An Approximate Method for Pitch-Damping Prediction 

by James E. Danberg and Paul Weinacht 

ARL-TR-3007 July 2003 

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Army Research Laboratory 

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ARL-TR-3007 July 2003 

An Approximate Method for Pitch-Damping Prediction 

James E. Danberg and Paul Weinacht 

Weapons and Materials Research Directorate, ARL 

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An Approximate Method for Pitch-Damping Prediction 

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James E. Danberg and Paul Weinacht 

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

A simple method of estimating the pitch-damping coefficients for axisymmetric flight bodies has been developed. The 

method is suitable for preliminary design and parametric studies. The procedure is based on concepts from slender body 

theory, but the method significantly improves the accuracy of the results through the use of correlation functions. To derive 

the correlation functions, extensive use of a recently developed and validated computational fluid dynamics (CFD) method for 

predicting the pitch-damping coefficients has been employed. The CFD method provides important details necessary to derive 

the correlation functions that are unavailable from the current experimental database. The method has been validated for a 

variety of nose geometries, body lengths, and Mach numbers using both experimental and computationally generated 

databases. 

15. SUBJECT TERMS

computational fluid dynamics, projectile aerodynamics, flight mechanics 

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19a. NAME OF RESPONSIBLE PERSON

Paul Weinacht 

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iii

Contents 

List of Figures iv

List of Tables v

1. Introduction 1

1.1 Slender Body Theory Relationships................................................................................2 

1.2 Relationships Between Coefficients................................................................................4 

2. Formulation of an Approximate Model 9

2.1 G(x) on Cylinder (Carry-Over Transformation) ...........................................................10 

2.2 Effect of Nose Length ...................................................................................................14 

2.3 Correlation for the Nose Region ...................................................................................15 

2.4 Pitch Damping Force, Nq C (Sacks’s Relationship)......................................................16 

2.5 Pitch-Damping Moment Coefficient Sum.....................................................................17 

2.6 Reynolds Number Effects .............................................................................................19 

3. Application of the Approximate Model 20

4. Conclusion 30

5. References 32

List of Abbreviations 34
iv

List of Figures 

Figure 1. Schematic of ANSR rocket configuration.......................................................................2 

Figure 2. Comparison of predicted normal force coefficient slope Nα

C distribution with 

slender body theory for the ANSR.............................................................................................5 

Figure 3. Comparison of PNS and slender body theory predictions of pitch-damping force 

coefficients over body, ANSR, Mach 2.5, Xcg/D = 5.038. ........................................................5 

Figure 4. Comparison of pitch-damping force coefficient Nα& C with integral of normal force 

coefficient over body, Mach 2.5, ANSR, CFD predictions.......................................................8 

Figure 5. Comparison between Nq

C and Nα mα C − C & (equation 11), Mach 2.5, ANSR, Xcg/D 

= 5.038, CFD predictions..........................................................................................................9 

Figure 6. G(x) over the ANSR for various Mach numbers...........................................................10 

Figure 7. Correlation of G(x) on the cylindrical afterbody, ANSR..............................................11 

Figure 8. Similarity factors S(M) and T(M). ................................................................................13 

Figure 9. Correlation of G25(X25) with computed data, ANSR.....................................................13 

Figure 10. Effect of nose length on G(x), tangent-ogive/cylinder, Mach 2.5...............................14 

Figure 11. Variation of G(nose) with Mach number and cone angle for conical noses. ..............15 

Figure 12. Distribution of H(x) along body for ANSR.................................................................17 

Figure 13. Comparison of PNS and approximate method predictions of pitch-damping force 

coefficients over body, ANSR, Mach 2.5, Xcg/D = 5.038. ......................................................18 

Figure 14. Comparison of PNS and approximate method predictions of pitch-damping 

moment coefficients over body, ANSR, Mach 2.5, Xcg/D = 5.038. ........................................18 

Figure 15. Comparison of slender body theory, PNS, and approximate method predictions of 

pitch-damping moment coefficient sum over body, ANSR, Mach 2.5, Xcg/D = 5.038..........20 

Figure 16. Correlation between approximate method predictions and experimental data for 

mq mα& C + C , ANSR, L/D = 5, 7, and 9, and Mach 1.3, 1.8, and 2.5......................................21 

Figure 17. Correlation between approximate method predictions and CFD computations for 

mq mα& C + C , ANSR, L/D = 5, 7, and 9, and Mach 1.3, 1.8, and 2.5.......................................22 

Figure 18. Correlation between approximate method predictions and CFD computations for 

mq mα& C + C , cone cylinder and tangent ogive cylinder bodies, L/D = 5, 7, and 9. ................23 

Figure 19. Correlation between approximate method predictions and CFD computations for 

mq mα& C + C , cone cylinder and tangent ogive cylinder bodies, L/D = 12, 15, and 20. ..........24 
v

Figure 20. Correlation between approximate method predictions and CFD computations for 

mα& C , cone cylinder and tangent ogive cylinder bodies, L/D = 5, 7, and 9. ............................24 

Figure 21. Correlation between approximate method predictions and CFD computations for 

mα& C , cone cylinder and tangent ogive cylinder bodies, L/D = 12, 15, and 20. ......................25 

Figure 22. Correlation between approximate method predictions and CFD computations for 

mq C , cone cylinder and tangent ogive cylinder bodies, L/D = 5, 7, and 9. ............................25 

Figure 23. Correlation between approximate method predictions and CFD computations for 

mq C , cone cylinder and tangent ogive cylinder bodies, L/D = 12, 15, and 20. ......................26 

Figure 24. Comparison of predicted Mach number variation of mq mα& C + C obtained from 

approximate method and CFD computations, cone cylinder bodies, Lnose/D = 2, L/D = 5, 

7, and 9.....................................................................................................................................27 

Figure 25. Comparison of predicted Mach number variation of mq mα& C + C obtained from 

approximate method and CFD computations, tangent ogive cylinder bodies, Lnose/D = 2, 

L/D = 5, 7, and 9. .....................................................................................................................27 

Figure 26. Comparison of predicted Mach number variation of mq mα& C + C obtained from 

approximate method and CFD computations, cone cylinder bodies, Lnose/D = 2, L/D = 12, 

15, and 20.................................................................................................................................28 

Figure 27. Comparison of predicted Mach number variation of mq mα& C + C obtained from 

approximate method and CFD computations, tangent ogive cylinder bodies, Lnose/D = 2, 

L/D = 12, 15, and 20. ...............................................................................................................28 

Figure 28. Comparison of predicted effect of nose length on mq mα& C + C obtained from 

approximate method and CFD computations, tangent ogive cylinder bodies, L/D = 7. .........29 

Figure 29. Comparison of predicted effect of nose length on mq mα& C + C obtained from 

approximate method and CFD computations, cone cylinder bodies, L/D = 7.........................30 

List of Tables 

Table 1. G25 as a function of X25...................................................................................................12 

Table 2. Coefficients of the fifth-order polynomial fit of equation 27. ........................................16 

Table 3. Center of gravity locations as a function of body length................................................23 
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

Computational fluid dynamics (CFD) has been shown to provide accurate predictions of the 

pitch-damping coefficients for a variety of projectile and missile geometries (1–6). In addition to 

the global coefficients for the complete configuration, the CFD results allow important details, 

such as the distribution of the force and moment along the body, to be examined. These details 

would be difficult to investigate experimentally. For some preliminary design and parametric 

studies, it may be convenient to have a simpler way of estimating the pitch-damping 

characteristics of flight bodies. Such models do currently exist (7–9), although they are currently 

being reexamined and updated (10, 11), in part, due to the recent availability of computationally 

derived pitch-damping data. This report documents the development of an engineering model 

for predicting the pitch-damping characteristics of axisymmetric flight bodies in supersonic flow. 

The engineering model is based, in part, on slender body theory and some of the relationships 

between the aerodynamic coefficients that can be derived from it. 

There are two primary pitch-damping force coefficients, Nq

C and Nα& C , that account for forces 

produced by the angular rate of the body q and angular rate α& associated with the angle of 

attack. For rectilinear flight, the angular rate of the body and the angular rate due to angle of 

attack are closely related. In this case, the aerodynamic effect represented by the individual 

coefficients Nq

C and Nα& C can be combined and treated as a single coefficient, the pitch-damping 

force coefficient sum, Nq Nα& C + C . Despite this simplification, the individual pitch-damping 

coefficients may still be required for some aerodynamic analyses. It is therefore desirable to 

develop a method of predicting the individual pitch-damping coefficients from which the pitchdamping coefficient sum can be easily computed. In most cases, the pitch-damping moment 

coefficients, mq C , mα& C , and mq

C + mα& C are of primary interest because of their importance to 

the aerodynamic damping. However, these coefficients are easily obtained once the 

corresponding force coefficients are determined. 

By applying slender body theory, it becomes apparent that the pitch-damping force coefficient 

due to rate of angle of attack change, Nα& C , is of primary importance because of its relationship 

to the normal force and all the other damping coefficients. Correlation functions that are 

evaluated from CFD computation are introduced into the slender body theory equation for Nα& C

to form the basis for an engineering model to estimate the pitch-damping coefficients for 

axisymmetric bodies. The model is then applied to a variety of axisymmetric body geometries 

over a range of supersonic Mach numbers, and comparisons are made with CFD predictions for 

the same body geometries to evaluate the effectiveness of the correlation functions.
2

An extensive database of pitch-damping coefficient data for a variety of axisymmetric body 

geometries and flight conditions was generated using results from a recently developed 

computational procedure for predicting the static and pitch-damping coefficients (1–3). The 

technique makes use of a novel rotating coordinate frame to compute the aerodynamics 

associated with the various pitch-damping coefficients. The technique has been embodied into 

several codes including a Parabolized Navier-Stokes (PNS) technique (1–3) and time-marching 

Navier-Stokes approach (F3D) (4). The original studies focused on the Army-Navy Spinner 

Rocket (ANSR) shown in Figure 1. High quality aerodynamic range data were used to validate 

the computational procedure. For the current study, the database has been expanded to include 

additional Mach numbers (Mach 1.8–4.5) and longer body lengths (up to length-to-diameter ratio 

[L/D] = 20) as well as a variety of nose geometries. An important feature of the computational 

data is that the distribution of the aerodynamic forces and moments along the body due to both 

the static and dynamic aerodynamic coefficients is obtained. It is unlikely that this type of data 

could be obtained from experimental means. 

Figure 1. Schematic of ANSR rocket configuration. 

1.1 Slender Body Theory Relationships 

The slender body theory suggested by Munk (12) and applied by Wood and Murphy (13) among 

others, allows for simple, though approximate, evaluation of both the static and dynamic 

aerodynamic coefficients needed to characterize the pitch-damping characteristics of a flight 

body. The slender body theory allows the forces and moment to be computed from the local 

body diameter, Dˆ . For an axisymmetric body, the slender body theory results for the static and 

dynamic force and moment coefficients can be summarized in the following equations:
3

Static Coefficients

Normal force coefficient slope: 

∫ = =

x

0

2

2

N Dˆ dx 2

dx

Dˆ d 2 C α . (1) 

Moment coefficient slope: 

∫ = −

x

0

2

m cg dx

dx

Dˆ d 2 C (x x) α . (2) 

Damping Force Coefficients

Force coefficient slope due to angular rate associated with angle of attack: 

∫ =

x

0

2

N D dx ˆ C 2 α& . (3) 

Force coefficient slope due to transverse angular velocity: 

[ ] D (x x) ˆ D dx 2 ˆ (x x)2

dx

d C cg

2

x

0

2

Nq = − cg − = − − ∫ . (4) 

Damping Moment Coefficients

Moment coefficient slope due to angular rate associated with angle of attack: 

∫ = −

x

0

2

m cg D dx ˆ C (x x)2 α& . (5) 

Moment coefficient slope due to transverse angular velocity: 

[ ] ∫ = − − −

x

0

2

mq cg cg D dx ˆ (x x)2

dx

d C . (6) (x x)

It should be noted that since the upper limit of integration in equations 1–6 is the local axial 

distance from the nose, the integrated aerodynamic coefficients represent the accumulated force 

or moment from the nose to local axial position on the body. By substituting the total body 

length as the upper limit of integration allows the aerodynamic coefficients to be evaluated for 

the complete body. Thus, these integrals allow both the global aerodynamic coefficients and 

their distribution along the body to be determined. 

The pitch-damping moment coefficient sum mq mα& C + C is of considerable practical importance 

in assessing stability and is easily determined from the sum of equations 5 and 6. 
4

2

cg

2

mq m D (x x) ˆ C + C = −2 − α& . (7) 

The slender body theory allows all of the aerodynamic coefficients of interest to be related to the 

local cross-sectional area of the body as previously seen. Because the aerodynamic coefficients 

are related to a local property of the body, the theory cannot account for such known effects as 

“carry-over lift,” which is produced in a transition region downstream of the junction between 

the nose and the cylinder. As a consequence, the normal force coefficient predicted by the 

slender body theory is known to be incorrect except where the body diameter is rapidly 

changing. Exactly the same slender body formulas can be obtained from a more sophisticated 

analysis such as that of Bryson (14). These relations can also be found in the standard reference 

by Nielsen (15). Bryson’s analysis is a general application of the apparent additional mass 

concept to bodies of more general cross-sectional geometries. 

Figure 2 shows comparisons of the slender body and PNS predictions of the normal force 

coefficient slope for the ANSR. This figure reveals several important and well-known 

characteristics of slender body theory. First, on the nose of the body where the body diameter is 

changing, slender body theory predicts the normal force fairly well. However, slender body 

theory fails to predict the carry-over lift on the cylinder behind the nose. The carry-over lift is 

significant as it accounts for more than 30% of the normal force. The normal force coefficient 

also exhibits a Mach number dependence not found in the slender body theory. Since the normal 

force is independent of the center of gravity location, the same data applies to shorter bodies 

truncated at any appropriate length. 

Similar comparisons are obtained when the slender body theory results for the pitch-damping 

force coefficients are compared with PNS predictions as shown in Figure 3. Again, the pitchdamping force coefficients are relatively well predicted on the nose, but a significant difference 

is again seen on the cylinder behind the nose. For both the normal force and pitch-damping force 

coefficients, slender body theory only provides a qualitative prediction of the force distributions 

though the error is likely greater than that desired for many preliminary design applications. 

1.2 Relationships Between Coefficients 

One of the implications of the application of slender body theory is that many of the 

aerodynamic coefficients appear to be related to one another. In particular, equation 1 provides a 

direct algebraic relationship between α CN and Dˆ . Because of this relationship, all of the 

coefficients previously shown can be related to Nα C in some manner. 

If this view is applied to the pitching moment coefficient, the following is obtained: 

∫ = −

x

0

N

m cg dx

dx

dC

C (x x) α

α . (8) 
5

0

1

2

3

4

0 4 8 12 16 20

X/D

Normal Force Coefficient 

M=1.8

M=2.5

M=3.5

M=4.5

Slender Body Theory

Figure 2. Comparison of predicted normal force coefficient slope Nα

C distribution with slender body 

theory for the ANSR. 

-12

-8

-4

0

4

8

12

16

0123456789

X/D

Pitch-Damping Force Coefficients 

Slender Body Theory

PNS

Nα& C

Nq

C

Figure 3. Comparison of PNS and slender body theory predictions of pitch-damping force coefficients 

over body, ANSR, Mach 2.5, Xcg/D = 5.038. 
6

This relationship is just the definition of the moment in terms of the distribution of forces written 

in coefficient form. This suggests that the slender body equations can be used to derive 

relationships between variables even though the slender body equations in terms of the body 

geometry are inaccurate. (It should be noted that this form of the pitching moment includes only 

the contribution from the normal force and ignores the small contribution from the asymmetrical 

longitudinal forces acting on the nose. This contribution can be shown to produce a small 

restoring moment. In the results presented here, the computed values of mα C from the more 

elaborate CFD computation that includes this additional contribution have been used, rather than 

the value of mα C from equation 8.) 

Similarly, the pitch-damping force coefficient Nα& C becomes 

∫ =

x

0

CN C N dx α& α . (9) 

In this form, the pitch-damping force coefficient Nα& C is related to the area under the normal 

force coefficient curve rather than twice the volume of the body according to slender body 

theory. This will be shown to be inaccurate, but forms a starting point for an approximation. 

The pitch-damping force coefficient Nq C can be obtained by differentiating the integrand of 

equation 4 by parts, i.e., 

∫ ∫ = − −

x

0

2

cg

x

0

2

Nq

dx

dx

Dˆ d 2 D dx (x x) ˆ C . (10) 2

Both terms of equation 10 can be replaced by the corresponding slender body coefficients, 

Nq Nα mα C = C − C & . (11) 

Equation 11 was derived by Sacks (16) using the Blasius method of calculating the forces and 

moments on a slender body from the cross-flow potential. This result might have been more 

useful if he had not used the integrated form of the equation for Nq C (equation 4) which gives 

rather poor results. Equation 11 is also somewhat remarkable because it shows that the two 

dynamic derivatives Nq

C and Nα& C are related to each other and the difference between the two 

coefficients is the pitching moment slope mα C which is a static aerodynamic coefficient. It is 

also important to note that the derivation of equation 11 presented here applies only to 

axisymmetric bodies, though it is possible to show that this relationship holds for bodies with 

more general cross sections (14, 16).
7

The pitch-damping moment coefficient becomes 

= ∫ − = ∫ − x

0

cg N

x

0

N

m cg dx (x x)C dx

dx

dC

C (x x) α

α

α

&

& . (12) 

This form is in direct analogy with the pitching moment and normal force relationship. 

The integrand of the damping moment due to transverse angular rate in equation 6 can also be 

expanded into two parts, which can be identified as follows: 

mq mα 2mα C = C − C & . (13) 

The static pitching second moment slope 2mα C represents an additional term defined as 

∫ = −

x

0

2 N

2m cg dx

dx

dC

C (x x) α

α . (14) 

The relationship shown in equation 13 is analogous to the form for the force coefficients shown 

in equation 11. The damping moment mq C can also be expressed as a function of Nq

C in a 

manner analogous to the damping moment mα& C : 

∫ = −

x

0

Nq

mq cg dx

dx

dC

C . (15) (x x)

Finally, the sum of the pitch-damping moment coefficients becomes 

2

m N cg mq

C + C = −C (x − x) α& α . (16) 

Note that all the coefficient relationships (equations 8, 9, and 11–13) are consistent with the 

translation of the center of gravity relations, as are the slender body theory equations. 

In this form, all the aerodynamic coefficients of interest are related in some manner to the normal 

force distribution rather than the local cross-sectional area of the body. Part of the motivation for 

casting the aerodynamic coefficients in this form is that it may be possible to overcome some of 

the shortcomings of slender body theory, such as the lack of history effects, because some of the 

history effects are already contained in the normal force distribution. Second, the normal force 

distribution is readily obtainable from current fast design approaches such as AP98 (7) and 

DATCOM (8). Thus, a method for predicting the pitch-damping coefficients based on the 

normal force distribution can be easily used in conjunction with existing fast design methods 

eliminating the need to correlate the normal force distribution itself.
8

Figure 4 shows a comparison of the pitch-damping force coefficient Nα& C computed using 

equation 9 with the CFD predictions of Nα& C . In applying equation 9, the predicted normal 

force distribution from CFD was used. The application of equation 9 appears to overestimate 

Nα& C by about 60% compared with the CFD predictions. However, the distribution of the 

force along the body is similar for both results. 

0

5

10

15

20

25

0123456789

X/D

Pitch-Damping Force Coefficient 

∫

x

0

CN dx

α

Nα& C

Figure 4. Comparison of pitch-damping force coefficient Nα& C with integral of normal force 

coefficient over body, Mach 2.5, ANSR, CFD predictions. 

Figure 5 shows a comparison of the pitch-damping force coefficient Nq C computed using 

Sacks’s relation (equation 11) with the CFD predictions of Nq

C . The pitch-damping coefficient 

distribution required in equation 11 was obtained from CFD. Over the nose region, equation 11 

yields an accurate prediction of Nq C , though the accuracy is degraded somewhat over the rear of 

the body. Equation 11 does provide enhanced accuracy compared with the direction application 

of slender body theory shown in Figure 3. (Weinacht and Danberg [17] provide a more complete 

discussion of the relative accuracy of Sacks’s relation.) 

The results shown in Figures 4 and 5 demonstrate that casting the original slender body theory 

results in terms of the normal force coefficient does allow some improvement in the predictive 

capability. However, further effort is required to establish a predictive capability using these 

relations with sufficient accuracy for design applications.
9

-10

-5

0

5

10

0123456789

X/D

Pitch-Damping Force Coefficient 

Nα mα

C −C &

Nq

C

Figure 5. Comparison between NqC and Nα mα C − C & (equation 11), Mach 2.5, ANSR, Xcg/D = 5.038, 

CFD predictions. 

2. Formulation of an Approximate Model

The critical element in the evaluation of the pitch-damping coefficients is the approximation of 

Nα& C . As can be seen from equations 11, 12, and 15, once Nα& C is determined, the damping 

coefficients Nq C , mα& C , and mq C can be determined using these equations. The relationship 

given by equation 9 overestimates Nα& C by about 60% at 9 calibers as is shown in Figure 4. 

Both curves, however, are similar in general form. A function, G(x), is introduced into equation 

9 multiplying Nα C to correct for this discrepancy. 

∫ =

x

0

CN G(x)CN dx α& α , (17) 

or alternatively, the function G(x) is defined: 

dx

dC

C

1 G(x) N

N

α

α

& = . (18)
10

By applying equation 12, the pitch-damping moment coefficient mα& C can be expressed in terms 

of the function G(x). 

∫ ∫ = − = −

x

0

cg N

x

0

N

m cg dx (x x)G(x)C dx

dx

dC

C (x x) α

α

α

&

& . (19) 

The function G(x) can be evaluated from CFD derived values of Nα C and Nα& C using equation 18. 

Figure 6 shows G(x) for the ANSR in the Mach number range from 1.8 to 4.5. Two distinct regions 

are seen in Figure 6 corresponding to the ogival nose (0 < x < 2) and cylindrical afterbody (x > 2). 

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

X/D

G(x)

M=1.8

M=2.5

M=3.5

M=4.5

Figure 6. G(x) over the ANSR for various Mach numbers. 

On the nose, G(x) is relatively small except near the nose tip. The large values of G(x) very 

close to the nose tip are believed to be caused by numerical start-up transients associated with the 

conical starting procedure used in the PNS approach. These transients have little effect on the 

computed coefficients because both Nα C and Nα& C go to zero very rapidly on the nose as the tip 

is approached. Because Nα& C is so small on the nose, the characterization of G(x) for the nose 

region is somewhat less important than for the cylindrical afterbody. 

2.1 G(x) on Cylinder (Carry-Over Transformation) 

Compared to the nose region, the more significant variation in G(x) occurs in the carry-over 

region after the nose as is shown in Figure 6. Rather than attempt to develop an analytic function 

to describe G(x) in terms of Mach number and x, the possibility of defining a coordinate 
11

transformation to relate at least a part of G(x,M) to a single function, e.g., G25 = G(X25,M = 2.5), 

is considered. 

Figure 6 suggests that, for x > 2 but before G(x) reaches a maximum, G(x) is nearly linear with 

x. In this region, G(x) can be approximated as a linear function for each Mach number, as shown 

in Figure 7. Furthermore, each curve appears to intersect at a common point, G(0) = G(x = 0). 

This suggests G(x) can be represented in the form shown in equation 20: 

 ) G(x,M) = T(M)[G(X25 ) − G(0)] + G(0 , (20) 

where, in the linear region, 

x

T(M)

S(M) X25 = ; (21) 

25 linear

25

linear

d X

dG

dx

dG

S(M)





















 





= . (22) 

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

X/D

G(x)

M=1.8

M=2.5

M=3.5

M=4.5

G(0)=-0.331

Figure 7. Correlation of G(x) on the cylindrical afterbody, ANSR. 
12

On the cylindrical body, G25 is obtained from a table look-up and interpolation based on the 

transformation of x given by equation 21. An abbreviation of the reference data is given in Table 

1. Local values of G are obtained from equation 20 where G(0) = !0.331. 

Table 1. G25 as a function of X25. 

X25 G25(X25) 

0. −0.331 

1. −0.127 

2. 0.076 

3. 0.281 

4. 0.486 

5. 0.674 

6. 0.796 

7. 0.842 

8. 0.835 

9. 0.799 

10. 0.763 

11. 0.730 

12. 0.703 

13. 0.680 

14. 0.658 

15. 0.636 

16. 0.615 

17. 0.595 

18. 0.576 

19. 0.558 

20. 0.543 

The similarity factor T accounts for the decrease in the maximum G(x) with Mach number. The 

x coordinate transformation factor S is equal to the ratio of the slopes of the linear regions. 

Based on the PNS data for the ANSR, the curve fits were obtained for the S(M) and T(M) and 

are shown in equations 23 and 24. The variation of the factors S(M) and T(M) with Mach 

number is also shown in Figure 8. 

2 M

1 0.0718

M

1 S(M) = 0.0688 + 2.302 + . (23) 

2 M

1 0.510

M

1 T(M) = 0.615 +1.176 − . (24) 

Figure 9 shows the resulting transformation of the G(x) distributions of Figure 6. The data in the 

carry-over region are very well represented by a single curve, although there is some divergence 

in the different Mach number curves beyond the peak in the data. Thus, some error in the final 

results for large L/Dref bodies must be accepted. 
13

0

0.5

1

1.5

2

2.5

12345

Mach Number

S(M),T(M)

S(M)

T(M)

Figure 8. Similarity factors S(M) and T(M). 

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

X25

G25(X25)

Data

G25

Figure 9. Correlation of G25(X25) with computed data, ANSR. 
14

2.2 Effect of Nose Length 

The previous analysis has considered only a fixed nose length of 2 calibers. When nose length is 

varied, changes in the distribution of G(x) become apparent. Figure 10 shows the distribution of 

G(x) for a tangent-ogive/cylinder configuration with nose lengths of 2, 4, 6, and 8 calibers at 

Mach 2.5. While the shape of the curves behind the nose/cylinder junction remain relatively the 

same, the curve shift upwards and to the right with increasing nose length. The shift of the curve 

to the right with increasing nose length might be expected because of the shift in the location of 

the nose/cylinder junction. The upward shift in G(x) is perhaps less intuitive. To account for the 

shift in the curve with increasing nose length, the following correction was added to the original 

correlation previously shown. 

2 G(0) D

L G(x,M) T(M) G (X ) G(0) 0.0183 nose 25 25 +



 



 



 



 = − + − ; (25) 





 



 



 



 = − − 2

D

L

x 0.5833

T(M)

S(M) X nose 25 . (26) 

When the L/D ratio of the nose Lnose/D approaches 2, the correlation reverts to the original form. 

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

X/D

G(x)

8-Caliber Nose

6-Caliber Nose

4-Caliber Nose

2-Caliber Nose

Figure 10. Effect of nose length on G(x), tangent-ogive/cylinder, Mach 2.5. 
15

2.3 Correlation for the Nose Region 

In the previous discussion, it was pointed out that the nose region has a relatively small effect on 

the overall prediction of Nα& C . However, to complete the correlation, a suitable evaluation of 

G(x) for the nose region must be performed. In the course of developing and testing the 

correlation, it was found that the slope of the nose had a significant effect on the G(x) on the 

nose, but a much less apparent effect on the carry-over region on the cylinder. To investigate 

this further, a parametric series of CFD computations for various conical bodies with cone angles 

from 2° to 28° was performed. In each case, the cone length was sufficient to establish many 

calibers of constant or near constant G(x) following the tip region transient. Figure 11 shows the 

results as a function of both the cone angle, θ C , and Mach number. The initial trend of G(nose) 

decreasing with Mach number at small values of cone angle is reversed at larger cone angles 

beginning after 8°. Using a curve-fitting procedure, a closed-form expression for G(nose) for the 

conical noses has been obtained. This allows a straight-forward evaluation of G(nose) for the 

conical noses in terms of the Mach number and the cone half-angle θ C in degrees. The values of 

the coefficients mi and bi are listed in Table 2. 

i

i 5

i 0

i

i 5

i 0

i

i G(nose) M∑m θ ∑b θ

=

=

=

=

= + . (27) 

For conical bodies, the correlation function G(nose) yields a very accurate evaluation of Nα& C . 

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

Cone Angle - Deg

G

Mach 2

Mach 3

Mach 4

Mach 5

Figure 11. Variation of G(nose) with Mach number and cone angle for conical noses. 
16

Table 2. Coefficients of the fifth-order polynomial fit of 

equation 27. 

i bi mi

0 1.06724 −0.0137712 

1 9.73099 H 10−2 −6.22894 H 10−2

2 −2.92027 H 10−2

 9.7226 H 10−3

3 2.13167 H 10−3 −5.82041 H 10−4

4 −6.54121 H 10−5

 1.59364 H 10−5

5 7.34183 H 10−7 −1.64649 H 10−7

This is due, in part, to the fact that G(x) for the conical nose is independent of the location along 

the nose and is a simple function of Mach number and cone angle. For other nose types, such as 

ogival noses, G(x) appears to vary along the nose (Figures 6 and 10). However, it was found that 

a suitable approximation for G(x) on an ogive nose could be obtained by using the G(nose) for a 

conical nose whose nose tip angle was equal to the slope at the tip of the ogive. For a tangent 

ogive nose, it can be shown that the local slope at the tip of the nose is equal to twice the cone 

angle for a conical nose of equivalent length. Thus, typically slopes for ogive noses are about 

20° and G(nose) is relatively small. 

2.4 Pitch Damping Force, NqC (Sacks’s Relationship) 

The pitch damping force due to the transverse angular rate NqC can be evaluated in terms of 

Nα& C and the static moment coefficient, mα C , from the relation derived by Sacks (i.e., equation 

11). The PNS data provides the first accurate test of equation 11 using higher level models. The 

following residual, H, is computed for the L/D = 20 ANSR case: 

N m Nq

H(x) = C − C − C α& α , (28) 

and this is shown in Figure 12. For the first 4–5 calibers, the residual is negligible, but for long 

bodies, the error in equation 11 becomes more important. The difference, H, can be 

approximated easily as straight lines of slope, 0.35

dx

dH = . The intercept on the axis, xH, of these 

lines is found to be proportional to the same x transformation factor as in the function G(x), i.e., 

xH = 4.35S(M)/T(M), where 4.35 is the intercept for the Mach number 2.5 case. The effect of 

nose length was investigated for H(x). Although some effect of nose length was apparent, the 

effect was less than the variability in the approximation previously described. Thus, no 

correction for nose length was applied. 

The pitch-damping force and moment coefficients Nq

C and mq C can be computed by 

appropriately adding the correction H(x) to equations 11 and 13. 

 ) C CN Cm H(x Nq = − − α& α . (29)
17

0

2

4

6

8

0 5 10 15 20

X/D

H(x)

M=1.8

M=2.5

M=3.5

M=4.5

Figure 12. Distribution of H(x) along body for ANSR. 

dx

dx

dH C C C (x x)

x

xH

mq = m − 2m − ∫ cg − α& α . (30) 

It should be noted that the correction term H(x) is added to improve the correlation, although it is 

relatively small. Direct application of equations 11 and 13 provides reasonable estimates of the 

coefficients Nq

C and mq C . (Weinacht and Danberg [17] further discuss direct application of 

these relations.) 

Figures 13 and 14 show the results of applying the approximate method to the L/D = 9 ANSR 

configuration at Mach 2.5. The comparisons with the PNS results are very good with errors less 

than 3%. The approximate method is effective in substantially reducing the errors that result 

from applying slender body theory. The results are typical of those that are obtained using the 

approximate method for other configurations, although somewhat larger errors may be 

encountered. 

2.5 Pitch-Damping Moment Coefficient Sum 

The pitch-damping moment coefficient sum can be easily obtained by adding equations 19 and 

30. However, the sum of these two equations can be recast into a form, shown in equation 31, 

that better illustrates the effect of the correlation terms.
18

-12

-8

-4

0

4

8

12

16

0123456789

X/D

Pitch-Damping Force Coefficients 

Approximate Method

PNS

Nα& C

Nq

C

Figure 13. Comparison of PNS and approximate method predictions of pitch-damping 

force coefficients over body, ANSR, Mach 2.5, Xcg/D = 5.038. 

-50

-40

-30

-20

-10

0

10

0123456789

X/D

Pitch-Damping Moment Coefficients 

Approximate Method

PNS

mq

C

mα& C

Figure 14. Comparison of PNS and approximate method predictions of pitch-damping 

moment coefficients over body, ANSR, Mach 2.5, Xcg/D = 5.038. 
19

dx

dx

d H C C (x x) C 2(1 G)(x x)C dx (x x) cg

x

xH

N

x

0

N cg

2

mq

+ m = − cg − − − − − − α& α ∫ α ∫ . (31) 

The first term on the right-hand side is the slender body theory result using the normal force 

coefficient slope (equation 16). Note that in either form of the slender body theory (equations 7 

or 16), the pitch-damping moment coefficient goes to zero at the center of gravity. The second 

term of equation 31 takes into account upstream effects and it is this term that prevents the 

moment sum from going to zero at the center of gravity. This is illustrated with Figure 15, which 

presented results for the 9-caliber ANSR. The large difference between the slender body theory 

and the PNS results are most significant around the center of gravity. Note also that the 

integrand of the second term of equation 31 changes sign at the center of gravity. By 

appropriately locating the center of gravity, this integral can be made to cancel out for a given 

body producing apparently good (though fortuitous) predictions of the pitch-damping coefficient 

sum using slender body theory for some center of gravity locations. The last integral term 

accounts for the discrepancy in Sacks’s relationships (equations 11 and 13), and its contribution 

is typically smaller than the second term. Although equation 31 is more complicated than 

slender body theory, it does account for the effect of the force carry-over from the nose onto the 

cylindrical afterbody that is necessary to produce an accurate evaluation of the pitch-damping 

moment coefficient sum. 

2.6 Reynolds Number Effects 

The effect of the Reynolds number on the correlation parameters was also considered during the 

course of the study because of its possible effect on the carry-over lift. The ANSR configuration 

was used as the reference configuration for the study. For the results presented here, all the 

Navier-Stokes results are computed at standard sea level conditions. As a consequence, the 

actual Reynolds number, for each geometry, varies linearly with Mach number. In the 

investigation of Reynolds number effects, the Reynolds number was changed by constant factors 

(Rer = 0.5, 2, 5, 10) holding the flight velocity and Mach number unchanged. This can be 

interpreted as a variation in the reference diameter from 10 to 200 mm for the ANSR, which has 

a nominal diameter of 20 mm. The magnitude of the Reynolds number = 454,000MRer. The 

results show some variation near the maximum in the carry-over region of the G curve but the 

magnitude of the effect on the pitch-damping moment coefficient was only between 2% and 4% 

and is not considered further.
20

-80

-60

-40

-20

0

0123456789

X/D

Pitch-Damping Moment Coefficient Sum

Modified SBT,Eq. 16

SBT,Eq. 7

PNS

ApproximateMethod

Figure 15. Comparison of slender body theory, PNS, and approximate method predictions 

of pitch-damping moment coefficient sum over body, ANSR, Mach 

2.5, Xcg/D = 5.038. 

3. Application of the Approximate Model

The accuracy of the approximate model was assessed by comparison with both experimental data 

and computational predictions. An important element in the experimental comparisons is the 

accuracy of the approximate method relative to the typical experimental error from range 

experiments. Due to the limited experimental data available, the computational predictions were 

utilized to perform a more complete evaluation of the model for a variety of geometries and 

flight conditions not available in the current experimental database. Errors were quantified in 

terms of the individual error ∆ for each one-to-one comparison and the overall root mean square 

(rms) deviation, σ, for complete datasets containing N individual comparisons. 

m Data mq

mq m Correlation mq m Data

(C C )

(C C ) (C C )

α

α α

&

& &

+

+ − +

∆ = . (29) 

= ∑∆

N

1

2

N

1

σ . (30)
21

The experimental database for the ANSR rocket is probably the highest quality experimental 

database for evaluating the pitch-damping predictions for axisymmetric flight bodies. The 

database includes three different body lengths; 5-, 7-, and 9-caliber bodies; with three center of 

gravity locations for each body length at Mach 1.3, 1.8, and 2.5. Figure 16 shows a comparison 

of the predicted pitch-damping moment coefficient sum using the approximate model with the 

experimental database. In general, no consistent bias in the predicted data is observed. The rms 

derivation, σ, is 18% for the complete dataset; however, this error includes the variability in the 

experimental data. The experimental variability is evident because multiple firings were 

performed for the same geometry and flight conditions and multivalued data were obtained for 

the same flight conditions. Figure 17 shows the comparisons between the approximate model 

with CFD predictions. The approximate model appears to show a small consistent 

overprediction of the pitch-damping moment coefficient sum for this dataset. Here, the rms 

derivation, σ, is 11% and is perhaps better representative of the overall accuracy of the model. 

This overprediction is not evident in the comparisons with experimental data due to the 

experimental uncertainty. 

-80

-60

-40

-20

0

0 -20 -40 -60 -80

Experiment

Approximate Method

Exact Correlation

Mach 1.3

Mach 1.8

Mach 2.5

mq mα& C + C

Figure 16. Correlation between approximate method predictions and experimental 

data for mq mα& C + C , ANSR, L/D = 5, 7, and 9, and Mach 1.3, 1.8, 

and 2.5. 
22

-80

-60

-40

-20

0

0 -20 -40 -60 -80

CFD

Approximate Method

Exact Correlation

Mach 1.3

Mach 1.8

Mach 2.5

mq mα& C + C

Figure 17. Correlation between approximate method predictions and CFD 

computations for mq mα& C + C , ANSR, L/D = 5, 7, and 9, and 

Mach 1.3, 1.8, and 2.5. 

The accuracy of the approximate model was also evaluated by comparing the results of the 

model with CFD predictions of the pitch-damping moment coefficient sum for a series of tangent 

ogive/cylinder and cone/cylinder body geometries. In this study, a variety of nose lengths were 

considered: 2, 2.5, 3, 4, 5, 6, and 8 calibers and body lengths from 5 to 20 calibers. All bodies 

had at least 1 caliber of cylindrical afterbody. Thus, not all nose lengths were considered for 

bodies less than 9 calibers in length. Mach numbers of 2, 3, 4, and 5 were also considered. For 

each body length, a fixed center of gravity location for each body length was used as shown in 

Table 3. Realistically, the center of gravity location should be expected to move rearward for 

larger nose lengths for monolithic bodies, although fixing the nose length at a specified location 

does allow the aerodynamic effects to be isolated from the effects produced by center of gravity 

shift. 

Figures 18 and 19 compare the results of the application of the approximate method with the 

CFD predictions for the tangent ogive cylinder and cone cylinder bodies for the pitch-damping 

moment sum, mq

C + mα& C . In general, the results of applying the approximate method compare 

well with the CFD predictions with relatively little bias in the results across the range of 

parameters tested. 

For the tangent ogive cylinder body, the maximum error between the approximate method and 

the CFD predictions was 25% with an overall rms deviation, σ, of 9%. For the cone cylinder 
23

Table 3. Center of gravity locations 

as a function of body length. 

L/D Xcg/D 

 5. 3.0 

 7. 4.037 

 9. 5.038 

12. 6.565 

15. 8.063 

20. 10.565 

-80

-60

-40

-20

0

0 -20 -40 -60 -80

CFD

Approximate Method

Exact Correlation

L/D=5 Tangent Ogive

L/D=7 Tangent Ogive

L/D=9 Tangent Ogive

L/D=5 Cone Cylinder

L/D=7 Cone Cylinder

L/D=9 Cone Cylinder

mq mα& C + C

Figure 18. Correlation between approximate method predictions and CFD 

computations for mq mα& C + C , cone cylinder and tangent ogive 

cylinder bodies, L/D = 5, 7, and 9. 

body, the maximum error between the approximate method and the CFD predictions was 21% 

with an overall rms deviation, σ, of 7%. In general, the largest errors were encountered for the 

smaller L/D bodies. 

The comparisons were also made for the individual pitch-damping coefficients mα& C and 

mq C and are shown in Figures 20–23. The comparisons for the pitch-damping coefficient mα& C

show a greater variability than for mq C . As discussed previously, Nα& C and mα& C are computed 

from the normal force distribution and the correlation function G(x) using equations 17 and 19. 

Nq

C and mq C are then computed from equations 29 and 30 using the computed values of 
24

-400

-300

-200

-100

0

0 -100 -200 -300 -400

CFD

Approximate Method

Exact Correlation

L/D=12 Tangent Ogive

L/D=15 Tangent Ogive

L/D=20 Tangent Ogive

L/D=12 Cone Cylinder

L/D=15 Cone Cylinder

L/D=20 Cone Cylinder

mq mα& C +C

Figure 19. Correlation between approximate method predictions and CFD 

computations for mq mα& C + C , cone cylinder and tangent ogive 

cylinder bodies, L/D = 12, 15, and 20. 

-20

-16

-12

-8

-4

0

0 -4 -8 -12 -16 -20

CFD

Approximate Method

Exact Correlation

L/D=5 Tangent Ogive

L/D=7 Tangent Ogive

L/D=9 Tangent Ogive

L/D=5 Cone Cylinder

L/D=7 Cone Cylinder

L/D=9 Cone Cylinder

mα& C

Figure 20. Correlation between approximate method predictions and CFD 

computations for mα& C , cone cylinder and tangent ogive cylinder 

bodies, L/D = 5, 7, and 9. 
25

-100

-80

-60

-40

-20

0

0 -20 -40 -60 -80 -100

CFD

Approximate Method

Exact Correlation

L/D=12 Tangent Ogive

L/D=15 Tangent Ogive

L/D=20 Tangent Ogive

L/D=12 Cone Cylinder

L/D=15 Cone Cylinder

L/D=20 Cone Cylinder

mα& C

Figure 21. Correlation between approximate method predictions and CFD 

computations for mα& C , cone cylinder and tangent ogive cylinder 

bodies, L/D = 12, 15, and 20. 

-60

-40

-20

0

0 -20 -40 -60

CFD

Approximate Method

Exact Correlation

L/D=5 Tangent Ogive

L/D=7 Tangent Ogive

L/D=9 Tangent Ogive

L/D=5 Cone Cylinder

L/D=7 Cone Cylinder

L/D=9 Cone Cylinder

mq

C

Figure 22. Correlation between approximate method predictions and CFD 

computations for mq

C , cone cylinder and tangent ogive cylinder 

bodies, L/D = 5, 7, and 9. 
26

-400

-300

-200

-100

0

0 -100 -200 -300 -400

CFD

Approximate Method

Exact Correlation

L/D=12 Tangent Ogive

L/D=15 Tangent Ogive

L/D=20 Tangent Ogive

L/D=12 Cone Cylinder

L/D=15 Cone Cylinder

L/D=20 Cone Cylinder

mq

C

Figure 23. Correlation between approximate method predictions and CFD 

computations for mq

C , cone cylinder and tangent ogive cylinder bodies, 

L/D = 12, 15, and 20. 

Nα& C and mα& C . The variability shown in Figures 20 and 21 are a measure of the effectiveness of 

the correlation function G(x). On the other hand, the reduced variability in Figures 22 and 23 

indicates that relatively less error is obtained by applying equations 29 and 30. Most of the 

variability shown in Figures 22 and 23 results from the prediction of mα& C which is one of the 

inputs used in equation 30. The additional terms in equations 29 and 30 appear to increase the 

magnitude of the coefficients without adding much additional error. An additional implication of 

these comparisons is that the variability of mq mα& C + C should be greater than mq

C but less than 

mα& C since the sum is obtained by adding mq

C and mα& C . 

The results presented in Figures 18 and 19 give a global assessment of the effectiveness of the 

approximate method. It is somewhat difficult from this data presentation to determine how well 

the model can be expected to perform when individual design variables are changed. Figures 

24–27 show comparisons of the approximate method with CFD results for tangent ogive cylinder 

and cone cylinder bodies with a nose length of 2 calibers for Mach numbers between 2 and 5 and 

for body lengths of 5, 7, 9, 12, 15, and 20 calibers. The dominant effect shown in this set of 

figures is the increase in mq mα& C + C with increasing body length. A weaker variation in 

mq mα& C + C is observed with changing Mach number. The variation in the Mach number effect 
27

-80

-70

-60

-50

-40

-30

-20

-10

0

123456

Mach Number

Pitch-Damping Moment Coefficient Sum 

CFD

Approximate Method

L/D=9

L/D=7

L/D=5

Cone Cylinder

Figure 24. Comparison of predicted Mach number variation of mq mα& C + C

obtained from approximate method and CFD computations, cone cylinder 

bodies, Lnose/D = 2, L/D = 5, 7, and 9. 

-80

-70

-60

-50

-40

-30

-20

-10

0

123456

Mach Number

Pitch-Damping Moment Coefficient Sum 

CFD

Approximate Method

L/D=9

L/D=7

L/D=5

Tangent Ogive Cylinder

Figure 25. Comparison of predicted Mach number variation of mq mα& C + C obtained 

from approximate method and CFD computations, tangent ogive cylinder 

bodies, Lnose/D = 2, L/D = 5, 7, and 9. 
28

-400

-300

-200

-100

0

123456

Mach Number

Pitch-Damping Moment Coefficient Sum 

CFD

Approximate Method L/D=20

L/D=15

L/D=12

Cone Cylinder

Figure 26. Comparison of predicted Mach number variation of mq mα& C + C obtained 

from approximate method and CFD computations, cone cylinder bodies, 

Lnose/D = 2, L/D = 12, 15, and 20. 

-400

-300

-200

-100

0

123456

Mach Number

Pitch-Damping Moment Coefficient Sum 

CFD

Approximate Method L/D=20

L/D=15

L/D=12

Tangent Ogive Cylinder

Figure 27. Comparison of predicted Mach number variation of mq mα& C + C obtained 

from approximate method and CFD computations, tangent ogive cylinder 

bodies, Lnose/D = 2, L/D = 12, 15, and 20. 
29

with increasing body length is well predicted by the approximate method compared with the 

CFD results. For the shorter body lengths, mq mα& C + C shows a decreasing trend with Mach 

number. However, as the body length increases this trend is reversed. For the nose and body 

lengths considered here, there is relatively little difference in the results between the two nose 

tips. 

The effect of nose length on the predicted pitch-damping moment coefficient sum is shown in 

Figures 28 and 29 for an L/D = 7 body for both the tangent ogive cylinder and cone cylinder 

bodies. Both the approximate method and the CFD results show a similar decreasing trend of 

mq mα& C + C with increasing nose length. The approximate method also correctly predicts the 

relative differences between the conical and tangent ogive noses. Although some differences 

between the approximate method and the CFD results are apparent for the larger nose lengths, it 

should be noted that the 6-caliber nose is a rather extreme case because the cylindrical afterbody 

is only 1 caliber in length. Not withstanding these differences, the approximate method appears 

to be capable of predicting the variation of mq mα& C + C with many of the design variables of 

interest. 

-40

-35

-30

-25

-20

-15

-10

-5

0

123456

Mach Number

Pitch-Damping Moment Coefficient Sum 

CFD

Approximate Method

Lnose/D=6

Lnose/D=2

Lnose/D=4

Tangent Ogive Cylinder

Figure 28. Comparison of predicted effect of nose length on mq mα& C + C obtained from 

approximate method and CFD computations, tangent ogive cylinder bodies, 

L/D = 7.
30

-40

-35

-30

-25

-20

-15

-10

-5

0

123456

Mach Number

Pitch-Damping Moment Coefficient Sum 

CFD

Approximate Method

Lnose/D=6

Lnose/D=2

Lnose/D=4

Cone Cylinder

Figure 29. Comparison of predicted effect of nose length on mq mα& C + C obtained from 

approximate method and CFD computations, cone cylinder bodies, L/D = 7. 

4. Conclusion

A semi-empirical method based on the slender body theory has been developed to predict the 

pitch-damping coefficients for axisymmetric projectile configurations. The method is based 

heavily on the slender body theory and some of the relationships between the aerodynamic 

coefficients that can be derived from the theory. Some slender body formulas are modified by 

the addition of correction factors that are derived from CFD predictions. 

The CFD results allow a detailed investigation of the distribution of the pitch-damping force and 

moment along the body. The ability to accurately compute the distribution of these forces and 

moments provides information on how the body geometry (nose shape, carry-over region, center 

of gravity location, and overall length) produces the overall coefficients. Such an investigation 

would be very difficult to perform with experimental methods. The distribution also provides 

insight into reasons for the discrepancies observed in the slender body theory, which is an 

entirely local theory. Specifically, the slender body theory neglects the region immediately 

downstream of the nose (the carry-over region), which can be described with an integral that 

depends on upstream events. The validity of Sacks’s relationship Nq Nα mα C = C − C & is 

demonstrated except at large L/D’s where a small correction is needed.
31

A modification to the slender body pitch damping theory is proposed in which the damping force 

coefficient Nα& C is related to the integral of the normal force coefficient slope, Nα C . A new 

function, G(x,M), is introduced to modify Nα C to accurately predict the damping coefficient. 

CFD results are used to determine the factor G on the nose and cylindrical afterbody of an 

axisymmetric projectile in the supersonic velocity regime. Once Nα& C is determined, all the 

other damping force and moments can be directly calculated. One of the contributions achieved 

by basing Nα& C on Nα C and the factor G is to account for the carry-over of the nose normal force 

onto the cylinder. G is significantly affected by Mach number and an approximate 

transformation technique is shown to give acceptable results at supersonic speeds. In calculating 

the pitch damping coefficients, it is assumed that Nα C can be determined from an auxiliary 

method such as CFD or fast-design methods. 

The technique has been applied to several axisymmetric projectile configurations. Comparisons 

have been made to the ANSR which is a family of secant ogive cylinder bodies of varying body 

length. Comparisons of the technique are made with both experimental and CFD data for this 

configuration to assess the accuracy of the method. To further assess the model, an extensive 

computational database was generated using CFD for a family of cone cylinder and tangent 

ogive cylinder bodies. Parametric variations of the body length (5–20 calibers), nose length (2–8 

calibers) and Mach numbers (Mach 2–5) were performed to construct the database. 

The model is simple and could be easily integrated into existing fast-design methods where its 

performance should be acceptable for preliminary design purposes. Extension of the modified 

slender body theory to bodies with fins or flares is currently being pursued.
32

5. References 

1. Weinacht, P. Prediction of Pitch-Damping Aerodynamic Derivatives Using Navier-Stokes 

Computational Techniques. Ph.D. Dissertation, University of Delaware, Newark, DE, 

May 1996. 

2. 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. 

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

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

4. Weinacht, P. Prediction of Pitch-Damping of Projectiles at Low Supersonic and Transonic 

Velocities. American Institute of Aeronautics and Astronautics: Reston, VA, January 1998;

AIAA Paper 97-0421. 

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

American Institute of Aeronautics and Astronautics: Reston, VA, September 1991; AIAA 

Paper 91-3339. 

6. Weinacht, P.; Sturek, W. B. Navier-Stokes Predictions of Pitch-Damping of Finned 

Projectiles Using Steady Coning Motion. American Institute of Aeronautics and 

Astronautics: Reston, VA, August 1990; AIAA Paper 90-3088. 

7. Moore, F. G.; McInville, R. M.; Hymer, T. The 1998 Version of the NSWC Aeroprediction 

Code: Part I—Summary of New Theoretical Methodology; NSWCDD/TR-98/1; Naval 

Surface Warfare Center: Dahlgren, VA, April 1998. 

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

Conventional Missiles Using Component Build-Up Techniques. American Institute of 

Aeronautics and Astronautics: Reston, VA, January 1984; AIAA Paper 84-0388. 

9. 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, VT, August 1969. 

10. Moore, F. G.; McInville, R. M.; Robinson, D. I. A Simplified Method for Predicting 

Aerodynamics of Multi-Fin Weapons; NSWCDD/TR-99/19; Naval Surface Warfare Center: 

Dahlgren, VA, March 1999. 

11. 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.
33

12. Munk, M. The Aerodynamic Forces on Airship Hulls; NACA Report No. 184; National 

Advisory Committee for Aeronautics: Washington, DC, 1924. 

13. Wood, R. M.; Murphy, C. H. Aerodynamic Derivatives for Both Steady and Nonsteady 

Motion of Slender Bodies. Journal of the Aeronautical Sciences 1955, 22 (12), 870–871. 

14. Bryson, A. E., Jr. Stability Derivatives for a Slender Missile With Application to a Wing-BodyVertical-Tail Configuration. Journal of the Aeronautical Sciences 1953, 20 (5), 297–308. 

15. Nielsen, J. N. Missile Aerodynamics; McGraw-Hill: New York, 1960; Chapter 10, 

pp 363–371. 

16. 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. 

17. Weinacht, P.; Danberg, J. E. Evaluation of Sacks’ Pitch-Damping Relationships. American 

Institute of Aeronautics and Astronautics: Reston, VA, August 2003; AIAA Paper 

2003-5467. 
34

List of Abbreviations 

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 Static pitching second moment coefficient slope with respect to angle of attack 

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 sum 

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 sum 

Dˆ Local body diameter normalized by reference diameter 

D Reference diameter
35

F Force 

G(x) Correlation function used in computing Nα& C

H Correlation function used in computing (x) Nq

C

L / D Body length-to-diameter ratio 

Lnose / D Nose length-to-reference diameter ratio 

M Mach number 

M Moment 

N Number of data points used to compute overall root mean square (rms) deviation 

q Transverse angular rate of body 

Rer Ratio of Reynolds number to baseline Reynolds number 

S Reference area, ref 4

D S

2

ref

π

=

V Reference velocity 

x Axial coordinate normalized by reference diameter 

x Axial location of center of gravity cg normalized by reference diameter 

x Dummy variable of integration associated with axial coordinate x 

X Dimensional axial coordinate normalized by reference diameter 

X Dimensional axial location of center of cg gravity normalized by reference diameter 

Greek Symbols

α Angle of attack 

α& Angular rate associated with angle of attack 

∆ Individual error 

ρ Freestream density 

σ Overall root mean square deviation 
36

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