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USAARDCOM ltr, 27 Feb 1984

THIS PAGE IS UNCLASSIFIED

IL I I

MEMORANDUM REPORT NO. 1L538FEBRUARY 1964

I A MVETHOD FOR ESTIMATING DAINGER AREAS DUE TOI

R1 RICOCH"-lETI NG PROJECilLES

D. J. Durin

f'f RDT A& E Proiect No. !M523801IA287

if BALLI STIC RESEARCH LABORATO~IRIS

IA&BE-RDEEN PROVING GROU!ND. M ARYL AN D

DDC AVoILMU MTY NOTICE

Qaalified requesters may obtain copies of this report from DDC.

Foreign annouicemipnt and dissemination of thls report by DDC is Inot authorized.

The findings in this report are not to te construe.as an official Department of the AiW position.

U

BAI-J.ZSTZ MUMS L0~T~&

Abardesa Proving 0e.-ima, !Mar~ inxr

P-0 M~ay 1964

fo~r

F^Mistic Research Ihboratories Fawrandum R~eport No. 1538 exititled"tA M~thd feir Eatisming, Danger Areas Dw~ to Richocheting Proj'ecItilas'by D. J. Dumne end 11. D. Dotson, Jr. dated Februa~ry 196L~.

page 14, 8th% linze fxsvLi o-t-t- of pzC_- scu-of the i XcLtCCSt V8Xizbles 7 az 6 re usee Co ouute

th-e traieetovyj coordiniates'

Page 14. 5th line frog bottmik of Pago sh-ould read:'1tbe tr,.JeetorY coordinates fill a 3 dimensionalvolume vhs em",pelý defines the"

BALLISTIC RESEARiCHILBTRjz

MMMO)'JM MCRT~'1110. IL535

FEBRULARY 1964

A IWOD FR ESMMM ~aM RTASVUtTO RCOCEMG 'R0'Pr-!IE

D. j.DuzI1 D. DtýAjr

Termnal -alist-cs sbor~urComptingLatoator

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-A BB RD-E N R V IN G0 RC U D, N- R YL . N

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BALI.I3TIC RESEARCH LABORATORIES

!4fl'oPAIIDU REORT No. 1538B

DJIhrnn/WGDot son, jr./j.k-Aberdeen Peovtng Ground, M~d.Feuruary 19064

A METHOD FOR ESTIMATING ]akNG--RQ AREAS DUETO RICOCHET"ING PEOJECTIES

ABSTRACT

The rit--ehet studies of Sirichoff ant Hitchcocýk are suzarized. A theory is

developed for ricoc-het of projectiles eff inclined pierce,. and this theory i7s

amiliea to obtain a. procedure for the estimatio., o:f danfter areas dr- t' ricochetiAng

projectiles.

f

kt

I. INTRODUCTION

In several instances over the past few years it nr:z been necessary to estimatc

the danger area resulting from the ricochet oi projectili.s off various surfaces

durln, outdoor training exercises or te•sting operations. Of -articular interest

wa:s the lateral extent of the danger zone. To supply these esVtrates, an analyti-

cal 'rocedure was worked out and is described in the present report.

Based upon a large collection of expc;-imental data on ricochet off land

ýurfaces, the follot z.g empirica" fur-maias ýere published by G. Bir-hýfflin 1945

(BRL Report 1o. 535):

1. 0 0.lul c

c

Where (see Figure 1): 0 = angle of impact, Iju magnitude of velocity of

impact, 0 = angle of ricochet, lvi = magnitude of velocity of ricochet, and

@c = the critical angle for the projectile/soil (or impact material) combination

sld is defined as that angle of impact above whicb ricochet occurs less than half

the time.

2In lcy47, H. P. Hitchcock published a large collection of additional experi-

mental data on ricochet off- land surfaces (BRL Repot, t No. 629). Cunsiderimg these

additional 6ata, D. J. Dumn obtained the following modifi~d Birkhoff formulas:

-1 .- (3)

iui c

0 (02>9c).4 : 3. -2. (o- <

3-5 . -2.8(4)

Graphs of Equations (3) and (4) are presented in Figure 2. These modified

Birkhoff formulas are currently used at BRL for computation of velocity and angle

(cZ ricochet from velocity and angle of impact. The critical angle 9c is a function

oi the nose shape, iwpact velocity, hardness of the projectile nose and hardness

of the ricochet surface. The value of Q is obtained by cearching through the

ricochet literature for an experimental value cor-veponding to the appropriate

5

FIGURE 1.

1.09375 8c

S/i(8c, 0.7 oC)

'5,

0 t0 8C0.625 oc oG

FIGURE 2

7

owl -w-

set of parameters. Notice in Equations. (3) in,- (,), or in Fieire 2, that both

lvi and 0 increase with an increase in 9-- as•-•,ing given values of the impact

conditions lul and 0. Consequently, for computation of dan.-r zones, 0 shouldC

be chosen as large as realism pcrmits in order to avoid underestimating the size

of the dauigr zone.

II. THEORY OF PCOCIIT -L .-CINED PWTES

Constdcr the impact of a given projectile on an inclined plane composed of q

given material. We assume that the criticetl angle, 0c) for the projectile/impp-+

material combination has been determined (or appropriately chosen). The point of

impact is taken as the origin (0, 0, O) of an xyz rectangular coordinate system,

Figure 3. The y axis is an extension of the radius-vector from the center of the

earth to the origin (0, 0, 0), and the xz plane is perpendicular to the y exis.

The x axis is chosen so that the impact velocity, u, of the projectile is contained

in the xy plane; and the z axis is then chosen so that the xyz system ;ril- be right-

handed. Unit vectors along the x, y, and z axes are denoted by i- J, It, respectively.

Hence, ye have

u- u + (5)

Si I -- u + u2( )x y

We assume that ux and u arx known from the output of a two-dimensional trajectoryX y _

computation terminating at (0, 0, 0). Hence, Jul is also known. The orienta-

tion of the inclined plant: is specified by a unit vector & perpendicular to The

inclined plane at (0, O, 0). We have

_% _ý.. ._X _ -11 = 11 y i±J 4 ri zk %

x y z

The two components nx and my will be arbitr-rily specified., and nz 4.1, then be

determined, except for sign, from Eqtation (8). Adopting the convention that

nz • 0 (i-!ich amounts to considerinr; half of a syznctrical dirtributicn of possible

orientations of the inclince plane -- i.e. symmetrical about the xy plane) we have

8

1�y

n -�

V'I., *J

k-Z

FIGURE 3.

9

T1e orientation of the inclined plane cean, of course, be varied by varying nx

and n . For any particular orientation, our k:novn quantities are Ux, u y (and

hencei in ), ny,, n., and 9 . In terms of triese known quantities, we mustcaV -'a-e (1) the magnitude, Ill , of the velocity v of ricochet, (2) the

angle a that the vector v makes with the xz p:Lune, and (3) the angle ' ihat the

projection of v on the xz plane makes with the x axis (see Figure 3). The

initiai conditions (i.e. initial velocity and angle of elevation) for the ricochet

trajectory are • and a, ana the range of the ricochet trajectory must be

plotted so as to form the angle V witn the x axis.

We will begin by determining the angle of impact, 9, that the .rector u makes

with the inclined plane (see Figure 3). We have (see Figure 4):

-4 n=

rl"' 11nI cos(90 ++Q)

-- in ) (10)

But, since u =u i + uyj and since n = nxi + nyj - nzk , we have

u n = u n + u n ii)xx •y

Equations (10) end (11) yield the following colition for 0 in terms of the known

quanti+•es ux, uy, nx , sny

0 = sin- 1 - x x+un (12)

in general ye hawr 'i o. nx - 0. u < 0. n > 0. so that 9 will generally turn

out to be a positive angle.We now know IuI , 9, axd ; and so we can compute IvI and 0 from the

modified Bi-k-hoif formulae, Equations (3) bonet (4). Of course, this 0 is the anglethat the ve(,tor V' makes with the inclined plane. Hence, we can now add jIV

and 0 to our list of known quantities. To determine the angles a ana *, we need

to find the components o.' the vector v in the x, y; ead z directions. We have

v = v i + v + v k (13)Sx y Z

and so

10

V 1t "* *".' | |\x y z

Since *I 2 is Knovn, Equation (14) is one •oilition for the determination of tLe

urkajowns vx, v., v . We need twn more conditions. We vil assme t .at the vector-4x z A- .S-a f3

J lies in the '.lane dete-nf'.c;d by a and n (see Fi,,are 4). .This assumption can

be ststec

x 0 (s

Since

"I x n = A uy 0

nx ny nz

an! since v is given by Equation (13), Equaticn (15) reduces to

x z y y z x z',Y y )(

Rlferring to Figure 4, -we have

u v -- I cos(@+0) (17)

Una •,.1 since ii. v = ux vx + u Vy ,v we have from Equation (17)

vux + v !i i1 cos(+0) (+)

The only ',nknowrr in •,,iat'ons (if) and (1) are v: .y. , v ; and (16) ana (25.)

are linear in v Vy, vz. Hence, it is easy to solve (16) and (13) simultaneously

for v end v in terms of vz and known quantities. The results are as follows:

%•- ~ nu¢ .- + u• n u Iv I Cos +e*¢1,_x AY

nu [i I'#l !" cos(G+ )-uv (+n A -n

zy l x :x _!, (2

7Y 2

n~ lulz

Iv

n

u

Intersection of Inclined Planeand Plane detemined by u and in

FIGURE 4.

12

Substituting Equations (19) and (20) for v and v in Equation (1h), we obtainY

an equation in which the orly unknown if vz. This equ:ation is easily solved to

give

I I Si, (0+ 0 (21

Iinz I )

v and vy can now be computza by substituting this solution for vz i=) Equasio

(19) and (20). Finally, we have (see Figure 3):

a sin- 'IV/ I I ) (-2

tan-1 (v/-v (23

Ill. ESTIMATION OF DANGER AREAS

The theory developed in Section II can be applied to estimate danger areas

due to ricocheting projectiles. For -:he particular projectile, gun and line of

P;re under consideration, the velocity of impact l and angle of fall are known

or can be computed for any range, R. Consider a particular range, R.V We

suppose the projectile strikes an inclined plane at R1 * As the orientation

of the inclined plane is varied, by varying the values of n* and n , the

various ricoc'e+. -rnges Rre deterrn._med by means of the theoz-, of Section 11 an.!

the rest coordinates after ricochet are plotted in the x-z plane. This procedur

is repeated for other pertinent values of R, say RP, R), ..... R , and for each

range the set of rest coordinates is plotted. An envelope of the family of set

points is sketched in to form the boundary of the danger area. In the interest

of safety, it is generally assumed that the --oJectile flys J' -.c well

after ricochet as it did before ricochet -- i.e. the same ballistics are used

to run the ricochet trajectories as were used to run the original trajectories.

Cne exception to this is that windshields are assumed tc break off after impact

on hard arufaces. Lu tin y euse, however, it is assumed that the flight of theprojectile after impact is stable.

15

An example of a danger area, constructed according to the ebove procedure,

is given in Figure 5.

iv. URTICAL DflC kP-.•A

In certain cases a training area may be located in the vicinity of an airfield

and training exercises may be in progress -while aircraft are taking off or landing.

it is of inte'rest to determi.ne Tlie v•.!, al a:.c-a er the aircr~tb blt auzc

of ricocheting projectiles. This ra,,blen •. ý special case of the literal ricochet

problem already described except for the following specializations. The values

of the ricochet variables I and 0 are uzed to compute the zenith coordinates,

instead of the rest coordinates, ofter ricochet. As before, the coordinates are

ý-.;,ated for each location and oricntation of the ricochet plane. The totality of

the zenith coordinates fill a 3 dimensional volume -whose envelope defi-n,: +hp

bounday of the vertical danger area. The boundary is now a surlace inStead of a

line. The section of the envelope in the vertical plarne which contains the line

of fire is ordinarily of most interest since this section includes the highest

zenith points.

D. J. DUNN

W. G. DOTSON, JR.

11+

I FIG. 5LATER~AL RICOCHET ~'%PIA OF THE 1O5MM M392

APDS-T PROJECTILE OFF A SAND 00R CLAY SURFACE

OA :LINE OF FIREOABCO a POSSIBLE RICOCHET AREA

A NOTE, THE RICOCH'ET AREA SHOWN LI'S TO2400- ...... .... .. THE R:GHT OF THE RIGHT LIMIT LINE OF

r1RE. A SIMILAR AREA LIES TO THE LEFIOF THE LEFT LIMIT LINE OF FIRE.

200 400WO

RIH DFETIN MTR

V. HEFE2ENCES

1. Birl"ho ±, C-. *...chctSRL Report No. Mar ch 19- -

2. Hitchcock, H. P. Effects of Ricochet on the I4ption of •.o3~ceiles. BTLReport No. 629, February 19hQ,.

16

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