Jump Due to Aerodynamic Asymmetry of a Projectile with a Varying Spin Rate G. R. Cooper*

Weapons and Materials Research Directorate U.S. Army Research Laboratory

Aberdeen Proving Ground, Maryland The linear theory for spinning projectiles is extended to account for aerodynamic asymmetry of a free flight

projectile with varying roll rate. Assuming roll is caused by differentially canted controls, allows the rolling motion to be governed by two parameters. Aerodynamic jump resulting from the asymmetry is studied as a function of these parameters and is found to be represented in closed form by using Confluent Hypergeometric functions. A simple rational approximation is given and compared, to the closed form solution, over a pertinent range of the governing parameters. Inquiries regarding jump due to asymmetry are addressed and compared to previous results which are limited in their formulation since the closed form solution was apparently unknown..

Nomenclature

pa Integration constant x

2LDD

p I2SDCρa =

va Integration constant m2

SDCρa 0xv =

pb Integration constant x

3LP

p I4SDCρb =

vb Integration constant v

0v a

Dθgb =

iC Projectile aerodynamic coefficients D Projectile characteristic length (diameter) FF Force component caused by asymmetry g Gravitational constant G Scaled gravitational constant 0VDgG =

Y

X

II

Mass moments of inertia

pK Constant defined as ( )

x

2LPxX0

p Im4DCmIC2SDρ

K+

−=

N~M~L

Applied moments about projectile mass center expressed in the no-roll frame

MM Moment component caused by asymmetry m Projectile mass

r~q~p

Angular velocity components vector of projectile in the no-roll frame

s Dummy integration variable S Surface area 4DπS 2=

wvu

Mass center velocity components in the body reference frame

*Research Physicist, ARL, Member of AIAA

American Institute of Aeronautics and Astronautics

1

AIAA Atmospheric Flight Mechanics Conference and Exhibit16 - 19 August 2004, Providence, Rhode Island

AIAA 2004-5057

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

0V Forward velocity of projectile

w~v~u

Mass center velocity components in the no-roll reference frame

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

zyx

Position vector of body center of mass in an inertial reference frame

Ζ Substitution variable defined as pKΖ ∞φ′= α Longitudinal aerodynamic angle of attack β Lateral aerodynamic angle of attack

K

J

ΓΓ

Components of aerodynamic jump I

I

KJ

Γ Substitution variable defined as ∞′′= φφ-1Γ 0 Π Complex aerodynamic jump caused by asymmetry

ψθφ

Euler roll, pitch and yaw angles of the projectile

∞

0

φ′φ′

Euler roll rates pp

00

KDaVDp

=φ′φ′

∞

0

Bφ Euler roll angle of configuration asymmetry ρ Air density

I. Projectile Dynamics Model This paper extends the work of Murphy / Bradley1 and Fansler / Schmidt2 describing jump caused by slight configuration asymmetries. Some of their results are briefly repeated here, for completeness, followed by a presentation of a closed form analytic solution. The discussion then continues with a simple rational approximation which is then compared to the analytic solution.

Flight mechanics of most projectile configurations can be captured using a rigid body 6 degrees of freedom dynamic model3. The degrees of freedom are three position components of the projectile mass center and three Euler orientation angles of the body. Figures 1 and 2 show two helpful schematics so that the degrees of freedom are seen to be related according to the following equations of motion3,

IIr

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

−++−

φφ

φφφ

φφφ

wvu

ccccssscccsscscsc

θθ

ψψθψ

ψψθψψ

⎢⎢⎢

⎣

⎡

−=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

φ

φ

ssssssccsscc

zyx

θ

ψθψθ

θψθ

&

&

&

⎢⎢⎢

⎣

⎡

−=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧φ

φφ

φ

φ

c/cs0c0

cts1

ψθ

θ

θ

&

&

&

; (1)

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

φ

φ

r~q~p

1/c00010t01

rqp

/cst

θ

θ

θ

θ

; (2)

I

Figure 1. Position Coordinate Definition. Kr

BKr

BJr

BIr

yx

w

V

u

v

z−

IJ

r

American Institute of Aeronautics and Astr

2

IIr

ψ

Jr

r

φ

θ

I

IK

Figure 2. Pro

jectile Orientation

onautics

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

wvu

0pqp0r

qr0

Z/mY/mX/m

wvu

&

&

&

; (3)

. (4) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−−

−−

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧−

rqp

I0pqp0r

qr0

NML

Irqp

1

&

&

&

Forces in the body frame that appear in Eq. (3) contain contributions from weight (W), and air loads (A),

. (5) ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

A

A

A

W

W

W

ZYX

ZYX

ZYX

The weight force resolved into projectile body coordinates is given by Eq. (6),

. (6) ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧−=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

θφ

θφ

θ

W

W

W

cccss

mgZYX

The air loads are split in two components, the standard aerodynamic forces and the Magnus forces,

. (7) ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

AM

AM

AM

AS

AS

AS

A

A

A

ZYX

ZYX

ZYX

Equation (8) gives the standard air loads acting at the aerodynamic center of pressure,

( )

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+

+

++

−=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

VwCC

VvCCV

wvCC

qZYX

NAZ0

NAY0

2

22

X2X0

a

AS

AS

AS

; (8)

where

2222a D)πwvρ(u

81q ++= ; (9)

222 wvuV ++= . (10) The Magnus aerodynamic force acts at the Magnus center of pressure,m

. (12) ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

UA

UA

UA

SA

SA

SA

NML

NML

NML

The aerodynamic moments caused by standard and Magnus air loads are computed with a cross product between the distance vector from the mass center to the force application point and the force itself. An unsteady aerodynamic damping moment is also present, which provides a damping source for angular motion,

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧ +

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

2VDCr

2VDCq

2VpDCC

DqNML

MQ

MQ

LPDD

a

UA

UA

UA

. (13)

All aerodynamic coefficients and the center of pressures are a function of the Mach number of the projectile mass center. The dynamic model previously described is nonlinear due to both three dimensional rotational kinematics expressions and the presence of complex aerodynamic forces. The applicability of the equations of motion shown prior, have been validated over the past 60 years at aeroballistic ranges throughout the world.5

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II. Reduction to Linear Theory Useful performance data regarding trajectory prediction and the stability of projectiles forced early ballisticians

to investigated mathematical simplifications to the equations of motion. Over time a set of simplified and solvable, yet accurate linear differential equations emerged which, today is commonly termed “projectile linear theory.”

The governing equations previously developed are expressed in the body reference frame. In linear theory, the lateral, translational and rotational velocity components are transformed to a nonrolling reference frame. The nonrolling frame or so-called fixed plane frame proceeds with only precession and nutation rotations from an inertial reference frame. Components of linear and angular body velocities in the fixed plane frame can be computed from the body frame components of the same vector through a single axis rotational transformation. For example, the body frame components of the projectile mass center velocity are transformed to the fixed plane frame by

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

φφ−φφ=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

wvu

cossin0sincos0

001

w~v~u~

. (14)

Note that the M~

superscript indicates the vector components relative to the fixed plane reference frame. Projectile linear theory, makes a change of variables from station line velocity component, , to total velocity, , as described in the next two equations.

u V

222222 w~v~uwvuV ++=++= ; (15)

V

dtw~dw~

dtv~dv~

dtduu

Vdtdww

dtdvv

dtduu

dtdV ++

=++

= ; (16)

A further change of variable from time, , to dimensionless arc length, s , is also preferred and following Murphy

t4, gives the dimensionless arc length.

dtVD1s

t

0∫= . (17)

Equations (18) and (19) relate time and arc length derivatives of a given quantityζ . Dotted terms refer to time derivatives, and primed terms denote dimensionless arc length derivatives,

ζVDζ ′⎟⎠⎞

⎜⎝⎛=& ; (18)

⎥⎦⎤

⎢⎣⎡ ′

′+′′⎟

⎠⎞

⎜⎝⎛= ζ

VVζ

VDζ

2&& . (19)

Linear theory makes several assumptions regarding the relative size of different quantities to further simplify the analysis. Euler angles are small so θθ ≈)sin( , 1)cos( ≈θ , ψ≈ψ)sin( , and and the aerodynamic angle of attack is small so that

1)cos( ≈ψVw~α = and Vv~β = . The projectile is mass balanced such that

and0III YZXZXY === YZZYYYYZZ IIIII ≡=⇒= . Quantities and φ are large compared to and , such that products of small quantities and their derivatives are negligible. Application of these

assumptions results in,

Vvr,q,ψ,θ, w

Dx =′ ; (20)

ψDv~VDy +=′ ; (21)

θDw~VDz −=′ ; (22)

pVD

=φ′ ; (23)

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4

q~VDθ =′ ; (24)

r~VDψ =′ ; (25)

VθDgV

2mρSDC

V XO −−=′ ; (26)

p4I

CDSρVI2CDSρp

X

LP3

X

LDD2

+=′ ; (27)

( )( )( )( ) ⎪

⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

φ+φφ+φ−

+φ+φφ+φ

+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−

−−

−−

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

′′′′

DcosVMMDsinVMMGsinVFF

cosVFF

r~q~w~v~

EFDB

DC

FEDC

DB

0DA0D00A

r~q~w~v~

B

B

B

B

; (28)

( )

( )

⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

−

−

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

VIIDpI16CDρπ

I8SLSLCDρπ

VI16SLSLCDpρπ

m8CDρπ

FECBA

y

x

y

MQ5

y

CGCOPNA4

y

CGMAGYPA5

NA3

. (29)

Aside from the fact that appears in some of the above coefficients, the dynamics now is expressed with linear ordinary differential equations.

V

III. Solution to the Linear Motion Linear theory offers physical insight into the flight dynamics since closed form solutions can be readily

obtained4. Since changes slowly with respect to the other variables, it is thus considered constant,V 0VV ≈ , when it appears as a coefficient in all dynamic equations except its own. Moreover, pitch attitude of the projectile is regarded as constant in the velocity equation, thus decoupling the velocity equation. The epicyclic motion, Eq. (28), together with the roll dynamics, Eq. (27), are uncoupled and form a linear system of equations. In projectile linear theory, the Magnus force in Eqs. (24) and (25) is typically regarded as small so that in further manipulation of the equations, all Magnus forces will be dropped. However, it is important to retain Magnus moments due to the fact that a cross product between Magnus force and its respective moment arm is not necessarily small.

The solution to the differential equation Eq. (26), for the forward velocity, is

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+

−= 1

sa2e

absa2eVsV V

V

VV20 ; (30)

When , the velocity solution reduces to the familiar exponential decay form.00 =θ 4 The roll dynamic equation is a nonhomogeneous linear differential equation with the following solution

for 00 =θ :

( ) Vab

asbe

abVa

pspVp

pp

Vp

0p0 +

−⎟⎟⎠

⎞⎜⎜⎝

⎛

++= ; (31)

Noting that implies,

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5

( )( )

sK

1e

p

sKp

∞

−0∞ φ′+

−φ′−φ′=φ (32)

Neglect the product of damping and the product since the density ratio is assumed small. Defining EA( ) 0Vw~iv~ +≡ξ enables Eqns. (28) to be reduced to a single differential with the following form4,

( ) ( ) ( ) ( )0

B VGEiFFEMMCEAAE −++−′−−′′+−= φφ

ξξξi

e , (33) by assuming is small and ignored. p

Rather than solve for the lateral translation and rotational velocity components, via. Eq. (33), a more direct way to obtain the effects of asymmetry is to solve the swerve differential equations. The lateral translation and rotational velocity components are contained in the attitude differential equations and the attitudes are contained within the swerve differential equations.

IV. Swerving Motion Swerving motion along the earth-fixed and axes results from a combination of the normal aerodynamic

forces, as the projectile pitches and yaws, plus the forces and moments due to the configuration asymmetry. Differentiating Eqs. (22) and (23) with respect to nondimensional arc length and using the definition of

IJ IK

ξ with Eq. (33) leads to the following expression,

( ) ( ) ( ) ( ) ( ) ξξφφ ′′−′−++−−=′′+′′+ + AAAE

VGCiFFCMMAziy

DCEA

0

i Be (34)

For a stable projectile, the swerve caused by epicyclical vibration decays as the projectile progresses downrange and does not affect the long-term lateral motion. However, the assumption that the projectile is configurationally asymmetric causes an integrated effect that contributes to the long-term lateral motion of the projectile. Linear theory shows this center of mass motion contains terms that are bounded with arc length plus terms that are linear with s and with the inclusion of gravity the solution of Eq. (34) will have even higher order diverging terms. These higher order terms are typically denoted as gravity drop. The linear terms are called jump terms, which are caused by initial conditions at the gun muzzle, forces caused by asymmetry, and aerodynamic characteristics. Ignoring gravity and evaluating the following limits formally defines aerodynamic jump

s

Ks

Js

ΓsD

z(s)ΓsD

y(s) limlim ==∞→∞→

. (35)

The total aerodynamic jump vector Γ is expressed as the sum of two vectors. The first vector represents the muzzle conditions and the second results from asymmetry subjected to a varying roll rate:

( )( )( ) ⎭

⎬⎫

⎩⎨⎧

ΠΠ

+⎭⎬⎫

⎩⎨⎧

+−−−

+=

⎭⎬⎫

⎩⎨⎧ΓΓ

ImRe

DqEwsinVFFDrEvcosVFF

VCEAA

00B0

00B0

0K

J

φφ (36)

for which

τστ σφ dde∫ ∫

⎟⎠⎞⎜

⎝⎛

∞→+−

−=Πs

s0 0

is1lim

CEAFFCMMA . (37)

The quantity is the contribution to the jump vector attributed to the assumed asymmetry of the projectile. Appendix Integral shows that

Π

( ⎥⎦

⎤⎢⎣

⎡ΓΖ−Ζ−

Ζ−Γ

+Ζ+

−−=Π i,i21;F

i1i

CEAFFCMMA

11 ) (38)

where, ∞′′= φφ0-1Γ , p0 KΖ φ′= and is the confluent hypergeometric5 function. Apparently neither Murphy1

nor Fansler2 were aware of Eq. (38) which makes their limiting and asymptotic analysis, and, 11F

1=Γ 11<<−Γ , unnecessary for calculations of jump due to asymmetric configurations.

For constant rolling motion and for comparison purposes the following definition introduced by Murphy1 will be used here so that of Eq. (I6) of Appendix Integral now becomes

0=ΓAAiˆ Π−=Π ΖΦ−=Φ iˆ

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(39) ( )( )

( ) ( ΖΓii,Ζ1;2FΖi1ΖΓi1

di1ˆ

111

1

0

y1lny

−−−−=

ΖΓ−=Φ

−

−−ΓΖ−∫ ye i

)Polar plots of the results from Eq. (39) for discrete values of Γ and a continuous variation of are given in Fig.

3, for Ζ

Φ̂ , and Fig. 4, for the argument of . Φ̂

Successive integration by parts of the last equation shows

( )( )( )

( )( )∫ −−ΓΖ−′ΖΓ

+

−ΓΖ

+ΓΓ−

−ΓΖ

Γ+

−ΓΓ

−=Φ

1

0

y1lny22

523

dyFi

112

1i

11ˆ

ie (40)

where

( )( )( ) 1n0n0 FFF and

11y1yiF −′=+−ΓΖ

−=

Hence in the limit of large,

, ∞→Ζ01

1ˆφφ′′

=Γ−

→Φ ∞

. (41)

Magnitude of

-1

0

1

2

3

4

5

6

7

8

9

0 10 20 30 4

Φ̂

Φ̂

1Γ =

43Γ −=

21Γ =

0Γ =

1−=Γ 3−=Γ

Furthermore Eq. (39) can be expanded using a Kummer series giving the following expression

0

ZFigure 3. Magnitude of Φ as a function of ˆ Ζ

( )( ) ( )( )( )

( )( )( )( ) L−Ζ−Ζ−Ζ−Ζ−

ΓΖ

Ζ−Ζ−ΓΖ

+Ζ−Ζ−

ΓΖ−

Ζ−ΓΖ

−=Φ

i4i3i2i1

i2i1i

i2i1i1i1ˆ

44

322

(42) The limit of Eq. (42)

+Ζ− i3

3

∞→Ζ is

1for 1

132 <ΓΓ−

=Γ+Γ (43)

This last result, in light of the general limiting case (41

1ˆ ++Γ+=Φ L

of Eq. ), suggests the Kummer series, Eq. (42), when

transformed to a continued fraction may produce an accurate approximation to Φ̂ for extended values of Γ . To continue

vestigation th assumption was made th t a reasonable approximation has the following representation the in e a

)(((( ))))))a71a31a21a11a0ˆ Γ+Γ+Γ+Γ+Γ=Φ L

After finding the coefficients results in the rational approximation

Phase Angle of

40

60

80

100

120

0 10 20 30Z

Arg

umen

t Deg

rees

1Γ =

21Γ =

43Γ −=

0Γ =

1Γ −=

Φ̂

Figure 4. Argument of Φ̂ as a function

40

of Ζ (44) a7a2a1,a0, L

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

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

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

1680821313

i7Ζi6Ζi5Ζ4Γi6Ζi5Ζ6ΖΓi5Ζ4ΖΓΖ

7iΖ6iΖ5iΖ4iΖ3iΖ2iΖiΖi

Γi8Ζi7Ζi6Ζi5Ζi4Ζi3Ζi2ΖΖi

Γi7Ζi6Ζi5Ζi4Ζi3Ζi2ΖΖi3

8i53ΖΖi7Ζi6Ζi5Ζi4Ζi3Ζi2Ζ3Ζi

10i5ΖΖi7Ζi6Ζi5Ζi4Ζi3Ζi2ΖΖi

ˆ

2343

222

222222

2222223

3222

425

+−+ΖΖ+Ζ

+Γ+++−++++−

Ζ+++++++

+++++++

−++++++

+Γ−+++++++

−Γ−+++++++

=Φ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ii

(45)

Taking the limit of this expression, for , yields ∞→Ζ

Γ11Φ̂−

→ which indicates Eq. (45)

is a reasonable approximation when . Figure 5 shows some

comparisons between the exact solution, Eq. (39), and its’ rational approximation Eq. (45) for various values of . It is noteworthy that even for the case where the comparison shows good agreement for .

1Γ ≠

Γ1Γ =

20Ζ <

V. Conclusion Previous efforts describing jump

due to asymmetry did not develop the close form solution presented here. All of the analysis is based on projectile linear theory which leads to an esolution is well approximated, for the argcontinued fraction expansion of the closed fFansler/Schmidt2 results that will prove usef

Comparison of Eq. (39) and Eq. (45)

0

2

4

6

8

10

0 10 20 30

Φ̂

1Γ (39); Eq. =

1Γ ; (45) Eq. =

3-Γ;Eq.(45)Eq.(39)

= 43Γ; Eq.(45)Eq.(39)

−=

40

.

Let ∞0 φ′φ′=Γ , -1 p0 KΖ φ′= and

( ) =τφ

Then

lim

s1limΦ

s

s=

∞→

∞→

=

Ignoring the limiting process for the mom

(ieΦs0n

ΓΖi

=

− ∑=

Integrating the last expression and taking

American I

ZFigure 5. A comparison of the Exact vs. a rational approximation of Φ̂

xpression based on the confluent hypergeometric function . This uments used here, with a simple rational expression obtained from a orm solution. This is a further extension of the Murphy / Bradley

11F

1 and ul for analysis and design purposes.

Appendix then Eq. (32) will be written as, pKsτ =

. (I1) ⎥⎦⎤

⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛ −−+ 1τeΓτΖ

( ) ( )[ ] dτeτss1

s

0

1eΓτΖi

0 0

τ

s τdτdσe σi

∫ −+

⎟⎠⎞⎜

⎝⎛

−−

∫ ∫φ

(I2)

ent allows the last equal sign to be written as,

) ( ) ( ) dτeτs!nΓΖ (I3) τnΖi

s

0

n−∫ −

the limit ∞→s causes Eq. (I3) to become,

nstitute of Aeronautics and Astronautics

8

( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡+= ∑

=

−

1n

nΓΖi

!n Ζi-nΓΖi

ΖieΦ (I4)

where after writing the summation as two sums over even and odd values of respectively becomes, n

( ) ( )( )

( ) ( )( ) ⎥

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

+

−+=

∑ ∫

∑ ∫

=

−+

=

−−−

0n

1

0

Ζ)i(2n1n2n

1n

1

0

Ζ)i1(2nn2n

ΓΖi

dtt!12n

ΓΖ1i

dtt!2nΓΖ1

Ζi

eΦ (I5)

Using a taylor series expansions followed with partial integration gives by Milton Abramowitz and Iren A. Stegun5,

( ) ( )

( )ΓΖiΖ,i21;FΖi1

ΓΖi

dyy1eΓΖidtteΓ

ΖiΦ

11

1

0

ΖiyΓΖi-1

0

Ζi1tΓΖi

−−−

+=

−+=+= ∫∫ −−−

(I6)

References 1 C. H. Murphy, J. W. Bradley, “ Jump Due to Aerodynamic Asymmetry of a Missile with Varying Roll Rate,”

BRL R. 1077, U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground AD 219312, MAY 1959. 2 K. S. Fansler, E. M. Schmidt, “Trajectory Perturbations of Asymmetric Fin-Stabilized Projectiles Caused by

Muzzle Blast,” Journal of Spacecraft and Rockets, Vol. 15, No. 1, pp. 62-64, 1978 3 McCoy, R. L. Modern Exterior Ballistics: the Launch and Flight Dynamics of Symmetric Projectiles, Schiffer

Publishing Ltd., Atglen, PA, 1999. 4 Murphy, C. H. “Free Flight Motion of Symmetric Missiles,” BRL Report No. 1216, U.S. Army Ballistic

Research Laboratories, Aberdeen Proving Ground, MD, 1963. 5 Handbook of Mathematical Functions… National Bureau of Standards Applied Mathematics Series 55, 1967,

Edited by Milton Abramowitz and Iren A. Stegun

American Institute of Aeronautics and Astronautics

9