& REPORT NO. 1314

EQUATIONS OF MOTION FOR A

MODIFIED POINT MASS TRAJECTORY

by

Robert F. LieskeMary L. Reiter

March 1966

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BALLISTIC RESEARCH LABORATORIESABERDEEN PROVING GROUND, MARYLAND

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BALLISTIC RESEARCH LAI<ORATORIES

REPORT NO. 1314

MARCH 1966

Distribution of this document is unlimited.

EQUATIONS OF MOTION

FOR A

MODIFIED POINT MASS

TRAJECTORY

Robert F. LieskeMary L. Reiter

Computing Laboratory

RDT & E Project No. IP5238O1A87

ABERDEEN PROVING GROUND, MARYLAND

BALLISTIC RESEARCH LABORATORIES

REPORT NO. 1314

RFLieske/MLReiter/vm

Aberdeen Proving Ground, Md.March 1966

EQUATIONS OF MOTION

FOR A

MODIFIED POINT MASS

TRAJECTORY

ABSTRACT

A modified point mass mathematical model which incorp )rates an

estimate of the yaw of repose, has been developed to represent the

flight of a spin stabilized, dynamically stable, artillery shell. This

improved mathematical model has the desirable feature of representing

the effects of the significant variables of yaw of repose and axial spin

along the trajectory.

3

TABLE OF CONTENTS

Page

Abstract .............. ................... 3.

Table of Symbols ........................ 7

Introduction ...... ..... ..... ........................... 11

Basic Laws of Motion of a Rigid Body ...... ............. .Z

Estimate for Yaw of Repose ....... .................. .... 13

Utilization ............................ 19

Conclusions ...... ....... ... ........................... 2Z

References ..... ............... .... .............. .. .23

Distribution List ................. ........................ Z5

5

SKVIWS PAME WAS StANE T132701 WAS NOT 711.MW

TAB.tE OF SYMBOLS

T errm Definition Units

A Axial moment of inertia lb-ft2

AZ Azimuth of line of fire (clockwise from north) mil

B Transverse moment of inertia lb-ft2

C Ballistic coefficient for standard mass lb/in2s

d Reference diameter of projectile ft

E Position of projectile with respect to spherical ftEarth surface

g Acceleration due to gravity ft/s ec 2

Acceleration due to gravity (surface) ft/sec2

"go2H Total angular momentum lb-ft rad/sec

I Unit vector in the direction of v

KA Spin damping moment coefficient

K Drag force coefficient

KDa Yaw drag -- rce coefficient

K F Magnus force coefficient

KH Damping moment coefficient

K1L Lift force coefficient

KM Overturning nmoment coefficientMJ

-,KS Pitching force coefficientS

K TMagnus mornent coefficient

Term Definition Units

1 Lift factor

L Latitude of launch deg

M Mach number

m Projectile mass lb

ms Standard projectile mass lb

N Axial spin rad/sec

Q Yaw drag factor

r Distance between center of Earth and projectile ft

R Effective radius of Earth ft

Time sec

u Velocity of projectile with respect to ground ft/sec

v Velocity of the projectile with respect to air ft/sec

w Velocity of the air with respect to ground ft/sec I

x Unit vector along the longitudinal axis ofthe projectile

X Position of the projectile with respect to a ftground-fixed coordinate system

a Angle of yaw of projectile rad

ae Approximation for yaw of repose rad

A Coriolis acceleration due to rotation of the ft/sec2

Earth

Air density as a function of E (where E is lb/ft3

a component of E)

8

Term Definition Units

Q Angular velocity of the Earth rad/sec

First derivative with respect to time -/Sec

Second derivative with respect to time -- 2sec

9

I'

?'ftVICvSs-A WASi W-IM 7MweZ WznAS WTs iir nXM

INTRODUCTION

The mathermatical model for rigid body trajectory simulation, as

reported in BRL Report No. 1244, closely matches the results of

physical experiments of spin--stabilized projectiles over a spectrum

of test conditions. However, its usefulness, as in most rigid body

simulations, is hindered by the time required to numerically solve

the system of differential equations. This report describes the der-

ivation of a mathematical model which will not be as time consuming

to solve as the rigid body system, but which will represent as 'losely

as possible the center of gravity motion of the projectile by utilizing

a force system, axial spin and an estimate of the yaw of repose.

I

*11

I 1

itt

BASIC LAWS OF MOTION OF A RIGID BODY

The frame of reference for all vectors to be presented is a ground-

fixed, right-handed Cartesian coordinate system with unit vectors

(I, 2, 3). Assume the 2 axis to be parallel to the vector g, and g

to have the same direction as 1 X 3 = - 2.

Assume that the body can be considered a solid of revolution. Then

spin-stabilized projectiles can be oriented by choosing a unit vector x

along the axis of symmetry, pointing from tail to nose. N is the mag-

nitude of angular velocity parallel to x. N is positive if it results from

a rotation causing a right-handed screw to advance in the direction of

X.

The following set of simultaneous differential equations of motion

for a spin-stabilized projectile is developed in BRL Report 1244.

The equation of motion of the center of mass is:

(1.1) C -m(Ko + az2 KD ) vv+pd 2K [ vX (xv) ]m Do0 a m L W _.

-pd K_ v + pd3 KN (xX v)+ g +Am _- -P -0 -

m

The total angular momentum of the body can be expressed as the

sum of two vectors in the ground-fixed coordinate system:

(a) The angular momentum about x.

(b) The total angular momentum about an axis perperdicular to x.

The angular momentum about x can be represented by the vector

A N x and the angular momentum about an axis perpendicular to x by

the vector B (x X k).

12

Let H denote the total angular momentum of the body. The vector

representation of H is:

(1.2) H= ANx + B (xX;)

The vector rate of change of angular mo:n, .itum is the sum of the

applied moments.

(1.3) T. = ANx + ANc_ + B (x X

3 4pd3KMv (vX x) - pd4K1 1 v (xX _i1

+pd4KTN [xX (xX v)]pdKAN vxT A~

ESTIMATE FOR YAW OF REPOSE

An estimate for the yaw of repose will be derived from the rigid

body system of differential equations of motion for the purpose of rep-

resenting the effects of yaw.

The following conditions are assumed:

(1) The projectile can be represented sufficiently well as a

body of revolution.

(2) The projectile i-.• dynamically stable.

(3) Initial yaw is assumed small, i. e., it has no sigr.ificant

effect on the trajectory.

Examination of equation (1. 1) shows the magnitude I x X v

as being present in the lift term and the Magnus term, the lift term

being more significant. Examination of equation (1. 3) shows its

presence also in the corresponding terms of H.

If I v /v, then xX I I - sin a. To a first order approximation,

13

this is a ; however, for computational purposes it is a more desirable

quantity than a. To get the proper orientation of this quantity, the

following vector will be defined:

(2.1) ae-I X(xXI)= x -cos aI

Obviously, ae I = 0. a represents vector yaw * directed from I-+e --e

toward x. The effect of a e on the trajectory is generally small under

the assumptions stated. Furthermore, it will be assumed that je is

negligible. This implies , is small; moreover, the following approx-

imations are warranted:

(2.2) C =cosai

(2.3) cos a

The separation of H into ,:cmnponents parallel and perpendicular to

x yields:

4(2.4) Al= -pd K ANv

3(2.5) ANk + B (xX x) pd3 K V(VvX x)

p 4 K4- pd4K v (xX i) + pd K N [xX (X Av)]

To determine ae' first replace x and its derivatives in (1. 1) and

(2. 5) using equations (2.1), (2. 2) and (Z. 3).

* NOTE: Iel a a. However, as mentioned earlier, ;n magnitude

%e I is a first-order approximation for yaw. a is in the plane

determined by x and I, in the direction from I toward x. Hence, a

will be referred to as vector yaw.

1411

The resulting equations are:

-pd 2(KD + aKD )v + pd 2 K 2v e pd 3 Ksv cos a(2. 6) D1 o D + L-

m m m

+ pd 3 KFNV (a ×I) Nag + A

(2.7) AN cos al+ B cos a(a 2i ) + B Cos2alIXY)

3 2 4

=pd KMv (I Xa.) +pdK Nv [cosa (ae+ c)s aI)-

v cos a [(e + cos a )Ix I

Cross multiplication by I of both sides of equations (2. 6) and (2. 7)

and, with A negligible in comparison to g, the simultaneous solution

of the resulting equations gives the following for aie

2 2 2 ix!+Pd V 8 [-ANcos a (IXI+Bcos agIpK( Ix ]

d4K '1" / {Zgd 7 KFKTN 2 2z

fl T5 rF S O

+ pdK v 2s [- A os

22 3d42 722

+pdl KLv [pd KMv + B Cos a I

15

jt --

Iit

IIFor f urther substitution into (2. 8) the following are required:

(2. •) 1 v v

(210)= [t-1~ /i /v [x x ]/v

(21v =-(z I/v + 1(.) 21 /v2

+V/v-( ir )I/v-(2 1)/ v

(2.12) V/vQ-' )_I/v [1 X xI)]/v

(2. 12) will be considered negligible since i can be approximated

by V, and U is essentially parallel to .

With this assumption,

(2.13)Ce= f-AK Ncosa (vX'i +md T. cos a [v×{i- g

+ pd3K cosa </m)] - KL'-U ".

ZB cosa. I" - Pd4 Ku cos 2 a v]2

r 3 4 5 2 2lpd KLKMv + pd K FK TN cos a v

+KLBcos •L o

In the denominator of equation (2. 13) the v term predominates

therefore, K B cos a ( - . will be considered negligible.

The ratio of this term to the remaining terms in the denominator is of

the order of (g /v 4). In the numerator, [I- ('" I) I] is the component

of perpendicular to the projectile flight in the air coordinate system.

16

For most spin stabilized trajectories I)- I)1 is no more than

the magnitude of g; hence, consider

(2.14) KL [V- (. I)I][ 2Bcos a(v. I) -pd K cos av ]

2 d4K 2

K [ 2 B cos a (• I p Kcos a v2 ]

For artillery shells, this term is generally bounded by

(2. 15) KL jg j 2 B (10g1) ZOKL Bg2

Now

(2.16' 'AK N'(vX A+ d2 KN X g

*1 L' Z ' mdZ KT[ vXv{isin €

Nv {- AKL+ md K L pdd + AKgLsin

where cos = g .v

g v

For artillery shells, it is usually true that

2 .17(2.17) fAK + md K IK. pdv < < AK gsinIL T L 01Lg

17

Comparison of (2. 15) and (2. 17) shows that the ratio of (2. 14) to

(2.16) is less than the order of (Zog /Nv). For high spin, (2. 14) will

be considered negligible.

Let

(2. 18) v u- w

(2. 19) v=u.-w

If the effects of the pitching force (KS term) and w are assumed

insignificant, and cos a can be approximatecd by 1, equation (2. 13) is

reduced to

(-AKL N (vXd) + md +m KTN [x vX -g)] I(2. Z0 ae

ýe3 4 5 2pd3K K v + pd5K K N vLpdKF TI

18

UTILIZATION

The primary goal of the development of a.e was the acquisition of a

mathematical model which would incorporate the effects of yaw, but

would not require the computing time of a complete r'..gid body simula-

tion. The following representation was devised to incorporate ae in a

modified point-mass mathematical model. This representation includes

auxiliary equations necessary for the numerical solution of the differen-

tial equation of motion of the center of mass.

The equation of motion of the center of mass is:

(3.1)*u= - Pm {D +KD [K Q ] vv144C m o a

s

+PdZ KL vZ le +.g + Ain

IT+ _L KF N Q(a eXv.mn

where: C = ballistic coefficient for standard masss

1 = lift factor

m = projectile mass

m = standard projectile masss

Q= yaw drag factor

*Note: If Q and 1 are set to zero (0) in equation (3. 1), this system

reduces to the classical point mass equations cf motion.

19

IThe axial spin is:

t 41,3.2z) N=- YO p K ANyA

The approximation for the yaw of repose is:

(3.3) a =a - a ) (v×X a (•) (Xg)

AKL Nwhere: a =

a Pd3KLKKv + pd5 K K N v2

rn KTNT

4 3 2 2pd KLKM v + pd3K FK TNv

The velocity of the projectile with respect to air is:

(3.4) v= u -vw

The position of the projectile with respect to ground-fixed Cartesian

coordinate system is:

(3.5) X u dt

The approximation for the position of the projectile with respect to

spherical Earth surface is:

"X 1

(3.6) E X +R. - (Rz - X 2/2

x3

where R- effective radius of Earth.

20

The approximation of the force of gravity is:

(3.• 7)-go-90 - + R-32

r -x [X 1 ++R)2 +x 32 ] 1/

where go = value of gravity at point of launch

r = distance between center of Earth and projectile

The Coriolis acceleration due to rotation of the Earth is :

X1 U2 -XZu 3

(3.8) A X LU X u1z 1 3u3

For the northern hemisphere, the X's are defined by the following

equations. [For the southern hemisphere replace L by ( L). ]

2 02 cos L sinAZ

2 Q sin L

3= 2 12 cos L cos AZ

where: 0 = Angular velocity of the Earth (radians/sec)L= Latitude of launch point

AZ= Azimuth of fire measured clockwise from north

The orientation of yaw (') is the angle between the plane containing

both vand ae and a vertical plane containing v. It is measured clock-

wise from the vertical plane. If desired, T is given by the expression:

-1 a (Vla- " r )v 1(3.9) T = tan e3 ) v-

[(VI 1t 2 v e I vZa 3 v 3 a)Zv

21

m m m m m m m I *

The dimensionless aerodynamic coefficients are functions of nmany

dimensionless power products, including the dimensionless shape

parameters, Reynolds number and Mach nunmber. Aerodynamic coeffi-

cients are defined with reference to a specific set of shape parameters

and may be expressed as functions of Mach number.

CONCLUSIONS

A modified, point-mass mathematical model has been developed

which incorporates an estimate of the yaw of repose. This improved

mathematical model has the desirable feature of representing the- effects

of the significant variables of yaw of repose and axial spin along the tra-

jectory. By the incorporation of this yaw of repose, the factors (ballis-

tic coefficient, yaw drag factor and lift factor) used to match empirical

results have been found to vary little from theoretically determined

values over the spectrum of conditions.

In comparisons between the rigid body mathematical model, the

most complete representation available, and point-mass representa-

tion, the modified point-mass model accounted for better than 90

percent of the discrepancies in the time-dependent variables range,

height and deflectior. These comparisons were made for spin-

stabilized artillery Lourds currently being used by the U. S. Army.

The modified point-mass model solution required only approximately

twice the computation time of the point-mass solution while the rigid

body solution requites about one hundred to one thousand times that

of the point-mass solution.

22

REFERENCES

1. Lieske, R. F., and McCoy, R. L., Equations of Motion of a Rigid

Projectile, BRL Report No. 1244, March 1964.

2. McShane, E. J., Kelley, J. L., and Reno, F., Exterior Ballistics.

University of Denver Press, 1953.

23

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3. REPORT TITLE

EQUATIONS OF MOTION FOR A MODIFIED POINT MASS TRAJECTORY

4. DESCRIPTIVE NOTES (Type of report and inclusive date&)

5. AUTHOR(S) (Last name. first name, initial)

Lieske, Robert F. and Reiter, Mary L.

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March 1966 26= 2Be. CONTnACT OR GRANT NO. 9s. ORIGINATOR'S REPORT NUMSER(S)

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13. ABSTRACT

A modified point mass mathematical model,which incorporates an estimate of theyaw of repose, has been developed to represent the flight of a spin stablized,dynamically stable, artillery shell. This improved mathematical model has thedesirable feature of representing the effects of the significant variables ofyaw of repose and axial spin along the trajectory.

DD I JAN 64 1473 UnclassifiedSecurity Classification

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14.KEY WORDS LINK A LINK B LINK CK WORDS ROLE WT ROLEI WY ROLE WT

Equations of motionPoint mass mathematical modelEstimate of the yaw of reposeAxial spin

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