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i

NASA/CR—2012–217475

Missile Aerodynamics for Ascent and Re-entry

G.L. Watts

Jacobs ESSSA Group, Qualis Corporation, Huntsville, Alabama

J.W. McCarter

Jacobs ESSSA Group, DESE Research, Huntsville, Alabama

November 2012

National Aeronautics and

Space Administration

Marshall Space Flight Center • Huntsville, Alabama 35812

Prepared for Marshall Space Flight Center

under Contract NNM12AA41C
ii

Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076–1320

443–757–5802

This report is also available in electronic form at

<https://www2.sti.nasa.gov/login/wt/>

Acknowledgment

The authors wish to acknowledge the following Marshall Space Flight Center employees for their contributions to this Contractor Report at various times during the past 40 years: Werner K. Dahm (deceased), David Bacchus, 

John Hengel, Davy Haynes, John Blevins, Mark Seaford, and David Purinton.
iii

TABLE OF CONTENTS

INTRODUCTION .................................................................................................................. 1

STATIC COEFFICIENT MODEL ......................................................................................... 2

AERODYNAMIC DAMPING MODEL ............................................................................... 5

AERODYNAMIC FORCES AND MOMENTS .................................................................... 7

APPENDIX A—MATHEMATICAL EQUIVALENCY FOR DAMPING 

DERIVATIVES ........................................................................................... 9

APPENDIX B—PITCH DAMPING MOMENT FOR A CROSS-FLOW CYLINDER ...... 12
iv
v

LIST OF FIGURES

 1. M frame, P frame, and static coefficients ................................................................... 2

A.1 Example missile for damping derivative equivalency ................................................ 9

B.1 Cross-flow cylinder ................................................................................................... 12
vi
vii

LIST OF ACRONYMS AND ABBREVIATIONS 

B frame traditional body coordinate frame (body-fixed) with origin at the missile 

center-of-mass, X-axis forward, Y-axis starboard, and Z-axis completing 

the right-handed system

DOF degrees-of-freedom

MAVERIC Marshall Aerospace Vehicle Representation in “C,” a 6-DOF digital flight 

simulation used at MSFC

M frame missile coordinate frame with origin at the MRP, X-axis forward along 

the missile centerline, and X-Z plane oriented to contain the wind-relative 

velocity vector

MRP moment reference point for missile aerodynamics on the missile centerline

P frame MRP coordinate frame (body-fixed) with origin at the MRP and all three 

axes parallel to those of the B frame
viii

NOMENCLATURE

CAm axial force coefficient in the M frame

Clm rolling moment coefficient in the M frame

Clmd roll damping moment coefficient in the M frame

Clpm roll damping derivative in the M frame

Cmm pitching moment coefficient in the M frame

Cmmd pitch damping moment coefficient in the M frame

Cmqm pitch damping derivative in the M frame

CNm normal force coefficient in the M frame

CYm side force coefficient in the M frame

Cyawm yawing moment coefficient in the M frame

Cyawmd yaw damping moment coefficient in the M frame

Cyawrm yaw damping derivative in the M frame

Dref missile aerodynamic reference length (cylinder diameter)



Fb resultant aerodynamic force vector in the B frame



Fm resultant aerodynamic force vector in the M frame



Fp resultant aerodynamic force vector in the P frame



Mb resultant aerodynamic moment vector in the B frame



Mm resultant aerodynamic moment vector in the M frame



Mp resultant aerodynamic moment vector in the P frame

MRP

  b vector from the center of mass to the MRP

pm, qm, rm components of the missile angular rate in the M frame (radians/s)
ix

NOMENCLATURE (Continued)

Q missile dynamic pressure (1/2 rVR

2)

Sref missile aerodynamic reference area (cylinder cross-sectional area)

VR magnitude of the wind-relative velocity vector at center-of-mass



VR wind-relative velocity vector of the missile center-of-mass

atot total angle-of-attack, measured in the X-Z plane of the M frame

fA aerodynamic roll angle (rotation angle betwen the M and P frames)

r density of the atmosphere at the missile’s altitude
x
CONTRACTOR REPORT

MISSILE AERODYNAMICS FOR ASCENT AND RE-ENTRY

INTRODUCTION

This document presents equations for aerodynamic forces and moments to be used in 6-DOF

missile simulations such as MAVERIC. The Missile coordinate frame (M frame) and a frame 

parallel to the M frame were used for aerodynamics to allow convenient simulation of either the 

ascent phase of flight or a tumbling re-entry. The missile configuration chosen as an example is a 

cylinder with fixed fins and a nose cone. The equations include both the static aerodynamic 

coefficients and the aerodynamic damping derivatives. The inclusion of aerodynamic damping is 

essential for simulating a tumbling re-entry in which large angular rates may occur. The missile

aerodynamic reference area and length are the cylinder cross-sectional area (Sref) and diameter 

(Dref), respectively. 

Appendix A presents a mathematical equivalency for damping derivatives to give insight into 

aerodynamic damping. Appendix B provides additional insight into aerodynamic damping by 

presenting an analytical technique for calculating the damping moment of a cylinder at its centerof-mass in cross-flow conditions. 
2

STATIC COEFFICIENT MODEL

The static coefficients in this document are dimensionless and are defined in the M frame. The M 

frame, shown in Figure 1, can greatly simplify the missile aerodynamics, as will be discussed 

later in the document. The M frame naturally fits the situation of large angular excursions 

experienced by a tumbling missile and allows an analyst to more easily visualize the missile 

rotational dynamics. The M-frame origin is at the moment reference point (MRP) on the missile 

centerline with the X-axis pointing forward along the centerline. For purposes of illustrating the 

M frame, the wind-relative velocity vector, V�⃗

R , has been translated from the missile center-ofmass to the MRP. The Y-axis of the M frame is pointed in the direction of V�⃗

R × ı̂ , where îis 

a unit vector along the X-axis of the M frame. In other words, V�⃗

R (after being translated to the 

MRP) remains in the X-Z plane of the M frame regardless of the roll position of the missile. The 

total angle-of-attack, αtot, shown in Figure 1, is measured in the X-Z plane of the M frame and 

has a range of 0o to +180o

. 

Figure 1. M frame, P frame, and static coefficients. 
3

The six static coefficients, shown in Figure 1, are listed below. The final subscript “m” denotes 

the M frame.

 CAm − axial force coefficient

 CYm − side force coefficient

 CNm − normal force coefficient

 Clm − rolling moment coefficient

 Cmm − pitching moment coefficient

 Cyawm − yawing moment coefficient

The axial force coefficient, CAm , and the normal force coefficient, CNm , are positive in a 

direction opposite to that of the X-axis and Z-axis, respectively. The three moment coefficients 

are positive by the right-hand rule.

CAm , CNm , and Cmm are called the pitch plane (or longitudinal) coefficients. CYm , Clm , and 

Cyawm are called the lateral-directional (or lateral) coefficients and can be set equal to zero in the 

M frame for a missile that is axisymmetric about the centerline. The reason that the lateraldirectional coefficients can be set equal to zero for an axisymmetric missile (leaving only the 

axial force, normal force, and pitching moment) is that the X-Z plane of the M frame is a plane 

of symmetry containing the V�⃗

R vector. 

 

Figure 1 also shows a body-fixed P frame, which is called the MRP frame, with origin at the 

MRP and with axes parallel to those of a traditional Body frame (B frame, not shown). The 

origin of the B frame is at the missile center-of-mass with the X-axis forward, the Y-axis 

starboard, and the Z-axis completing the right-handed system.

The aerodynamic roll angle, φA , is the rotation angle between the M and P frames and will be 

defined mathematically in the next paragraph. The P-frame angle-of-attack, αp, and sideslip 

angle, β , in Figure 1, are not used in the formulation of static coefficients in the M frame. 
4

Static coefficients for a missile are, in general, a function of αtot, φA , and Mach number. For a 

missile that is axisymmetric about the centerline, the static coefficients in the M frame are only a 

function of αtot and Mach, because the aerodynamic properties do not change with φA . The 

equations for these three variables are:

αtot = cos−1 �

VRx

VR

� (1)

φA = tan−1 �

VRy

VRz

� (2)

MACH = 

VR

Vsound

 (3)

where VRx , VRy , VRz are components of V�⃗

R in the B frame, translated to the MRP

 Vsound is the speed of sound at the missile’s altitude

 αtot has a range of 0o to +180o and is never negative 

 φA has a range of -180o to +180o

It is imperative to note that singularities exist in the φA equation at αtot = 0o and 180o, and these

must be handled by specifying a value of φA (such as 0o

) for these two values of αtot. 

If aerodynamic uncertainties are to be modeled in a simulation, the dispersions should be applied 

to the static coefficients in the M frame. The dispersed static coefficients will then be used to 

calculate aerodynamic forces and moments on the missile in the M frame as described later in 

this document. 
5

AERODYNAMIC DAMPING MODEL

The missile aerodynamic damping model is based on traditional damping derivatives that are

dimensionless and are used to calculate aerodynamic moments caused by angular rates of the 

missile. See Appendix A for mathematical insight into damping derivatives. Missile damping 

derivatives are typically negative and, therefore, tend to reduce the angular rates.

Damping derivatives should be developed at (or near) the missile center-of-mass rather than at 

the MRP. See Appendix B for an explanation of the effects of longitudinal center-of-mass 

location upon damping. Damping derivatives should also be developed in a coordinate frame 

parallel to the M frame to properly separate the M-frame pitch and yaw damping effects. The

coordinate frame that will be used for calculating missile damping derivatives is parallel to the 

M frame and has its origin on the centerline at the same X-axis location as the center-of-mass. In 

using such a coordinate frame on the centerline, the effects upon damping of small lateral centerof-mass offsets from the centerline are ignored. 

Damping derivatives represent pure couples at the center-of-mass, and these couples may be 

applied at any other point. After being calculated on the centerline at the same X-axis location as 

the center-of-mass, the damping derivatives will be applied at the MRP in the M frame. Damping 

derivatives will be given names as though they were originally established in the M frame. 

Damping derivatives in this document are Clpm, Cmqm, and Cyawrm for roll, pitch, and yaw, 

respectively. These damping derivatives are positive in the M frame by the right-hand rule, and 

their mathematical definitions are:

Clpm = 

∂Clm

∂ �

pm Dref

2VR �

 (4)

Cmqm = 

∂Cmm

∂ �

qm Dref

2VR �

 (5)

Cyawrm = 

∂Cyawm

∂ �

rm Dref

2VR �

 (6)

The dimensionless terms in the denominators of equations (4), (5), and (6) do seem a bit strange, 

but their mathematical basis can be found in a close examination of the damping equations in 

Appendix A. 
6

These damping derivatives use M-frame angular rates (pm, qm, rm), which are obtained by 

transforming the body rates from the B frame to the M frame by a negative rotation ( -φA ) about 

the centerline. For more complicated vehicle configurations such as airplanes, aerodynamicists

sometimes use additional damping moment terms that are not discussed in this document. There 

also might be a change in an aerodynamic force due to an angular rate, such as the change in 

normal force caused by a pitch angular rate, but this type force will not be considered at this 

time.

At αtot = 0o, the pitch damping derivative for the missile is assumed to be equal to the yaw 

damping derivative (also applies for αtot =180o) because the missile shape approximates a 

vehicle that is axisymmetric about the longitudinal axis. The yaw damping derivative for the 

missile at αtot = 90o (cross-flow conditions) is assumed to be much less than the corresponding 

pitch damping derivative because of the difference in the orientations of the pitch and yaw axes 

with respect to the wind-relative velocity vector.

If aerodynamic uncertainties are to be modeled in a simulation, the dispersions should be applied 

to the damping derivatives in the M frame. The dispersed damping derivatives should then be 

converted into the form of dimensionless damping moment coefficients (Clmd , Cmmd , Cyawmd ) 

in the M frame by assuming a linear relationship between coefficients and angular rates: 

Clmd = 

pm Dref

2VR

∙ Clpm (7)

Cmmd = 

qm Dref

2VR

∙ Cmqm (8)

Cyawmd = 

rm Dref

2VR

∙ Cyawrm (9)

The damping moment coefficients in equations (7), (8), and (9) can be interpreted as the changes 

in the rolling, pitching, and yawing moment coefficients caused by damping. Damping moment 

coefficients will be used with the six static coefficients to calculate the aerodynamic forces and 

moments.
7

AERODYNAMIC FORCES AND MOMENTS

Aerodynamic forces and moments will first be calculated in the M frame at the MRP using the 

six static coefficients and three damping moment coefficients described in this document. The

vector equations for the aerodynamic forces and moments in the M frame are:

F⃗

m = �

FXm

FYm

FZm

� = Q Sref �

−CAm

 CYm

 – CNm

� (10)

M��⃗

m = �

MXm

MYm

MZm

� = Q Sref Dref �

Clm + Clmd

Cmm + Cmmd

Cyawm + Cyawmd� (11)

where Q is the missile dynamic pressure. These aerodynamic forces and moments must be 

transformed from the M frame to the P frame by a rotation about the centerline through the 

aerodynamic roll angle, φA . These transformations, in matrix notation, are:

�

FXp

FYp

FZp� = �

1 0 0

0 cos(φA) sin(φA)

0 −sin(φA) cos(φA)

� �

FXm

FYm

FZm

� (12)

�

MXp

MYp

MZp� = �

1 0 0

0 cos(φA) sin(φA)

0 −sin(φA) cos(φA)

� �

MXm

MYm

MZm

� (13)

These P-frame aerodynamic forces and moments must finally be transformed to the B frame at 

the missile center-of-mass, taking into account the location of the moment reference point 

(MRP ���������⃗

b ) relative to the center-of-mass. The vector equations for the aerodynamic forces and 

moments in the B frame are:

F⃗

b = �

FXb

FYb

FZb

� = �

FXp

FYp

FZp� (14)
8

M��⃗

b = �

MXb

MYb

MZb

� = �

MXp

MYp

MZp�

 

 + �

MRPXb

MRPYb

MRPZb

� × �

FXp

FYp

FZp� (15)

Carrying out the cross product in equation (15), the moment equations in the B frame become:

 MXb = MXp + MRPYb FZp − MRPZb FYp (16)

MYb = MYp + MRPZb FXp − MRPXb FZp (17)

MZb = MZp + MRPXb FYp − MRPYb FXp (18)

The force and moment equations in this section should be used to incorporate the aerodynamic 

models in this document into a 6-DOF simulation.
9

APPENDIX A— 

MATHEMATICAL EQUIVALENCY FOR DAMPING DERIVATIVES 

Nomenclature

Cfin normal force coefficient of fin based on fin projected area, Sfin

Cmqm pitch damping derivative

F⃗

fin normal force on fin

M��⃗

d damping moment caused by fin

M��⃗

fin total pitching moment caused by fin

Qd dynamic pressure for fin damping calculation 

Qfin dynamic pressure at fin

qm pitch angular rate (radians/s)

VR magnitude of the wind-relative velocity vector at center-of-mass

This Appendix provides insight into aerodynamic damping by showing that a pitch damping 

derivative, Cmqm, for a missile is equivalent to the negative of a normal force coefficient for a fin 

on the missile. The example used to develop the mathematics for this coefficient equivalency is a

simple missile (not necessarily realistic) with a horizontal fin at the rear to provide all the pitch 

damping (Figure A.1). The distance from the fin to the missile center-of-mass will be set equal to 

the missile reference length, Dref, solely for the purpose of obtaining a final result without 

inserting additional constants. For this derivation the M, P, and B frames (not shown) are 

assumed to be coincident. All three frames have their origins on the centerline at the center-ofmass with their Z-axes pointed downward. The equations are simplified by assuming cross-flow 

conditions. 

 

Figure A.1 Example missile for damping derivative equivalency. 
10

By inspection of Figure A.1, the dynamic pressure at the fin is:

Qfin = 

1

2 ρ �VR

2 + 2VR Dref qm + Dref

2 qm

2 � (A. 1)

The magnitude of the third term in parentheses in equation (A.1) is typically less than 1% of the 

magnitude of the second term, so the third term will be ignored. The fin dynamic pressure now 

becomes:

Qfin = 

1

2 ρ �VR

2 + 2 VR Dref qm � (A. 2)

The fin normal force is simply:

Ffin = Qfin Cfin Sfin (A. 3)

where Sfin is the fin projected area and Cfin is the normal force coefficient of the fin based on 

the fin projected area. The total pitching moment caused by the fin is the negative of the fin 

normal force times the moment arm, Dref, as follows: 

Mfin = − Qfin Cfin Sfin Dref (A. 4)

It will now be assumed that the fin projected area, Sfin, is 1/4 of the missile reference area, Sref. 

(This assumption permits the final result to be obtained without inserting additional constants). 

Plugging this 1/4 fin area relationship into equation (A.4) produces:

Mfin = − 1

4Qfin Cfin Sref Dref (A. 5)

Going back to equation (A.1) for the dynamic pressure at the fin, the first term inside the 

parentheses is for the static moment, so only the second term is needed for damping calculations. 

The dynamic pressure to be used for fin damping is therefore: 

Qd = ρ VR Dref qm (A. 6)

Substituting the right side of equation (A.6) for the Qfin term in equation (A.5) produces the fin 

damping moment as follows: 

Md = − 1

4 ρ VR qm Cfin Sref Dref

2 (A. 7)
11

The fin damping moment will now be expressed in a generic form using the following traditional

damping moment coefficient equation:

Md = 

1

2 ρ VR

2 Sref Dref Cmmd (A. 8)

where Cmmd is the pitch damping moment coefficient, which represents the change in the 

pitching moment coefficient caused by fin damping in this example. Combining equations (A.7) 

and (A.8) to eliminate Md produces the following equation that shows a linear relationship 

between Cmmd and qm for the simple example chosen:

Cmmd = − 

qmDref

2VR

∙ Cfin (A. 9)

Cmmd can also be expressed, in a general sense, as the following partial differential equation:

Cmmd = 

∂Cmm

∂qm

∙ qm (A. 10)

Combining equations (A.9) and (A.10) to eliminate Cmmd, and then solving for the normal force 

coefficient of the fin, Cfin, produces the following relationship: 

Cfin = − ∂Cmm

∂ �

qm Dref

2 VR �

 (A. 11)

The traditional equation that defines the pitch damping derivative, Cmqm, is: 

Cmqm = 

∂Cmm

∂ �

qm Dref

2 VR �

 (A. 12)

The identical expressions on the right side of equations (A.11) and (A.12) illustrate that for 

purposes of missile damping, a pitch damping derivative is equivalent to the negative of a normal 

force coefficient for a horizontal fin. 
12

APPENDIX B— 

PITCH DAMPING MOMENT FOR A CROSS-FLOW CYLINDER

Nomenclature

C cylinder cross-flow drag coefficient (The reference area for C is always the 

 cylinder cross-flow projected area being analyzed, whether that area is an

 incremental area or the total area; C is assumed to not vary with the size of

 the cylinder or the size of any incremental projected area) 

cg distance from center-of-mass to forward end of cylinder, expressed as a fraction 

 of cylinder length (forward end is right side of cylinder in figure B.1) 

D cylinder diameter (total projected area in cross-flow conditions = L D)

dx incremental distance along cylinder length (incremental projected area = dx D) 

F⃗

i incremental aerodynamic force on incremental projected area (F⃗i = Q C dx D)

L cylinder length

M��⃗

d total damping moment about center-of-mass

M��⃗

i incremental moment about center-of-mass

Q dynamic pressure at incremental projected area

qm pitch angular rate (radians/s)

VR magnitude of the wind-relative velocity vector at center-of-mass

X distance from center-of-mass to incremental projected area (positive forward) 

This Appendix provides additional insight into aerodynamic damping by presenting a derivation 

of the pitch damping moment for a cylinder in cross-flow conditions (Figure B.1). The effects of 

the longitudinal location of the center-of-mass are included and are very important. The cylinder 

is assumed to have the aerodynamic properties of an infinite cylinder (no end flow). For this 

derivation, the M, P and B frames (not shown) are assumed to be coincident. All three frames 

have their origins on the centerline at the center-of-mass with their Z-axes pointed downward. 

 

Figure B.1 Cross-flow cylinder. 
13

By inspection of Figure B.1, the dynamic pressure at the incremental projected area is:

Q = 

1

2 ρ(VR

2 − 2VRqmX + qm

2 X2) (B. 1)

Typically, the magnitude of the third term in parentheses in equation (B.1) is approximately 

1% of the magnitude of the second term, so the third term will be ignored. The dynamic 

pressure now becomes:

Q = 

1

2 ρ(VR

2 − 2VRqmX) (B. 2)

The incremental aerodynamic force on the incremental projected area is:

Fi = 

1

2 ρ(VR

2 − 2VRqmX) C D dx (B. 3)

The incremental moment about the center-of-mass caused by the incremental aerodynamic 

force is:

Mi = 

1

2 ρ(VR

2 − 2VRqmX) C X D dx (B. 4)

The first term in parentheses of equation (B.4) is for calculating the incremental static moment 

and is not needed in this derivation. The second term, −2VRqmX , is the term of interest in this 

example because it determines the incremental damping moment, Mdi, as follows:

Mdi = −ρVRqm C X2 D dx (B. 5)

Integrating equation (B.5) from the aft end to the forward end of the cylinder (from left to 

right in Figure B.1) produces the following equation for the total damping moment about the 

center-of-mass: 

Md = − 1

3 ρVRqm C D L3(1 − 3cg + 3cg2) (B. 6)

Equation (B.6) for the total damping moment is the final result for this example, and it shows 

the significant sensitivity of pitch damping to the longitudinal center-of-mass location and to 

the cylinder length.
xi

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Missile Aerodynamics for Ascent and Re-entry

G.L. Watts* and J.W. McCarter**

George C. Marshall Space Flight Center, Huntsville, AL 35812

Jacobs ESSSA Group, Huntsville, AL

National Aeronautics and Space Administration

Washington, DC 20546–0001

Unclassified-Unlimited

Subject Category 02

Availability: NASA CASI (443–757–5802)

Prepared for the Spacecraft & Vehicle Systems Dept., Engineering Directorate

COTR: Kimball Y. Ibrahim

*Qualis Corporation **DESE Research

M–1351

NNM12AA41C

Contractor Report October 2011–September 2012

NASA/CR—2012–217475

missile aerodynamics, aerodynamic damping, damping derivatives, missile coordinate frame, missile 

re-entry, 6 degrees-of-freedom

01–11–2012

U U U UU 28

Aerodynamic force and moment equations are developed for 6-DOF missile simulations of both the 

ascent phase of flight and a tumbling re-entry. The missile coordinate frame (M frame) and a frame parallel to the M frame were used for formulating the aerodynamic equations. The missile configuration chosen 

as an example is a cylinder with fixed fins and a nose cone. The equations include both the static aerodynamic coefficients and the aerodynamic damping derivatives. The inclusion of aerodynamic damping 

is essential for simulating a tumbling re-entry. Appended information provides insight into aerodynamic 

damping.

STI Help Desk at email: help@sti.nasa.gov

STI Help Desk at: 443–757–5802

National Aeronautics and

Space Administration

IS20

George C. Marshall Space Flight Center

Huntsville, Alabama 35812