks’ relations for winged vehicles and for other flight velocity regimes. 
25

5. References 

1. Sacks, A. H. Aerodynamic Forces, Moments, and Stability Derivatives for Slender Bodies of 

General Cross Section; NACA Technical Note 3283; National Advisory Committee for 

Aeronautics: Washington, DC, November 1954; p 27. 

2. Bryson, A. E., Jr. Stability Derivatives for a Slender Missile With Application to a WingBody-Vertical-Tail Configuration. Journal of the Aeronautical Sciences May 1953, 20 (5), 

297–308. 

3. Moore, F. G.; Hymer, T. C. The 2002 Version of the Aeroprediction Code: Part I – 

Summary of New Theoretical Methodology; NSWCDD/TR-01/108; Naval Surface Warfare 

Center: Dahlgren, VA, March 2002. 

4. Vukelich, S. R.; Jenkins, J. E. Missile DATCOM: Aerodynamic Prediction on Conventional 

Missiles Using Component Build-up Techniques; AIAA Paper 84-0388; American Institute 

of Aeronautics and Astronautics: Reston, VA, January 1984. 

5. Whyte, R. E. Spinner – A Computer Program for Predicting the Aerodynamic Coefficients 

of Spin Stabilized Projectiles; Class 2 Report No. 69APB3; General Electric Company: 

Burlington, VA, August 1969. 

6. Danberg, J. E.; Weinacht, P. Approximate Computation of Pitch-Damping Coefficients; 

AIAA Paper 2002-5048; American Institute of Aeronautics and Astronautics: Reston, VA, 

August 2002. 

7. Sigal, A. Correlation of the Damping in Pitch Stability Derivatives for Body-Tail 

Configurations; AIAA Paper 94-3482; American Institute of Aeronautics and Astronautics: 

Reston, VA, August 2002. 

8. Weinacht, P.; Sturek, W. B.; Schiff, L. B. Navier-Stokes Predictions of Pitch-Damping for 

Axisymmetric Projectiles. Journal of Spacecraft and Rockets 1997, 34 (6), 753–761. 

9. Weinacht, P. Navier-Stokes Predictions of the Individual Components of the Pitch-Damping 

Sum. Journal of Spacecraft and Rockets 1998, 35 (5), 598–605. 

10. Park, S. H.; Kim, Y.; Kwon, J. H. Prediction of Damping Coefficients Using the Unsteady 

Euler Equations. Journal of Spacecraft and Rockets 2003, 40 (3), 356–362. 

11. Schiff, L. B.; Steger, J. L. Numerical Simulation of Steady Supersonic Viscous Flow. AIAA 

Journal 1980, 18 (12), 1421–1430. 
26

12. Baldwin, B. S.; Lomax, H. Thin Layer Approximation and Algebraic Model for Separated 

Turbulent Flows; AIAA Paper 78-257; American Institute of Aeronautics and Astronautics: 

Reston, VA, January 1978. 

13. Beam, R.; Warming, R. F. An Implicit Factored Scheme for the Compressible NavierStokes Equations. AIAA Journal 1978, 16 (4), 85–129. 

14. Weinacht, P. Navier-Stokes Predictions of Pitch-Damping for a Family of Flared 

Projectiles; AIAA Paper 91-3339; American Institute of Aeronautics and Astronautics: 

Reston, VA, September 1991. 
27

List of Symbols, Abbreviations, and Acronyms 

a Speed of sound 

Cm Pitching moment coefficient, 

V S D

2

1

M

ref

2 ρ ∞

mα C Pitching moment coefficient slope with respect to angle of attack, ∂α

∂Cm

2mα C Pitching second moment coefficient slope with respect to angle of attack, ∂α

∂C2m

mα C Pitch-damping moment coefficient slope, 

⎟

⎠

⎞ ⎜

⎝

⎛

∂

∂

V

D

Cm

α

mq C Pitch-damping moment coefficient slope, 

⎟

⎠

⎞ ⎜

⎝

⎛

∂

∂

V

q D

Cm

mq mα C + C Pitch-damping moment coefficient 

Cn Side moment coefficient 

C Normal force coefficient, N

ref

2 V S

2

1

F

ρ ∞

Nα C Normal force coefficient slope with respect to angle of attack, ∂α

∂CN

Nα C Pitch-damping force coefficient slope, 

⎟

⎠

⎞ ⎜

⎝

⎛

∂

∂

V

D

CN

α

Nq

C Pitch-damping force slope, 

⎟

⎠

⎞ ⎜

⎝

⎛

∂

∂

V

q D

CN

Nq Nα C + C Pitch-damping force coefficient 
28

D Reference diameter 

e Total energy per unit volume 

Gˆ F, ˆ E, ˆ Flux vectors in transformed coordinates 

F Force 

Hˆ Source term in Navier-Stokes equations due to rotating coordinate frame 

Hc ˆ Source term in Navier-Stokes equations due to cylindrical coordinate formulation 

L Body length 

M Mach number 

M Moment 

p Pressure 

q Transverse angular rate of body as used in flight mechanics equations 

T q Transposed vector of dependent variables as used in Navier-Stokes equations 

r Radial coordinate 

Re Reynolds number, ∞ ρ ∞ μ ∞ a D /

R Helix radius o

S Reference area, ref 4

D S

2

ref

π

=

S

ˆ Viscous flux vector 

Sc

ˆ Viscous terms due to cylindrical coordinate formulation 

u, v,w Velocity components in the x,φ,r directions 

U Contravariant velocity components ,V,W

V Freestream velocity 

e e Ze X Earth-fixed coordinates ,Y ,

x Axial location along body from nose 

x Integration variable associated with axial location along body from nose 

x Axial location of center of gravity from nose cg
29

Greek Symbols

α Angle of attack 

α Angular rate associated with angle of attack 

β Yaw angle 

γ Cosine of total angle of attack 

δ Sine of total angle of attack 

∆ Deviation or error in Sacks’ relations for force 

∆ Deviation or error in Sacks’ relations for moment 

μ Viscosity 

ξ, Transformed coordinates in η,ζ the Navier-Stokes equations 

ρ Density 

φ Circumferential coordinate 

ω Angular velocity about longitudinal axis 

Ω Angular rate associated with coning and helical motions 

Subscripts

∞ Quantity evaluated at freestream conditions 
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